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Lernmodul Mathematik

Übersicht:

 

23.3 Partielle Ableitungen - Lösungen

1. Aufgabe

1)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& 4\cdot 2x^{2-1}+3y+0\\&=& 8x+3y\end{array}   \begin{array}{rcl}f'_{xx}(x,y) &=& 8+0 \\&=& 8\end{array}
    \begin{array}{rcl}f'_{xy}(x,y) &=& 0+3 \\&=& 3 \end{array}
\begin{array}{rcl}f'_y(x,y) &=& 0+3x+7\cdot 3y^{3-1}\\&=& 3x+21y^2\end{array}   \begin{array}{rcl}f'_{yx}(x,y) &=& 3+0 \\&=& 3\end{array}
    \begin{array}{rcl}f'_{yy}(x,y) &=& 0+21\cdot 2y^{2-1}\\&=& 42y\end{array}

 

2) 
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_a(a,b) &=& 3a^{3-1}+3b^2-15-0\\&=& 3a^2+3b^2-15\end{array}   \begin{array}{rcl}f'_{aa}(a,b) &=& 3\cdot 2a^{2-a}+0-0\\&=& 6a\end{array}
    \begin{array}{rcl}f'_{ab}(a,b) &=& 0+3\cdot 2b^{2-1}-0 \\&=& 6b\end{array}
\begin{array}{rcl}f'_b(a,b) &=& 0+3\cdot 2ab^{2-1}-0-12\\&=& 6ab-12\end{array}   \begin{array}{rcl}f'_{ba}(a,b) &=& 6b-0 \\&=& 6b\end{array}
    \begin{array}{rcl}f'_{bb}(a,b) &=& 6a-0\\&=& 6a\end{array}

 

3)
\begin{array}{rcl}\mathbb{D} &=& \mathbb{R}\times\mathbb{R} \\\\f(x,y) &=& \left(x-2y\right)\left(2x+y\right) \\&=& 2x^2-3xy-2y^2\end{array}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& 2\cdot2x^{2-1}-3y\\&=& 4x-3y\end{array}   \begin{array}{rcl}f'_{xx}(x,y) &=& 4\end{array}
    \begin{array}{rcl}f'_{xy}(x,y) &=& -3\end{array}
\begin{array}{rcl}f'_y(x,y) &=& -3x-2\cdot 2y^{2-1} \\&=& -3x-4y\end{array}   \begin{array}{rcl}f'_{yx}(x,y) &=& -3\end{array}
 

\begin{array}{rcl}f'_{yy}(x,y) &=& -4\end{array}

 

4)
\begin{array}{rcl}\mathbb{D} &=& \mathbb{R}\times\mathbb{R} \\\\f(x,y) &=& \left(x+3y\right)^2-2xy \\&=& x^2+6xy+9y^2-2xy \\&=& x^2+4xy+9y^2\end{array}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& 2x^{2-1}+4y+0\\&=& 2x+4y\end{array}   \begin{array}{rcl}f'_{xx}(x,y) &=& 2+0\\&=& 2\end{array}
    \begin{array}{rcl}f'_{xy}(x,y) &=& 0+4 \\&=& 4\end{array}
\begin{array}{rcl}f'_y(x,y) &=& 0+4x+9\cdot 2y^{2-1} \\&=& 4x+18y\end{array}   \begin{array}{rcl}f'_{yx}(x,y) &=& 4+0 \\&=& 4\end{array}
    \begin{array}{rcl}f'_{yy}(x,y) &=& 0+18\\&=& 18\end{array}

 

5)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& 4x^{3-1}-0+14y+0 \\&=& 4x^3+14y\end{array}   \begin{array}{rcl}f'_{xx}(x,y) &=& 4\cdot 3x^{3-1}+0 \\&=& 12x^2\end{array}
    \begin{array}{rcl}f'_{xy}(x,y) &=& 0+14 \\&=& 14\end{array}
\begin{array}{rcl}f'_y(x,y) &=& 0-53\cdot 3y^{3-1}+14x+0 \\&=& 14x-159y^2\end{array}   \begin{array}{rcl}f'_{yx}(x,y) &=& 14+0 \\&=& 14\end{array}
    \begin{array}{rcl}f'_{yy}(x,y) &=& 0-159\cdot 2y^{2-1} \\&=& -318y\end{array}

 

6)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& 3\cdot 5x^{5-1}-20\cdot 2x^{2-1}y-0 \\&=& 15x^4-40xy\end{array}   \begin{array}{rcl}f'_{xx}(x,y) &=& 15\cdot 4x^{4-1}-40y \\&=& 60x^3-40y\end{array}
    \begin{array}{rcl}f'_{xy}(x,y) &=& 0-20\cdot 2x^{2-1} \\&=& -40x\end{array}
\begin{array}{rcl}f'_y(x,y) &=& 0-20x^2-10\cdot 2y^{2-1} \\&=& -20x^2-20y\end{array}   \begin{array}{rcl}f'_{yx}(x,y) &=& -20\cdot 2x^{2-1}-0 \\&=& -40x\end{array}
    \begin{array}{rcl}f'_{yy}(x,y) &=& 0-20 \\&=& -20\end{array}

 

7)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_x(x,y) &=& 7x^{7-1}+\dfrac{5}{2}\cdot 2x^{2-1}y^3+0 \\&=& 7x^6+5xy^3\end{array}   \begin{array}{rcl}g'_{xx}(x,y) &=& 7\cdot 6x^{6-1}+5y^3 \\&=& 42x^5+5y^3\end{array}
    \begin{array}{rcl}g'_{xy}(x,y) &=& 0+5\cdot 3xy^{3-1} \\&=& 15xy^2\end{array}
\begin{array}{rcl}g'_y(x,y) &=& 0+\dfrac{5}{2}\cdot 3x^2y^{3-1}+13 \\&=& \dfrac{15}{2}x^2y^2+13\end{array}   \begin{array}{rcl}g'_{yx}(x,y) &=& \dfrac{15}{2}\cdot 2x^{2-1}y^2+0 \\&=& 15xy^2\end{array}
    \begin{array}{rcl}g'_{yy}(x,y) &=& \dfrac{15}{2}\cdot 2x^2y^{2-1}+0\\&=& 15x^2y\end{array}

 

8)
\begin{array}{rcl}\mathbb{D} &=& \mathbb{R}\times\mathbb{R} \\\\g(x,y) &=& \left(5x+22\right)^2+9x^2y+y^2 \\&=& 25x^2+220x+484+9x^2y+y^2\end{array}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_x(x,y) &=& 25\cdot 2x^{2-1}+220+0+9\cdot 2x^{2-1}y+0 \\&=& 50x+220+18xy\end{array}  

\begin{array}{rcl}g'_{xx}(x,y) &=& 50+0+18y\\&=& 50+18y\end{array}
    \begin{array}{rcl}g'_{xy}(x,y) &=& 0+0+18x \\&=& 18x\end{array}
\begin{array}{rcl}g'_y(x,y) &=& 0+0+0+9x^2+2y^{2-1} \\&=& 9x^2+2y\end{array}   \begin{array}{rcl}g'_{yx}(x,y) &=& 9\cdot 2x^{2-1}+0 \\&=& 18x\end{array}
    \begin{array}{rcl}g'_{yy}(x,y) &=& 0+2 \\&=& 2\end{array}

 

9)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_x(x,z) &=& 7\cdot 4x^{4-1}+8z^2-0 \\&=& 28x^3+8z^2\end{array}   \begin{array}{rcl}g'_{xx}(x,z) &=& 28\cdot 3x^{3-1}+0 \\&=& 84x^2\end{array}
    \begin{array}{rcl}g'_{xz}(x,z) &=& 0+8\cdot 2z^{2-1} \\&=& 16z\end{array}
\begin{array}{rcl}g'_z(x,z) &=& 0+8\cdot 2xz^{2-1}-12\cdot 3z^{3-1} \\&=& 16xz-36z^2\end{array}   \begin{array}{rcl}g'_{zx}(x,z) &=& 16z-0 \\&=& 16z\end{array}
    \begin{array}{rcl}g'_{zz}(x,z) &=& 16x-36\cdot 2z^{2-1} \\&=& 16x-72z\end{array}

 

10)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_b(b,c) &=& 0-8\cdot 3b^{3-1}+24c+0 \\&=& -24b^2+24c\end{array}   \begin{array}{rcl}g'_{bb}(b,c) &=& -24\cdot 2b^{2-1}+0 \\&=& -48b\end{array}
    \begin{array}{rcl}g'_{bc}(b,c) &=& 0+24 \\&=& 24\end{array}
\begin{array}{rcl}g'_c(b,c) &=& 0-0+24b+3\cdot 8c^{8-1} \\&=& 24b+24c^7\end{array}   \begin{array}{rcl}g'_{cb}(b,c) &=& 24+0 \\&=& 24\end{array}
    \begin{array}{rcl}g'_{cc}(b,c) &=& 0+24\cdot 7c^{7-1} \\&=& 168c^6\end{array}

 

11)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_b(b,c) &=& 3\cdot 2b^{2-1}-125c+0-5\cdot 2b^{2-1}c^2 \\&=& 6b-125c-10bc^2\end{array}   \begin{array}{rcl}f'_{bb}(b,c) &=& 6-0-10c^2 \\&=& 6-10c^2\end{array}
    \begin{array}{rcl}f'_{bc}(b,c) &=& 0-125-10\cdot 2bc^{2-1} \\&=& -125-20bc\end{array}
\begin{array}{rcl}f'_c(b,c) &=& 0-125b+7\cdot 2c^{2-1}-5\cdot 2b^2c^{2-1}\\&=& -125b+14c-10b^2c\end{array}   \begin{array}{rcl}f'_{cb}(b,c) &=& -125+0-10\cdot 2b^{2-1}c \\&=& -125-20bc\end{array}
    \begin{array}{rcl}f'_{cc}(b,c) &=& 0+14-10b^2 \\&=& 14-10b^2\end{array}

 

12)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}a'_x(x,y) &=& 5\cdot 3x^{3-1}y+10\cdot 2x^{2-1}y^2+2y^3+5y \\&=& 15x^2y+20xy^2+2y^3+5y\end{array}  

\begin{array}{rcl}a'_{xx}(x,y) &=& 15\cdot 2x^{2-1}y+20y^2+0+0 \\&=& 30xy+20y^2\end{array}
    \begin{array}{rcl}a'_{xy}(x,y) &=& 15x^2+20\cdot 2xy^{2-1}+2\cdot 3y^{3-1}+5 \\&=& 15x^2+40xy+6y^2+5\end{array}
\begin{array}{rcl}a'_y(x,y) &=& 5x^3+10\cdot 2x^2y^{2-1}+2\cdot 3xy^{3-1}+5x \\&=& 5x^3+20x^2y+6xy^2+5x\end{array}   \begin{array}{rcl}a'_{yx}(x,y) &=& 5\cdot 3x^{3-1}+20\cdot 2x^{2-1}y+6y^2+5 \\&=& 15x^2+40xy+6y^2+5\end{array}
    \begin{array}{rcl}a'_{yy}(x,y) &=& 0+20x^2+6\cdot 2xy^{2-1}+0 \\&=& 20x^2+12xy\end{array}

 

13)
\mathbb{D}=\mathbb{R}_0^+\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_x(x,y) &=& \dfrac{1}{2}x^{\frac{1}{2}-1}+5y+0 \\\\&=& \dfrac{1}{2}x^{-\frac{1}{2}}+5y \\\\&=& \dfrac{1}{2\sqrt{x}}+5y\end{array}   \begin{array}{rcl}g'_{xx}(x,y) &=& \dfrac{1}{2}\cdot \left(-\dfrac{1}{2}\right)x^{-\frac{1}{2}-1}+0 \\\\&=& -\dfrac{1}{4}x^{-\frac{3}{2}} \\\\&=& \dfrac{-1}{4\sqrt{x^3}}\end{array}
    \begin{array}{rcl}g'_{xy}(x,y) &=& 0+5 \\&=& 5\end{array}
\begin{array}{rcl}g'_y(x,y) &=& 0+5x+2y^{2-1} \\&=& 5x+2y\end{array}   \begin{array}{rcl}g'_{yx}(x,y) &=& 5+0 \\&=& 5\end{array}
    \begin{array}{rcl}g'_{yy}(x,y) &=& 0+2 \\&=& 2\end{array}

 

14)
\mathbb{D}=\mathbb{R}\times\mathbb{R}_0^+

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_x(x,y) &=& 2x^{2-1}y^{\frac{3}{2}}-0 \\\\&=& 2xy^{\frac{3}{2}} \\\\&=& 2x\sqrt{y^3}\end{array}   \begin{array}{rcl}g'_{xx}(x,y) &=& 2\sqrt{y^3}\end{array}
    \begin{array}{rcl}g'_{xy}(x,y) &=& 2\cdot\dfrac{3}{2}xy^{\frac{3}{2}-1} \\\\&=& 3xy^{\frac{1}{2}} \\\\&=& 3x\sqrt{y}\end{array}
\begin{array}{rcl}g'_y(x,y) &=& \dfrac{3}{2}x^2 y^{\frac{3}{2}-1}-0 \\\\&=& \dfrac{3}{2}x^2y^{\frac{1}{2}} \\\\&=& \dfrac{3}{2}x^2\sqrt{y}\end{array}   \begin{array}{rcl}g'_{yx}(x,y) &=& \dfrac{3}{2}\cdot 2x^{2-1}\sqrt{y} \\\\&=& 3x\sqrt{y}\end{array}
    \begin{array}{rcl}g'_{yy}(x,y) &=& \dfrac{3}{2}\cdot\dfrac{1}{2}x^2y^{\frac{1}{2}-1} \\\\&=& \dfrac{3}{4}x^2y^{-\frac{1}{2}} \\\\&=& \dfrac{3x^2}{4\sqrt{y}}\end{array}

 

15)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_d(d,h) &=& \dfrac{\pi}{4}\cdot 2d^{2-1}h \\\\&=& \dfrac{\pi}{2}dh\end{array}   \begin{array}{rcl}g'_{dd}(d,h) &=& \dfrac{\pi}{2}h\end{array}
    \begin{array}{rcl}g'_{dh}(d,h) &=& \dfrac{\pi}{2}d\end{array}
\begin{array}{rcl}g'_h(d,h) &=& \dfrac{\pi}{4}d^2\end{array}   \begin{array}{rcl}g'_{hd}(d,h) &=& 2\cdot\dfrac{\pi}{4}d^{2-1} \\&=& \dfrac{\pi}{2}d\end{array}
    \begin{array}{rcl}g'_{hh}(d,h) &=& 0\end{array}

 

16)
\mathbb{D}=\mathbb{R}\times\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y,z) &=& -12\cdot 2x^{2-1}-3\cdot 4x^{4-1}y+34y^5+0\\&=& -24x^{1}-12x^{3}y+34y^5\end{array}   \begin{array}{rcl}f'_{xx}(x,y,z) &=& -24-12\cdot 3x^{3-1}y+0\\&=& -24-36x^2y\end{array}
    \begin{array}{rcl}f'_{xy}(x,y,z) &=& 0-12x^3+34\cdot 5y^{5-1} \\&=& -12x^3+170y^4\end{array}
    \begin{array}{rcl}f'_{xz}(x,y,z) &=& 0 \\\end{array}
\begin{array}{rcl}f'_y(x,y,z) &=& 0-3x^4+34\cdot 5xy^{5-1}+0\\&=& -3x^4+170xy^4\end{array}   \begin{array}{rcl}f'_{yx}(x,y,z) &=& -3\cdot 4x^{4-1}+170y^4\\&=& -12x^3+170y^4\end{array}
    \begin{array}{rcl}f'_{yy}(x,y,z) &=& 680xy^3\\\end{array}
    \begin{array}{rcl}f'_{yz}(x,y,z) &=& 0 \\\end{array}
\begin{array}{rcl}f'_z(x,y,z) &=& 0-0+0+\dfrac{1}{2}\\&=& \dfrac{1}{2}\end{array}   \begin{array}{rcl}f'_{zx}(x,y,z) &=& 0 \\\end{array}
    \begin{array}{rcl}f'_{zy}(x,y,z) &=& 0 \\\end{array}
    \begin{array}{rcl}f'_{zz}(x,y,z) &=& 0 \\\end{array}

 

17)
\mathbb{D}=\mathbb{R}\times\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y,z) &=& \dfrac{13}{2}\cdot 2zx^{2-1}+3\cdot 12x^{12-1}y^7+0\\\\&=& 13zx+36x^{11}y^7\end{array}   \begin{array}{rcl}f'_{xx}(x,y,z) &=& 13z+36\cdot 11x^{11-1}y^7\\&=& 13z+396x^{10}y^7\end{array}
    \begin{array}{rcl}f'_{xy}(x,y,z) &=& 0+36\cdot 7x^{11}y^{7-1} \\&=& 252x^{11}y^6\end{array}
    \begin{array}{rcl}f'_{xz}(x,y,z) &=& 13x+0 \\&=& 13x\end{array}
\begin{array}{rcl}f'_y(x,y,z) &=& 0+3\cdot 7x^{12}y^{7-1}+0\\&=& 21x^{12}y^6\end{array}   \begin{array}{rcl}f'_{yx}(x,y,z) &=& 21\cdot 12x^{12-1}y^6\\&=& 252x^{11}y^6\end{array}
    \begin{array}{rcl}f'_{yy}(x,y,z) &=& 21\cdot 6x^{12}y^{6-1}\\&=& 126x^{12}y^5\end{array}
    \begin{array}{rcl}f'_{yz}(x,y,z) &=& 0 \\\end{array}
\begin{array}{rcl}f'_z(x,y,z) &=& \dfrac{13}{2}x^2+0+\sqrt{78}\\\\&=& \dfrac{13}{2}x^2+\sqrt{78}\end{array}   \begin{array}{rcl}f'_{zx}(x,y,z) &=& \dfrac{13}{2}\cdot 2x^{2-1} \\\\&=& 13x\end{array}
    \begin{array}{rcl}f'_{zy}(x,y,z) &=& 0\\\\\end{array}
    \begin{array}{rcl}f'_{zz}(x,y,z) &=& 0 \\\end{array}

 

18)
\mathbb{D}=\mathbb{R}\times\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_a(a,b,c) &=& \dfrac{\pi}{2}\cdot 2a^{2-1}b^3c+17\pi b^3+0\\\\&=& \pi a^1b^3c+17\pi b^3\end{array}   \begin{array}{rcl}f'_{aa}(a,b,c) &=& \pi b^3c+0\\&=& \pi b^3c\end{array}
    \begin{array}{rcl}f'_{ab}(a,b,c) &=& \pi\cdot 3ab^{3-1}c+17\pi\cdot 3b^{3-1}\\&=& 3\pi ab^{2}c+51\pi b^{2}\end{array}
    \begin{array}{rcl}f'_{ac}(a,b,c) &=& \pi ab^3+0\\&=& \pi ab^3\end{array}
\begin{array}{rcl}f'_b(a,b,c) &=& \dfrac{\pi}{2}\cdot 3a^2b^{3-1}c+17\pi\cdot 3ab^{3-1}+31\cdot 2c^2b^{2-1}\\\\&=& \dfrac{3\pi}{2}a^2b^{2}c+51\pi ab^{2}+62c^2b\end{array}   \begin{array}{rcl}f'_{ba}(a,b,c) &=& \dfrac{3\pi}{2}\cdot 2a^{2-1}b^{2}c+51\pi b^{2}+0\\\\&=& 3\pi ab^{2}c+51\pi b^{2}\\\end{array}
    \begin{array}{rcl}f'_{bb}(a,b,c) &=& \dfrac{3\pi}{2}\cdot 2a^2b^{2-1}c+2\cdot 51\pi ab^{2-1}+62c^2\\\\&=& 3\pi a^2bc+102\pi ab+62c^2\end{array}
    \begin{array}{rcl}f'_{bc}(a,b,c) &=& \dfrac{3\pi}{2}a^2b^{2}+0+62\cdot 2c^{2-1}b\\\\&=& \dfrac{3\pi}{2}a^2b^{2}+124cb\\\end{array}
\begin{array}{rcl}f'_c(a,b,c) &=& \dfrac{\pi}{2}a^2b^3+0+31\cdot 2c^{2-1}b^2\\\\&=& \dfrac{\pi}{2}a^2b^3+62cb^2\end{array}   \begin{array}{rcl}f'_{ca}(a,b,c) &=& \dfrac{\pi}{2}\cdot 2a^{2-1}b^3+0\\\\&=& \pi ab^3\end{array}
    \begin{array}{rcl}f'_{cb}(a,b,c) &=& \dfrac{\pi}{2}\cdot 3a^2b^{3-1}+62\cdot 2cb^{2-1}\\\\&=& \dfrac{3\pi}{2}a^2b^{2}+124cb\end{array}
    \begin{array}{rcl}f'_{cc}(a,b,c) &=& 0+62b^2\\&=& 62b^2\end{array}

 

19)
\mathbb{D}=\mathbb{R}\times\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y,z) &=& 3ab^2z+0-0\\&=& 3ab^2z\end{array}   \begin{array}{rcl}f'_{xx}(x,y,z) &=& 0\end{array}
    \begin{array}{rcl}f'_{xy}(x,y,z) &=& 0\end{array} 
    \begin{array}{rcl}f'_{xz}(x,y,z) &=& 3ab^2\end{array}
\begin{array}{rcl}f'_y(x,y,z) &=& 0+8abz-0\\&=& 8abz\end{array}   \begin{array}{rcl}f'_{yx}(x,y,z) &=& 0\end{array}
    \begin{array}{rcl}f'_{yy}(x,y,z) &=& 0\end{array}
    \begin{array}{rcl}f'_{yz}(x,y,z) &=& 8ab \\\end{array}
\begin{array}{rcl}f'_z(x,y,z) &=& 3ab^2x+8aby-0\\&=& ab\left(3bx+8y\right)\end{array}   \begin{array}{rcl}f'_{zx}(x,y,z) &=& 3ab^2\end{array}
    \begin{array}{rcl}f'_{zy}(x,y,z) &=& 8ab\end{array}
    \begin{array}{rcl}f'_{zz}(x,y,z) &=& 0 \\\end{array}

 

20)
\mathbb{D}=\mathbb{R}\times\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y,z) &=& \dfrac{3}{2}\cdot 2x^{2-1}y^{\frac{1}{2}}z^3+4\cdot\dfrac{1}{3}x^{\frac{1}{3}-1}y^2\\\\&=& 3x^1y^{\frac{1}{2}}z^3+\dfrac{4}{3}x^{-\frac{2}{3}}y^2\\\\&=& 3x\sqrt{y}z^3+\dfrac{4y^2}{3x^{\frac{2}{3}}}\end{array}   \begin{array}{rcl}f'_{xx}(x,y,z) &=& 3y^{\frac{1}{2}}z^3-\dfrac{4}{3}\cdot\dfrac{2}{3}x^{-\frac{2}{3}-1}y^2\\\\&=& 3y^{\frac{1}{2}}z^3-\dfrac{8}{9}x^{-\frac{5}{3}}y^2\end{array}
    \begin{array}{rcl}f'_{xy}(x,y,z) &=& 3\cdot\dfrac{1}{2}x^1y^{\frac{1}{2}-1}z^3+\dfrac{4}{3}\cdot 2x^{-\frac{2}{3}}y^{2-1}\\\\&=& \dfrac{3}{2}x^1y^{-\frac{1}{2}}z^3+\dfrac{8}{3}x^{-\frac{2}{3}}y\end{array}
    \begin{array}{rcl}f'_{xz}(x,y,z) &=& 3\cdot 3x^1y^{\frac{1}{2}}z^{3-1}+0\\\\&=& 9xy^{\frac{1}{2}}z^{2}\end{array}
\begin{array}{rcl}f'_y(x,y,z) &=& \frac{3}{2}\cdot\dfrac{1}{2}x^2y^{\frac{1}{2}-1}z^3+4\cdot 2x^{\frac{1}{3}}y^{2-1}\\\\&=& \dfrac{3}{4}x^2y^{-\frac{1}{2}}z^3+8x^{\frac{1}{3}}y^1\\\\&=& \dfrac{3x^2z^3}{4\sqrt{y}}+8x^{\frac{1}{3}}y\end{array}   \begin{array}{rcl}f'_{yx}(x,y,z) &=& \dfrac{3}{4}\cdot 2x^{2-1}y^{-\frac{1}{2}}z^3+8\cdot\dfrac{1}{3}x^{\frac{1}{3}-1}y^1\\\\&=& \dfrac{3}{2}xy^{-\frac{1}{2}}z^3+\dfrac{8}{3}x^{-\frac{2}{3}}y^1\end{array}
    \begin{array}{rcl}f'_{yy}(x,y,z) &=& -\dfrac{3}{4}\cdot\dfrac{1}{2}x^2y^{-\frac{1}{2}-1}z^3+8x^{\frac{1}{3}}\\\\&=& -\dfrac{3}{8}x^2y^{-\frac{3}{2}}z^3+8x^{\frac{1}{3}}\end{array}
    \begin{array}{rcl}f'_{yz}(x,y,z) &=& \dfrac{3}{4}\cdot 3x^2y^{-\frac{1}{2}}z^{3-1}+0\\\\&=& \dfrac{9}{4}x^2y^{-\frac{1}{2}}z^{2}\end{array}
\begin{array}{rcl}f'_z(x,y,z) &=& \dfrac{3}{2}\cdot 3x^2y^{\frac{1}{2}}z^{3-1}+0\\\\&=& \dfrac{9}{2}x^2y^{\frac{1}{2}}z^2\\\\&=& \dfrac{9}{2}x^2\sqrt{y}z^2\end{array}   \begin{array}{rcl}f'_{zx}(x,y,z) &=& \dfrac{9}{2}\cdot 2x^{2-1}y^{\frac{1}{2}}z^2\\\\&=& 9xy^{\frac{1}{2}}z^2\end{array}
    \begin{array}{rcl}f'_{zy}(x,y,z) &=& \dfrac{9}{2}\cdot\dfrac{1}{2}x^2y^{\frac{1}{2}-1}z^2\\\\&=& \dfrac{9}{4}x^2y^{-\frac{1}{2}}z^2\end{array}
    \begin{array}{rcl}f'_{zz}(x,y,z) &=& \dfrac{9}{2}\cdot 2x^2y^{\frac{1}{2}}z^{2-1}\\\\&=& 9x^2y^{\frac{1}{2}}z\\\\\end{array}

 

2. Aufgabe

Eine Bemerkung vorab: Bei diesen Funktionen werden für die partiellen Ableitungen teilweise auch die Produkt-, Quotienten- und Kettenregel benötigt. Dies ist jeweils unter der partiellen Ableitung notiert. Die übrigen "einfachen" Regeln werden hingegen nicht erwähnt. Für alle, die mathematisch bis hierhin gekommen sind, sollten Summen-, Faktor- etc. Regel keine Schwierigkeit mehr darstellen. 

1) 
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& 0+1\cdot e^{4xy}+x\cdot e^{4xy}\cdot 4y\\&=& e^{4xy}+4xy e^{4xy}\end{array}

Vorgehen: Produktregel, Kettenregel		   \begin{array}{rcl}f'_{xx}(x,y) &=& e^{4xy}\cdot 4y+4y\cdot e^{4xy}+4xy\cdot e^{4xy}\cdot 4y\\&=& 8ye^{4xy}+16xy^2e^{4xy}\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl}f'_{xy}(x,y) &=& e^{4xy}\cdot 4x+4x\cdot e^{4xy}+4xy\cdot e^{4xy}\cdot 4x \\&=& 8xe^{4xy}+16x^2ye^{4xy}\end{array}

Vorgehen: Produktregel, Kettenregel
\begin{array}{rcl}f'_y(x,y) &=& 0+x\cdot e^{4xy}\cdot 4x\\&=& 4x^2e^{4xy}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{yx}(x,y) &=& 4\cdot 2x^{2-1}\cdot e^{4xy}+4x^2\cdot e^{4xy}\cdot 4y\\&=& 8xe^{4xy}+16x^2ye^{4xy}\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl}f'_{yy}(x,y) &=& 4x^2\cdot e^{4xy}\cdot 4x\\&=& 16x^3e^{4xy}\end{array}

Vorgehen: Kettenregel

 

2)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& e^x+0-15y\\&=& e^x-15y\end{array}

   \begin{array}{rcl}f'_{xx}(x,y) &=& e^x-0\\&=& e^x\end{array}
    \begin{array}{rcl}f'_{xy}(x,y) &=& 0-15\\&=& -15\end{array}
\begin{array}{rcl}f'_y(x,y) &=& 0+3\cdot 2y^{2-1}-15x\\&=& 6y-15x\end{array}

   \begin{array}{rcl}f'_{yx}(x,y) &=& 0-15\\&=& -15\end{array}
    \begin{array}{rcl}f'_{yy}(x,y) &=& 6-0\\&=& 6\end{array}

 

3)
\mathbb{D}=\mathbb{R}_0^+\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& e^{3xy-23}\cdot 3y+y\cdot\dfrac{1}{2}x^{\frac{1}{2}-1}\\\\&=& 3ye^{3xy-23}+\dfrac{1}{2}yx^{-\frac{1}{2}}\\\\&=& 3ye^{3xy-23}+\dfrac{y}{2\sqrt{x}}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{xx}(x,y) &=& 3y\cdot e^{3xy-23}\cdot3y-\dfrac{1}{2}y\cdot\dfrac{1}{2}x^{-\frac{1}{2}-1}\\\\&=& 9y^2e^{3xy-23}-\dfrac{1}{4}yx^{-\frac{3}{2}}\\\\&=& 9y^2e^{3xy-23}-\dfrac{y}{4\sqrt{x^3}}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl}f'_{xy}(x,y) &=& 3\cdot e^{3xy-23}+3y\cdot e^{3xy-23}\cdot 3x+\dfrac{1}{2\sqrt{x}}\\\\&=& 3e^{3xy-23}+9yxe^{3xy-23}+\dfrac{1}{2\sqrt{x}}\end{array}

Vorgehen: Produktregel, Kettenregel
\begin{array}{rcl}f'_y(x,y) &=& e^{3xy-23}\cdot 3x+x^{\frac{1}{2}}\\&=& 3xe^{3xy-23}+\sqrt{x}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{yx}(x,y) &=& 3\cdot e^{3xy-23}+3x\cdot e^{3xy-23}\cdot 3y+\dfrac{1}{2} x^{\frac{1}{2}-1}\\&=& 3e^{3xy-23}+9xye^{3xy-23}+\dfrac{1}{2}x^{-\frac{1}{2}}\\&=& 3e^{3xy-23}+9xye^{3xy-23}+\dfrac{1}{2\sqrt{x}}\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl}f'_{yy}(x,y) &=& 3x\cdot e^{3xy-23}\cdot 3x+0\\&=& 9x^2e^{3xy-23}\end{array}

Vorgehen: Kettenregel

 


4)
\mathbb{D}=\mathbb{R}^+\times\mathbb{R}^+

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& \dfrac{1}{xy}\cdot y-0\\&=& \dfrac{1}{x}\\&=& x^{-1}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{xx}(x,y) &=& -1\cdot x^{-1-1}\\&=& -x^{-2}\\&=& -\dfrac{1}{x^2}\end{array}
    \begin{array}{rcl}f'_{xy}(x,y) &=& 0\end{array}
\begin{array}{rcl}f'_y(x,y) &=& \dfrac{1}{xy}\cdot x-0\\&=& \dfrac{1}{y}\\&=& y^{-1}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{yx}(x,y) &=& 0\end{array}
    \begin{array}{rcl}f'_{yy}(x,y) &=& -1\cdot y^{-1-1}\\&=& -y^{-2}\\&=& -\dfrac{1}{y^2}\end{array}

 

5)
\mathbb{D}=\left]-1;\infty\right[\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_t(t,z) &=& \dfrac{1}{1+t}\cdot 1\cdot z^2+5\cdot 2t^{2-1}z-0\\&=& \dfrac{z^2}{1+t}+10tz\\&=& z^2\cdot \left(1+t\right)^{-1}+10tz\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{tt}(t,z) &=& z^2\cdot (-1)\cdot \left(1+t\right)^{-1-1}\cdot 1+10z\\&=& -\dfrac{z^2}{\left(1+t\right)^2}+10z\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl}f'_{tz}(t,z) &=& \dfrac{2z^{2-1}}{1+t}+10t\\\\&=& \dfrac{2z}{1+t}+10t\end{array}
\begin{array}{rcl}f'_z(t,z) &=& \ln(1+t)\cdot 2\cdot z^{2-1}+5t^2-0\\&=& 2z\ln(1+t)+5t^2\end{array}

   \begin{array}{rcl}f'_{zt}(t,z) &=& 2z\cdot \dfrac{1}{1+t}\cdot 1+5\cdot 2t^{2-1}\\&=& \dfrac{2z}{1+t}+10t\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl}f'_{zz}(t,z) &=& 2\ln(1+t)+0\\&=& 2\ln(1+t)\end{array}

 


6)
\mathbb{D}=\mathbb{R}\times\mathbb{R} und x+y^2>0

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& \dfrac{1}{x+y^2}\cdot 1-e^{2xy}\cdot 2y+3\\&=& \dfrac{1}{x+y^2}-2ye^{2xy}+3\\&=& \left(x+y^2\right)^{-1}-2ye^{2xy}+3\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{xx}(x,y) &=& -1\left(x+y^2\right)^{-1-1}\cdot 1\\& & -2ye^{2xy}\cdot 2y\\\\&=& -\dfrac{1}{\left(x+y^2\right)^2}-4y^2e^{2xy}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl}f'_{xy}(x,y) &=& -1\left(x+y^2\right)^{-1-1}\cdot 2y\\& & -2\cdot e^{2xy}-2ye^{2xy}\cdot 2x\\\\&=& -\dfrac{2y}{\left(x+y^2\right)^2}+\left(-2-4xy\right)e^{2xy}\\\end{array}

Vorgehen: Produktregel, Kettenregel

\begin{array}{rcl}f'_y(x,y) &=& \dfrac{1}{x+y^2}\cdot 2y-e^{2xy}\cdot 2x+0\\&=& 2y\left(x+y^2\right)^{-1}-2xe^{2xy}\end{array}

Vorgehen: Kettenregel

   \begin{array}{rcl}f'_{yx}(x,y) &=& 2y\cdot (-1)\left(x+y^2\right)^{-1-1}\cdot 1\\& & -2\cdot e^{2xy}-2xe^{2xy}\cdot 2y\\\\&=& -\dfrac{2y}{\left(x+y^2\right)^2}+\left(-2-4xy\right)e^{2xy}\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl}f'_{yy}(x,y) &=& 2\cdot \left(x+y^2\right)^{-1}+2y\cdot(-1)\left(x+y^2\right)^{-1-1}\cdot 2y\\& & -0\cdot e^{2xy}-2xe^{2xy}\cdot 2x\\\\&=& \dfrac{2}{x+y^2}-\dfrac{4y^2}{\left(x+y^2\right)^2}-4x^2e^{2xy}\end{array}

Vorgehen: Kettenregel, Produktregel

 

7)
\mathbb{D}=\mathbb{R}\times \mathbb{R}^+

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& e^{3x}\cdot 3 \cdot\ln(y)-e^{2y}\cdot 5\\&=& 3e^{3x}\cdot\ln(y)-5e^{2y}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{xx}(x,y) &=& 3e^{3x}\cdot 3 \cdot\ln(y)-0\\&=& 9e^{3x}\cdot\ln(y)\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl}f'_{xy}(x,y) &=& 3e^{3x}\cdot\dfrac{1}{y}-5e^{2y}\cdot 2\\&=& \dfrac{3e^{3x}}{y}-10e^{2y}\end{array}

Vorgehen: Kettenregel
\begin{array}{rcl}f'_y(x,y) &=& e^{3x}\cdot\dfrac{1}{y}-e^{2y}\cdot 2 \cdot 5x\\&=& \dfrac{e^{3x}}{y}-10xe^{2y}\\&=& e^{3x}\cdot y^{-1}-10xe^{2y}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{yx}(x,y) &=& e^{3x}\cdot 3\cdot\dfrac{1}{y}-10e^{2y}\\&=& \dfrac{3e^{3x}}{y}-10e^{2y}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl}f'_{yy}(x,y) &=& e^{3x}\cdot (-1)\cdot y^{-1-1}-10xe^{2y}\cdot 2\\&=& -\dfrac{e^{3x}}{y^2}-20xe^{2y}\end{array}

Vorgehen: Kettenregel

 

8)
\mathbb{D}=\mathbb{R}^+\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_x(x,y) &=& 52e^y+\dfrac{y}{8}\cdot\dfrac{1}{x}\\\\&=& 52e^y+\dfrac{y}{8x}\\\\&=& 52e^y+\dfrac{1}{8}x^{-1}y\end{array}

   \begin{array}{rcl}g'_{xx}(x,y) &=& 0+\dfrac{1}{8}\cdot(-1)x^{-1-1}y\\\\&=& -\dfrac{y}{8x^2}\end{array}
    \begin{array}{rcl}g'_{xy}(x,y) &=& 52e^y+\dfrac{1}{8x}\end{array}
\begin{array}{rcl}g'_y(x,y) &=& 52xe^y+\dfrac{1}{8}\ln(x)\end{array}

   \begin{array}{rcl}g'_{yx}(x,y) &=& 52e^y+\dfrac{1}{8x}\end{array}
    \begin{array}{rcl}g'_{yy}(x,y) &=& 52xe^y+0\\&=& 52xe^y\end{array}

 

9)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_h(h,k) &=& 135\cos\left(h^2-k\right)\cdot 2h\\&=& 270h\cos\left(h^2-k\right)\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}g'_{hh}(h,k) &=& 270\cos\left(h^2-k\right)\\& & +270h\left(-\sin\left(h^2-k\right)\cdot 2h\right)\\&=& 270\cos\left(h^2-k\right)-540h^2\sin\left(h^2-k\right)\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl}g'_{hk}(h,k) &=& -270h\sin\left(h^2-k\right)\cdot 1\\&=& -270h\sin\left(h^2-k\right)\end{array}

Vorgehen: Kettenregel
\begin{array}{rcl}g'_k(h,k) &=& 135\cos\left(h^2-k\right)\cdot 1\\&=& 135\cos\left(h^2-k\right)\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}g'_{kh}(h,k) &=& -135\sin\left(h^2-k\right)\cdot 2h\\&=& -270h\sin\left(h^2-k\right)\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl}g'_{kk}(h,k) &=& -135\sin\left(h^2-k\right)\cdot 1\\&=& -135\sin\left(h^2-k\right)\end{array}

Vorgehen: Kettenregel

 

10)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& 4\cos(x)-\sin\left(-\dfrac{1}{4}x+y\right)\cdot\left(-\dfrac{1}{4}\right)\\&=& 4\cos(x)+\dfrac{1}{4}\sin\left(-\dfrac{1}{4}x+y\right)\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{xx}(x,y) &=&-4\sin(x)\\& & +\dfrac{1}{4}\cos\left(-\dfrac{1}{4}x+y\right)\cdot \left(-\dfrac{1}{4}\right)\\&=& -4\sin(x)-\dfrac{1}{16}\cos\left(-\dfrac{1}{4}x+y\right)\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl}f'_{xy}(x,y) &=& \dfrac{1}{4}\cos\left(-\dfrac{1}{4}x+y\right)\cdot 1\\&=& \dfrac{1}{4}\cos\left(-\dfrac{1}{4}x+y\right)\end{array}

Vorgehen: Kettenregel
\begin{array}{rcl}f'_y(x,y) &=& -\sin(-4y)\cdot(-4)-\sin\left(-\dfrac{1}{4}x+y\right)\cdot 1\\&=& 4\sin(-4y)-\sin\left(-\dfrac{1}{4}x+y\right)\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{yx}(x,y) &=& -\cos\left(-\dfrac{1}{4}x+y\right)\cdot \left(-\dfrac{1}{4}\right)\\&=& \dfrac{1}{4}\cos\left(-\dfrac{1}{4}x+y\right)\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl}f'_{yy}(x,y) &=& 4\cos(-4y)\cdot (-4)\\& & -\cos\left(-\dfrac{1}{4}x+y\right)\cdot 1\\&=&-16\cos(-4y)-\cos\left(-\dfrac{1}{4}x+y\right)\end{array}

Vorgehen: Kettenregel

 

11)
\mathbb{D}=\mathbb{R}\times\mathbb{R}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_a(a,b) &=& -b\sin(a)+\sin(b)-b\end{array}
   \begin{array}{rcl}g'_{aa}(a,b) &=& -b\cos(a)+0+0\\&=& -b\cos(a)\end{array}
    \begin{array}{rcl}g'_{ab}(a,b) &=& -\sin(a)+\cos(b)-1\end{array}
\begin{array}{rcl}g'_b(a,b) &=& \cos(a)+a\cos(b)-a\end{array}
   \begin{array}{rcl}g'_{ba}(a,b) &=& -\sin(a)+\cos(b)-1\end{array}
    \begin{array}{rcl}g'_{bb}(a,b) &=& 0-a\sin(b)+0\\&=& -a\sin(b)\end{array}

 

12)
\begin{array}{rcl}\mathbb{D} &=& \mathbb{R}\times\mathbb{R}\setminus_{\{0\}} \\\\g(x,y) &=& \dfrac{7\cos(x)}{10y}+19xy-0{,}851 \\&=& \dfrac{7}{10} \cos(x)\cdot y^{-1}+19xy-0{,}851\end{array}

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_x(x,y) &=& -\dfrac{7\sin(x)}{10y}+19y\\\\&=& -\dfrac{7}{10}\sin(x)\cdot y^{-1}+19y\end{array}

   \begin{array}{rcl}g'_{xx}(x,y) &=& -\dfrac{7\cos(x)}{10y}+0\\\\&=& -\dfrac{7\cos(x)}{10y}\end{array}
    \begin{array}{rcl}g'_{xy}(x,y) &=& -\dfrac{7}{10}\sin(x)\cdot(-1)y^{-1-1}+19\\\\&=& \dfrac{7}{10}\sin(x)\cdot y^{-2}+19\\\\&=& \dfrac{7\sin(x)}{10y^2}+19\end{array}
\begin{array}{rcl}g'_y(x,y) &=& \dfrac{7}{10}\cos(x)\cdot(-1)y^{-1-1}+19x\\\\&=& -\dfrac{7}{10}\cos(x)\cdot y^{-2}+19x\\\\&=& -\dfrac{7\cos(x)}{10y^2}+19x\end{array}

   \begin{array}{rcl}g'_{yx}(x,y) &=& \dfrac{7\sin(x)}{10y^2}+19\end{array}
    \begin{array}{rcl}g'_{yy}(x,y) &=& -\dfrac{7}{10}\cos(x)\cdot(-2)y^{-2-1}+0\\\\&=& \dfrac{7}{5}\cos(x)\cdot y^{-3}\\\\&=&\dfrac{7\cos(x)}{5y^3}\end{array}

 

13)

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}f'_x(x,y) &=& \dfrac{6\cdot\dfrac{1}{x}\cdot\left(x+9y\right)-6\ln(x)\cdot 1}{\left(x+9y\right)^2} \\\\&=&\dfrac{6\left(x+9y\right)-6x\ln(x)}{x\left(x+9y\right)^2} \\\\&=& \dfrac{6x-6x\ln(x)+54y}{x(x+9y)^2}\end{array}

Vorgehen: Quotientenregel		   \begin{array}{rcl}f'_{xx}(x,y) &=& \dfrac{\left(6-6\ln(x)-6x\cdot\frac{1}{x}\right)\cdot x\left(x+9y\right)^2-\left(6x-6x\ln(x)+54y\right)\cdot\left(1\cdot\left(x+9y\right)^2+x\cdot 2\left(x+9y\right)\cdot 1\right)}{x^2\left(x+9y\right)^4} \\\\&=& \dfrac{\left(6-6\ln(x)-6\right)\cdot x\left(x+9y\right)^2-\left(6x-6x\ln(x)+54y\right)\cdot\left(\left(x+9y\right)^2+2x(x+9y)\right)}{x^2\left(x+9y\right)^4} \\\\&=& \dfrac{(x+9y)\left[-6\ln(x)\cdot x\left(x+9y\right)-\left(6x-6x\ln(x)+54y\right)\cdot\left(\left(x+9y\right)+2x\right)\right]}{x^2\left(x+9y\right)^4} \\\\&=& \dfrac{-6x\ln(x)\cdot\left(x+9y\right)-\left(6x-6x\ln(x)+54y\right)\cdot\left(3x+9y\right)}{x^2\left(x+9y\right)^3} \\\\&=& \dfrac{-6x^2\ln(x)-54xy\ln(x)-18x^2+18x^2\ln(x)-162xy-54xy+54xy\ln(x)-486y^2}{x^2\left(x+9y\right)^3} \\\\&=& \dfrac{12x^2\ln(x)-18x^2-216xy-486y^2}{x^2\left(x+9y\right)^3}\end{array}

Vorgehen: Produktregel, Quotientenregel, Kettenregel
    \begin{array}{rcl}f'_{xy}(x,y) &=& \dfrac{54\cdot x\left(x+9y\right)^2-\left(6x-6x\ln(x)+54y\right)\cdot 2x\left(x+9y\right)^{2-1}\cdot 9}{x^2\left(x+9y\right)^4} \\\\&=& \dfrac{x\left(x+9y\right)\left[54\left(x+9y\right)-\left(6x-6x\ln(x)+54y\right)\cdot 18\right]}{x^2\left(x+9y\right)^4} \\\\&=& \dfrac{54x+486y-108x+108x\ln(x)-972y}{x\left(x+9y\right)^3} \\\\&=& \dfrac{-54x-486y+108x\ln(x)}{x\left(x+9y\right)^3}\end{array}

Vorgehen: Quotientenregel, Kettenregel
\begin{array}{rcl}f'_y(x,y) &=& \dfrac{0\cdot\left(x+9y\right)-6\ln(x)\cdot 9}{\left(x+9y\right)^2} \\\\&=& -\dfrac{54\ln(x)}{(x+9y)^2}\end{array}

Vorgehen: Quotientenregel		   \begin{array}{rcl}f'_{yx}(x,y) &=& \dfrac{-54\cdot\dfrac{1}{x}\cdot\left(x+9y\right)^2-\left(-54\ln(x)\cdot 2\left(x+9y\right)^{2-1}\cdot 1\right)}{\left(x+9y\right)^4} \\\\&=& \dfrac{-54\cdot\left(x+9y\right)^2+54x\ln(x)\cdot 2\left(x+9y\right)}{x\left(x+9y\right)^4} \\\\&=& \dfrac{(x+9y)\left[-54\left(x+9y\right)+108x\ln(x)\right]}{x\left(x+9y\right)^4} \\\\&=& \dfrac{-54x-486y+108x\ln(x)}{x\left(x+9y\right)^3}\end{array}

Vorgehen: Quotientenregel, Kettenregel
    \begin{array}{rcl}f'_{yy}(x,y) &=& \dfrac{0\cdot\left(x+9y\right)^2-\left(-54\ln(x)\cdot 2\left(x+9y\right)^{2-1}\cdot 9\right)}{\left(x+9y\right)^4} \\\\&=& \dfrac{54\ln(x)\cdot 18\left(x+9y\right)}{\left(x+9y\right)^4} \\\\&=& \dfrac{972\ln(x)}{\left(x+9y\right)^3}\end{array}

Vorgehen: Quotientenregel, Kettenregel

 

14)

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung 
\begin{array}{rcl}g'_r(r,t) &=& -\dfrac{16}{\cos^2(r)}+3te^{rt}\\\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}g'_{rr}(r,t) &=& \dfrac{0\cdot \cos^2(r)-\left(-16\cdot 2\cdot\cos(r)\cdot\left(-\sin(r)\right)\right)}{\cos^4(r)}+3t\cdot te^{rt} \\\\&=& -\dfrac{32\cos(r)\cdot\sin(r)}{\cos^4(r)}+3t^2e^{rt} \\\\&=& -\dfrac{32\sin(r)}{\cos^3(r)}+3t^2e^{rt}\end{array}

Vorgehen: Quotientenregel, Kettenregel
    \begin{array}{rcl}g'_{rt}(r,t) &=& 0+3\cdot e^{rt}+3t\cdot re^{rt} \\&=& 3e^{rt}+3rte^{rt} \\&=& 3e^{rt}\left(1+rt\right)\end{array}

Vorgehen: Produktregel, Kettenregel
\begin{array}{rcl}g'_t(r,t) &=& 0+3r\cdot e^{rt} \\&=& 3re^{rt}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}g'_{tr}(r,t) &=& 3\cdot e^{rt}+3r\cdot te^{rt} \\&=& 3e^{rt}+3rte^{rt} \\&=& 3e^{rt}\left(1+rt\right)\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl}g'_{tt}(r,t) &=& 3r\cdot re^{rt} \\&=& 3r^2e^{rt}\end{array}

Vorgehen: Kettenregel

 

15)

Partielle Ableitungen 1. Ordnung 

  

Partielle Ableitungen 2. Ordnung  
\begin{array}{rcl}f'_x(x,y) &=& \dfrac{5}{9}e^{\sin(x)+\sin(y)}\cdot \cos(x) \\\\&=& \dfrac{5}{9}\cos(x)e^{\sin(x)+\sin(y)}\end{array}


Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{xx}(x,y) &=& -\dfrac{5}{9}\sin(x)\cdot e^{\sin(x)+\sin(y)}+\dfrac{5}{9}\cos(x)e^{\sin(x)+\sin(y)}\cdot\cos(x) \\\\&=& -\dfrac{5}{9}\sin(x)e^{\sin(x)+\sin(y)}+\dfrac{5}{9}\cos^2(x)e^{\sin(x)+\sin(y)} \\\\&=& \dfrac{5}{9}e^{\sin(x)+\sin(y)}\left(-\sin(x)+\cos^2(x)\right)\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl}f'_{xy}(x,y) &=& \dfrac{5}{9}\cos(x)e^{\sin(x)+\sin(y)}\cdot\cos(y) \\\\&=& \dfrac{5}{9}\cos(x)\cos(y)e^{\sin(x)+\sin(y)}\end{array}

Vorgehen: Kettenregel
\begin{array}{rcl}f'_y(x,y) &=& \dfrac{5}{9}e^{\sin(x)+\sin(y)}\cdot \cos(y) \\\\&=& \dfrac{5}{9}\cos(y)e^{\sin(x)+\sin(y)}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl}f'_{yx}(x,y) &=& \dfrac{5}{9}\cos(y)e^{\sin(x)+\sin(y)}\cdot\cos(x) \\\\&=& \dfrac{5}{9}\cos(x)\cos(y)e^{\sin(x)+\sin(y)}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl}f'_{yy}(x,y) &=& -\dfrac{5}{9}\sin(y)\cdot e^{\sin(x)+\sin(y)}+\dfrac{5}{9}\cos(y)e^{\sin(x)+\sin(y)}\cdot\cos(y) \\\\&=& -\dfrac{5}{9}\sin(y)e^{\sin(x)+\sin(y)}+\dfrac{5}{9}\cos^2(y)e^{\sin(x)+\sin(y)} \\\\&=& \dfrac{5}{9}e^{\sin(x)+\sin(y)}\left(-\sin(y)+\cos^2(y)\right)\end{array}

Vorgehen: Produktregel, Kettenregel

 

16)
\mathbb{D}=\mathbb{R}\times\mathbb{R}\times\mathbb{R}^+

Partielle Ableitungen 1. Ordnung

 

Partielle Ableitungen 2. Ordnung

\begin{array}{rcl} f'_x(x,y,z) &=& y^2 e^{-3xy} \ln(z)+xy^2 e^{-3xy}\cdot (-3y)\ln(z) \cr \cr &=& y^2 e^{-3xy} \ln(z) - 3xy^3 e^{-3xy} \ln(z) \cr \cr&=& \left(y^2-3xy^3\right) e^{-3xy} \ln(z)\end{array}

Vorgehen: Produktregel, Kettenregel		   \begin{array}{rcl} f'_{xx}(x,y,z) &=& -3y^3 e^{-3xy} \ln(z) + \left(y^2-3xy^3\right) e^{-3xy}\cdot(-3y)\ln(z) \\\\&=& -3y^3 e^{-3xy} \ln(z) - 3y\left(y^2-3xy^3\right) e^{-3xy}\ln(z) \\\\&=& \left[-3y^3- 3y\left(y^2-3xy^3\right)\right] e^{-3xy} \ln(z) \\\\&=& \left[-3y^3-3y^3+9xy^4\right] e^{-3xy} \ln(z) \\\\&=& \left[9xy^4-6y^3\right] e^{-3xy} \ln(z)\end{array}

Vorgehen: Produktregel, Kettelregel

 

   \begin{array}{rcl} f'_{xy}(x,y,z) &=& \left(2y-9xy^2\right) e^{-3xy} \ln(z) + \left(y^2-3xy^3\right) e^{-3xy} \cdot (-3x) \ln(z) \\\\&=& \left(2y-9xy^2\right) e^{-3xy} \ln(z) - 3x\left(y^2-3xy^3\right) e^{-3xy} \ln(z) \\\\&=& \left[\left(2y-9xy^2\right) - 3x\left(y^2-3xy^3\right)\right] e^{-3xy} \ln(z) \\\\&=& \left[2y-9xy^2-3xy^2+9x^2y^3\right] e^{-3xy} \ln(z) \\\\&=& \left[9x^2y^3-12xy^2+2y\right] e^{-3xy} \ln(z)\end{array}

Vorgehen: Produktregel, Kettelregel
    \begin{array}{rcl} f'_{xz}(x,y,z) &=& \left(y^2-3xy^3\right) e^{-3xy}\cdot\dfrac{1}{z} \\\\&=& \dfrac{y^2-3xy^3}{z}e^{-3xy}\end{array}
\begin{array}{rcl} f'_y(x,y,z) &=& 2xy e^{-3xy} \ln(z) +xy^2 e^{-3xy}\cdot (-3x) \ln(z) \cr \cr &=& 2xy e^{-3xy} \ln(z) - 3x^2y^2 e^{-3xy} \ln(z) \cr \cr&=& \left(-3x^2y^2+2xy\right) e^{-3xy} \ln(z)\end{array}

Vorgehen: Produktregel, Kettenregel		   \begin{array}{rcl} f'_{yx}(x,y,z) &=& \left(-6xy^2+2y\right) e^{-3xy}\ln(z) + \left(-3x^2y^2+2xy\right) e^{-3xy} \cdot(-3y)\ln(z) \\\\&=& \left(-6xy^2+2y\right) e^{-3xy}\ln(z) - 3y\left(-3x^2y^2+2xy\right) e^{-3xy} \ln(z) \\\\&=& \left[\left(-6xy^2+2y\right) - 3y\left(-3x^2y^2+2xy\right)\right] e^{-3xy} \ln(z) \\\\&=& \left[-6xy^2+2y+9x^2y^3-6xy^2\right] e^{-3xy} \ln(z) \\\\&=& \left[9x^2y^3-12xy^2+2y\right] e^{-3xy} \ln(z)\end{array}

Vorgehen: Produktregel, Kettenregel

 

   \begin{array}{rcl} f'_{yy}(x,y,z) &=& \left(-6x^2y+2x\right) e^{-3xy}\ln(z) + \left(-3x^2y^2+2xy\right) e^{-3xy} \cdot (-3x) \ln(z) \\\\&=& \left(-6x^2y+2x\right) e^{-3xy}\ln(z) - 3x\left(-3x^2y^2+2xy\right) e^{-3xy} \ln(z) \\\\&=& \left[\left(-6x^2y+2x\right) - 3x\left(-3x^2y^2+2xy\right)\right] e^{-3xy} \ln(z) \\\\&=& \left[-6x^2y+2x+9x^3y^2-6x^2y\right] e^{-3xy} \ln(z) \\\\&=& \left[9x^3y^2-12x^2y+2x\right] e^{-3xy} \ln(z)\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl} f'_{yz}(x,y,z) &=& \left(-3x^2y^2+2xy\right) e^{-3xy}\cdot\dfrac{1}{z} \\\\&=& \dfrac{-3x^2y^2+2xy}{z}e^{-3xy} \\\\\end{array}
\begin{array}{rcl} f'_z(x,y,z) &=& xy^2 e^{-3xy}\cdot\dfrac{1}{z} \\\\ &=& \dfrac{xy^2 e^{-3xy}}{z} \\\\ &=& xy^2 e^{-3xy}\cdot z^{-1} \end{array}   \begin{array}{rcl} f'_{zx}(x,y,z) &=& \dfrac{y^2 e^{-3xy} +xy^2 e^{-3xy}\cdot (-3y)}{z}\\\\&=& \dfrac{y^2 e^{-3xy}-3xy^3 e^{-3xy}}{z}\\\\&=& \dfrac{-3xy^3+y^2}{z} e^{-3xy}\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl} f'_{zy}(x,y,z) &=& \dfrac{2xy e^{-3xy}+xy^2 e^{-3xy} \cdot(-3x)}{z} \\\\&=& \dfrac{2xy e^{-3xy}-3x^2y^2 e^{-3xy}}{z} \\\\&=& \dfrac{2xy-3x^2y^2}{z} e^{-3xy} \\\\\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl} f'_{zz}(x,y,z) &=& xy^2 e^{-3xy}\cdot (-1)\cdot z^{-1-1} \\\\ &=& -xy^2 e^{-3xy} z^{-2} \\\\&=& -\dfrac{xy^2}{z^2} e^{-3xy}\end{array}

 

17)
\begin{array}{rcl}\mathbb{D} &=& \mathbb{R}\times\mathbb{R}\times\mathbb{R} \text{ mit } x\neq -yz \\\\f(x,y,z) &=& 300 \cdot \sqrt{x+yz}-923 \\&=& 300(x+yz)^{\frac{1}{2}}-923\end{array}

Partielle Ableitungen 1. Ordnung
  Partielle Ableitungen 2. Ordnung
\begin{array}{rcl} f'_x(x,y,z) &=& 300\cdot \dfrac{1}{2}\left(x+yz\right)^{\frac{1}{2}-1} \cdot 1\cr \cr &=& 150\left(x+yz\right)^{-\frac{1}{2}} \cr \cr&=& \dfrac{150}{\sqrt{x+yz}} \end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl} f'_{xx}(x,y,z) &=& 150\cdot \left(-\dfrac{1}{2}\right)(x+yz)^{-\frac{1}{2}-1}\cdot 1 \\\\&=& -75(x+yz)^{-\frac{3}{2}} \\\\&=& -\dfrac{75}{\left(\sqrt{x+yz}\right)^3}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{xy}(x,y,z) &=& 150\cdot \left(-\dfrac{1}{2}\right)(x+yz)^{-\frac{1}{2}-1}\cdot z \\\\&=& -75z(x+yz)^{-\frac{3}{2}} \\\\&=& -\dfrac{75z}{\left(\sqrt{x+yz}\right)^3}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{xz}(x,y,z) &=& 150\cdot \left(-\dfrac{1}{2}\right)(x+yz)^{-\frac{1}{2}-1}\cdot y \\\\&=& -75y(x+yz)^{-\frac{3}{2}} \\\\&=& -\dfrac{75y}{\left(\sqrt{x+yz}\right)^3} \end{array}

Vorgehen: Kettenregel
\begin{array}{rcl} f'_y(x,y,z) &=& 300\cdot\dfrac{1}{2}\left(x+yz\right)^{\frac{1}{2}-1}\cdot z \cr \cr &=& 150z\left(x+yz\right)^{-\frac{1}{2}} \cr \cr&=& \dfrac{150z}{\sqrt{x+yz}} \end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl} f'_{yx}(x,y,z) &=& 150z\cdot \left(-\dfrac{1}{2}\right)(x+yz)^{-\frac{1}{2}-1}\cdot 1 \\\\ &=& -75z{(x+yz)^{-\frac{3}{2}}}\\\\&=& -\dfrac{75z}{\left(\sqrt{x+yz}\right)^3}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{yy}(x,y,z) &=& 150z\cdot \left(-\dfrac{1}{2}\right)(x+yz)^{-\frac{1}{2}-1}\cdot z \\\\ &=& -75z^2{(x+yz)^{-\frac{3}{2}}}\\\\&=& -\dfrac{75z^2}{\left(\sqrt{x+yz}\right)^3}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{yz}(x,y,z) &=& 150\left(x+yz\right)^{-\frac{1}{2}}+150z\cdot \left(-\dfrac{1}{2}\right)(x+yz)^{-\frac{1}{2}-1}\cdot y \\\\ &=& 150\left(x+yz\right)^{-\frac{1}{2}}-75yz{(x+yz)^{-\frac{3}{2}}}\\\\&=& \dfrac{150}{\sqrt{x+yz}}-\dfrac{75yz}{\left(\sqrt{x+yz}\right)^3}\end{array}

Vorgehen: Kettenregel
\begin{array}{rcl}f'_z(x,y,z) &=& 300\cdot\dfrac{1}{2}\left(x+yz\right)^{\frac{1}{2}-1}\cdot y \cr \cr&=& 150y\left(x+yz\right)^{-\frac{1}{2}} \cr \cr&=& \dfrac{150y}{\sqrt{x+yz}} \end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl} f'_{zx}(x,y,z) &=& 150y\cdot \left(-\dfrac{1}{2}\right)(x+yz)^{-\frac{1}{2}-1}\cdot 1 \\\\ &=& -75y{(x+yz)^{-\frac{3}{2}}}\\\\&=& -\dfrac{75y}{\left(\sqrt{x+yz}\right)^3}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{zy}(x,y,z) &=& 150\left(x+yz\right)^{-\frac{1}{2}}+150y\cdot \left(-\dfrac{1}{2}\right)(x+yz)^{-\frac{1}{2}-1}\cdot z \\\\ &=& 150\left(x+yz\right)^{-\frac{1}{2}}-75yz{(x+yz)^{-\frac{3}{2}}}\\\\&=& \dfrac{150}{\sqrt{x+yz}}-\dfrac{75yz}{\left(\sqrt{x+yz}\right)^3}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{zz}(x,y,z) &=& 150y\cdot \left(-\dfrac{1}{2}\right)(x+yz)^{-\frac{1}{2}-1}\cdot y \\\\ &=& -75y^2{(x+yz)^{-\frac{3}{2}}}\\\\&=& -\dfrac{75y^2}{\left(\sqrt{x+yz}\right)^3}\end{array}

Vorgehen: Kettenregel

 

18)
\mathbb{D}=\mathbb{R}\times\mathbb{R}\times\mathbb{R}, x und y nicht gleichzeitig 0

Partielle Ableitungen 1. Ordnung

  Partielle Ableitungen 2. Ordnung
\begin{array}{rcl} f'_x(x,y,z) &=& \dfrac{-8z^2\cdot (17x^2+y^2) - \left(-8xz^2\right) \cdot 34x}{\left(17x^2+y^2\right)^2} \\\\ &=& \dfrac{-136x^2z^2-8y^2z^2+272x^2z^2}{\left(17x^2+y^2\right)^2} \\\\&=& \dfrac{136x^2z^2-8y^2z^2}{\left(17x^2+y^2\right)^2} \end{array}

Vorgehen: Quotientenregel		   \begin{array}{rcl} f'_{xx}(x,y,z) &=& \dfrac{272xz^2 \cdot\left(17x^2+y^2\right)^2-\left(136x^2z^2-8y^2z^2\right)\cdot 2(17x^2+y^2)\cdot 34x}{\left(\left(17x^2+y^2\right)^2\right)^2} \\\\&=& \dfrac{\left(17x^2+y^2\right)\left[272xz^2 \cdot\left(17x^2+y^2\right)-\left(136x^2z^2-8y^2z^2\right)\cdot 2\cdot 34x\right]}{\left(17x^2+y^2\right)^4} \\\\&=& \dfrac{272xz^2 \cdot\left(17x^2+y^2\right)-\left(136x^2z^2-8y^2z^2\right)\cdot 2\cdot 34x}{\left(17x^2+y^2\right)^3} \\\\&=& \dfrac{4.624x^3z^2+272xy^2z^2-9.248x^3z^2+544xy^2z^2}{\left(17x^2+y^2\right)^3} \\\\&=& \dfrac{-4.624x^3z^2+816xy^2z^2}{(17x^2+y^2)^3}\end{array}

Vorgehen: Quotientenregel, Kettenregel
    \begin{array}{rcl} f'_{xy}(x,y,z) &=& \dfrac{-16yz^2 \cdot\left(17x^2+y^2\right)^2-\left(136x^2z^2-8y^2z^2\right)\cdot 2(17x^2+y^2)\cdot 2y}{\left(\left(17x^2+y^2\right)^2\right)^2} \\\\&=& \dfrac{\left(17x^2+y^2\right)\left[-16yz^2 \cdot\left(17x^2+y^2\right)-\left(136x^2z^2-8y^2z^2\right)\cdot 2\cdot 2y\right]}{\left(17x^2+y^2\right)^4} \\\\&=& \dfrac{-16yz^2 \cdot\left(17x^2+y^2\right)-\left(136x^2z^2-8y^2z^2\right)\cdot 2\cdot 2y}{\left(17x^2+y^2\right)^3} \\\\&=& \dfrac{-272x^2yz^2-16y^3z^2-544x^2yz^2+32y^3z^2}{\left(17x^2+y^2\right)^3} \\\\&=& \dfrac{-816x^2yz^2+16y^3z^2}{(17x^2+y^2)^3}\end{array}

Vorgehen: Quotientenregel, Kettenregel
    \begin{array}{rcl} f'_{xz}(x,y,z) &=& \dfrac{272x^2z-16y^2z}{\left(17x^2+y^2\right)^2}\end{array}
\begin{array}{rcl} f'_y(x,y,z) &=& \dfrac{0 \cdot \left(17x^2 + y^2\right) - \left(-8xz^2\right) \cdot 2y}{\left(17x^2+y^2\right)^2} \cr \cr &=& \dfrac{16xyz^2}{\left(17x^2+y^2\right)^2}\end{array}

Vorgehen: Quotientenregel		   \begin{array}{rcl} f'_{yx}(x,y,z) &=& \dfrac{16yz^2 \cdot\left(17x^2+y^2\right)^2-16xyz^2\cdot 2(17x^2+y^2)\cdot 34x}{\left(\left(17x^2+y^2\right)^2\right)^2} \\\\&=& \dfrac{\left(17x^2+y^2\right)\left[16yz^2 \cdot\left(17x^2+y^2\right)-16xyz^2\cdot 2\cdot 34x\right]}{\left(17x^2+y^2\right)^4} \\\\&=& \dfrac{16yz^2 \cdot\left(17x^2+y^2\right)-16xyz^2\cdot 2\cdot 34x}{\left(17x^2+y^2\right)^3} \\\\&=& \dfrac{272x^2yz^2+16y^3z^2-1.088x^2yz^2}{\left(17x^2+y^2\right)^3} \\\\&=& \dfrac{-816x^2yz^2+16y^3z^2}{\left(17x^2+y^2\right)^3} \end{array}

Vorgehen: Quotientenregel, Kettenregel
    \begin{array}{rcl} f'_{yy}(x,y,z) &=& \dfrac{16xz^2 \cdot\left(17x^2+y^2\right)^2-16xyz^2\cdot 2(17x^2+y^2)\cdot 2y}{\left(\left(17x^2+y^2\right)^2\right)^2} \\\\&=& \dfrac{\left(17x^2+y^2\right)\left[16xz^2 \cdot\left(17x^2+y^2\right)-16xyz^2\cdot 2\cdot 2y\right]}{\left(17x^2+y^2\right)^4} \\\\&=& \dfrac{16xz^2 \cdot\left(17x^2+y^2\right)-16xyz^2\cdot 2\cdot 2y}{\left(17x^2+y^2\right)^3} \\\\&=& \dfrac{272x^3z^2+16xy^2z^2-64xy^2z^2}{\left(17x^2+y^2\right)^3} \\\\&=& \dfrac{272x^3z^2-48xy^2z^2}{\left(17x^2+y^2\right)^3}\end{array}

Vorgehen: Quotientenregel, Kettenregel
    \begin{array}{rcl} f'_{yz}(x,y,z) &=& \dfrac{32xyz}{\left(17x^2+y^2\right)^2}\end{array}
\begin{array}{rcl}f'_z(x,y,z) &=& \dfrac{-16xz}{17x^2+y^2} \end{array}   \begin{array}{rcl} f'_{zx}(x,y,z) &=& \dfrac{-16z \cdot\left(17x^2+y^2\right)-(-16xz)\cdot 34x}{\left(17x^2+y^2\right)^2} \\\\&=& \dfrac{-272x^2z-16y^2z+544x^2z}{\left(17x^2+y^2\right)^2} \\\\&=& \dfrac{272x^2z-16y^2z}{\left(17x^2+y^2\right)^2}\end{array}

Vorgehen: Quotientenregel
    \begin{array}{rcl} f'_{zy}(x,y,z) &=& \dfrac{0\cdot\left(17x^2+y^2\right)-(-16xz)\cdot 2y}{\left(17x^2+y^2\right)^2} \\\\&=& \dfrac{32xyz}{\left(17x^2+y^2\right)^2}\end{array}

Vorgehen: Quotientenregel
    \begin{array}{rcl} f'_{zz}(x,y,z) &=& -\dfrac{16x}{17x^2+y^2}\end{array}

 

19)
\begin{array}{rcl}\mathbb{D} &=& \mathbb{R}\times\mathbb{R}\times\mathbb{R} \\\\f(x,y,z) &=& \dfrac{\pi x\sin\left(7y^5\right)}{e^{10xz+1}} \\&=& \pi x\sin\left(7y^5\right)e^{-(10xz+1)} \\&=& \pi x\sin\left(7y^5\right)e^{-10xz-1}\end{array}

Partielle Ableitungen 1. Ordnung
  Partielle Ableitungen 2. Ordnung
\begin{array}{rcl} f'_x(x,y,z) &=& \dfrac{\pi\sin\left(7y^5\right) \cdot e^{10xz+1} - \pi x\sin\left(7y^5\right) \cdot e^{10xz+1}\cdot 10z}{\left(e^{10xz+1}\right)^2} \cr \cr &=& \dfrac{\left(\pi-10\pi xz\right) \sin\left(7y^5\right) e^{10xz+1}}{\left(e^{10xz+1}\right)^2} \cr \cr&=& \dfrac{\left(\pi-10\pi xz\right) \sin\left(7y^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Quotientenregel, Kettenregel		  

\begin{array}{rcl} f'_{xx}(x,y,z) &=& \dfrac{-10\pi z\sin\left(7y^5\right)\cdot e^{10xz+1} - \left(\pi-10 \pi xz\right)\sin\left(7y^5\right)\cdot e^{10xz+1}\cdot 10z}{\left(e^{10xz+1}\right)^2} \\\\ &=& \dfrac{\left[-10\pi z-\left(\pi-10 \pi xz\right)\cdot 10z\right]\sin\left(7y^5\right)\cdot e^{10xz+1}}{\left(e^{10xz+1}\right)^2} \\\\&=& \dfrac{\left[-10\pi z-10\pi z+100\pi xz^2\right]\sin\left(7y^5\right)}{e^{10xz+1}} \\\\&=& \dfrac{\left[-20\pi z+100\pi xz^2\right]\sin\left(7y^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Quotientenregel, Kettenregel
    \begin{array}{rcl} f'_{xy}(x,y,z) &=& \dfrac{\left(\pi-10\pi xz\right) \cos\left(7y^5\right)\cdot 35y^4}{e^{10xz+1}} \\\\&=& \dfrac{\left(35\pi y^4-350\pi xy^4z\right) \cos\left(7y^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{xz}(x,y,z) &=& \dfrac{-10\pi x\sin\left(7y^5\right)\cdot e^{10xz+1} - \left(\pi-10 \pi xz\right)\sin\left(7y^5\right)\cdot e^{10xz+1}\cdot 10x}{\left(e^{10xz+1}\right)^2} \\\\ &=& \dfrac{\left[-10\pi x-\left(\pi-10 \pi xz\right)\cdot 10x\right]\sin\left(7y^5\right)\cdot e^{10xz+1}}{\left(e^{10xz+1}\right)^2} \\\\&=& \dfrac{\left[-10\pi x-10\pi x+100\pi x^2z\right]\sin\left(7y^5\right)}{e^{10xz+1}} \\\\&=& \dfrac{\left[-20\pi x+100\pi x^2z\right]\sin\left(7y^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Quotientenregel, Kettenregel
\begin{array}{rcl} f'_y(x,y,z) &=& \dfrac{\pi x\cos\left(7y^5\right)\cdot 35y^4}{e^{10xz+1}} \cr \cr &=& \dfrac{35\pi xy^4\cos\left(7y^5\right)}{e^{10xz+1}} \cr\cr&=& 35\pi xy^4\cos\left(7y^5\right)e^{-10xz-1}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl} f'_{yx}(x,y,z) &=& \dfrac{35\pi y^4\cos\left(7y^5\right)\cdot e^{10xz+1} - 35\pi xy^4\cos\left(7y^5\right)\cdot e^{10xz+1}\cdot 10z}{\left(e^{10xz+1}\right)^2} \\\\&=& \dfrac{\left(35\pi y^4-35\pi xy^4\cdot 10z\right)\cos\left(7y^5\right)\cdot e^{10xz+1}}{\left(e^{10xz+1}\right)^2} \\\\&=& \dfrac{\left(35\pi y^4z-350\pi xy^4z\right)\cos\left(7y^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Quotientenregel, Kettenregel
    \begin{array}{rcl} f'_{yy}(x,y,z) &=& \dfrac{140\pi xy^3\cdot\cos\left(7y^5\right) + 35\pi xy^4\left(-\sin\left(7x^5\right)\right)\cdot 35y^4}{e^{10xz+1}} \\\\&=& \dfrac{140\pi xy^3\cos\left(7y^5\right)-1.225\pi xy^8\sin\left(7x^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Produktregel, Kettenregel
    \begin{array}{rcl} f'_{yz}(x,y,z) &=& 35\pi xy^4\cos\left(7y^5\right)e^{-10xz-1}\cdot (-10x) \\\\&=& -350\pi x^2y^4\cos\left(7y^5\right)e^{-10xz-1} \\\\&=& -\dfrac{350\pi x^2y^4\cos\left(7y^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Kettenregel
\begin{array}{rcl} f'_z(x,y,z) &=& \pi x\sin\left(7y^5\right)e^{-10xz-1}\cdot (-10x) \\\\&=& -10\pi x^2\sin\left(7y^5\right)e^{-10xz-1} \\\\&=& -\dfrac{10\pi x^2\sin\left(7y^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl} f'_{zx}(x,y,z) &=& -\dfrac{20\pi x\sin\left(7y^5\right)\cdot e^{10xz+1} - 10\pi x^2\sin\left(7y^5\right)e^{10xz+1}\cdot 10z}{\left(e^{10xz+1}\right)^2} \\\\&=& -\dfrac{\left(20\pi x-10\pi x^2\cdot 10z\right)\sin\left(7y^5\right)e^{10xz+1}}{\left(e^{10xz+1}\right)^2} \\\\&=& \dfrac{\left(100\pi x^2z-20\pi x\right)\sin\left(7y^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Quotientenregel, Kettenregel
    \begin{array}{rcl} f'_{zy}(x,y,z) &=& -\dfrac{10\pi x^2\cos\left(7y^5\right)\cdot 35y^4}{e^{10xz+1}} \\\\&=& -\dfrac{350\pi x^2y^4\cos\left(7y^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{zz}(x,y,z) &=& -10\pi x^2\sin\left(7y^5\right)e^{-10xz-1}\cdot (-10x) \\\\&=& 100\pi x^3\sin\left(7y^5\right)e^{-10xz-1} \\\\&=& \dfrac{100\pi x^3\sin\left(7y^5\right)}{e^{10xz+1}}\end{array}

Vorgehen: Kettenregel

 

20)
\begin{array}{rcl} \mathbb{D} &=& \mathbb{R}\times\mathbb{R}\times\mathbb{R} \text{ mit } 16a^2-b^2+25c^2>0 \\\\f(a,b,c) &=& \dfrac{16a^2-b^2+25c^2}{\sqrt{16a^2-b^2+25c^2}} \\\\ &=& \sqrt{16a^2-b^2+25c^2} \\\\ &=& \left(16a^2-b^2+25c^2\right)^{\frac{1}{2}}\end{array}

Partielle Ableitungen 1. Ordnung
  Partielle Ableitungen 2. Ordnung
\begin{array}{rcl} f'_a(a,b,c) &=& \dfrac{1}{2}\left(16a^2-b^2+25c^2\right)^{\frac{1}{2}-1} \cdot 32a \\\\&=& 16a\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}} \\\\&=& \dfrac{16a}{\sqrt{16a^2-b^2+25c^2}}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl} f'_{aa}(a,b,c) &=& 16\cdot\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}}+16a\cdot\left(-\dfrac{1}{2}\right)\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}-1}\cdot 32a \\\\&=& 16\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}}-256a^2\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \left[16\left(16a^2-b^2+25c^2\right)-256a^2\right] \left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \left[256a^2-16b^2+400c^2-256a^2\right]\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \left[-16b^2+400c^2\right]\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \dfrac{-16b^2+400c^2}{\left(16a^2-b^2+25c^2\right)^{\frac{3}{2}}} \\\\&=& \dfrac{-16b^2+400c^2}{\left(\sqrt{16a^2-b^2+25c^2}\right)^3}\end{array}

Vorgehen: Produktregel, Kettenregel

 

   \begin{array}{rcl} f'_{ab}(a,b,c) &=& 16a\cdot\left(-\dfrac{1}{2}\right)\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}-1}\cdot (-2b) \\\\&=& 16ab\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \dfrac{16ab}{\left(16a^2-b^2+25c^2\right)^{\frac{3}{2}}} \\\\&=& \dfrac{16ab}{\left(\sqrt{16a^2-b^2+25c^2}\right)^3}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{ac}(a,b,c) &=& 16a\cdot\left(-\dfrac{1}{2}\right)\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}-1}\cdot 50c \\\\&=& -400ac\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& -\dfrac{400ac}{\left(16a^2-b^2+25c^2\right)^{\frac{3}{2}}} \\\\&=& -\dfrac{400ac}{\left(\sqrt{16a^2-b^2+25c^2}\right)^3}\end{array}

Vorgehen: Kettenregel
\begin{array}{rcl} f'_b(a,b,c) &=& \dfrac{1}{2}\left(16a^2-b^2+25c^2\right)^{\frac{1}{2}-1} \cdot (-2b) \\\\&=& -b\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}} \\\\&=& -\dfrac{b}{\sqrt{16a^2-b^2+25c^2}}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl} f'_{ba}(a,b,c) &=& -b\cdot\left(-\dfrac{1}{2}\right)\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}-1}\cdot 32a \\\\ &=& 16ab\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \dfrac{16ab}{\left(16a^2-b^2+25c^2\right)^{\frac{3}{2}}} \\\\&=& \dfrac{16ab}{\left(\sqrt{16a^2-b^2+25c^2}\right)^3}\end{array}

Vorgehen: Kettenregel

 

   \begin{array}{rcll} f'_{bb}(a,b,c) &=& -1\cdot\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}}-b\cdot\left(-\dfrac{1}{2}\right)\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}-1}\cdot (-2b) \\\\&=& -\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}}-b^2\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \left[-\left(16a^2-b^2+25c^2\right)-b^2\right] \left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \left[-16a^2+b^2-25c^2-b^2\right] \left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \left[-16a^2-25c^2\right] \left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& -\dfrac{16a^2+25c^2}{\left(16a^2-b^2+25c^2\right)^{\frac{3}{2}}} \\\\&=& -\dfrac{16a^2+25c^2}{\left(\sqrt{16a^2-b^2+25c^2}\right)^3}\end{array}

Vorgehen: Produktregel, Kettenregel

 

   \begin{array}{rcl} f'_{bc}(a,b,c) &=& -b\cdot\left(-\dfrac{1}{2}\right)\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}-1}\cdot 50c \\\\ &=& 25bc\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \dfrac{25bc}{\left(16a^2-b^2+25c^2\right)^{\frac{3}{2}}} \\\\&=& \dfrac{25bc}{\left(\sqrt{16a^2-b^2+25c^2}\right)^3}\end{array}

Vorgehen: Kettenregel
\begin{array}{rcl} f'_c(a,b,c) &=& \dfrac{1}{2}\left(16a^2-b^2+25c^2\right)^{\frac{1}{2}-1} \cdot 50c \\\\&=& 25c\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}} \\\\&=& \dfrac{25c}{\sqrt{16a^2-b^2+25c^2}}\end{array}

Vorgehen: Kettenregel		   \begin{array}{rcl} f'_{ca}(a,b,c) &=& 25c\cdot\left(-\dfrac{1}{2}\right)\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}-1}\cdot 32a \\\\&=& -400ac\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}-1} \\\\&=& -\dfrac{400ac}{\left(16a^2-b^2+25c^2\right)^{\frac{3}{2}}} \\\\&=& -\dfrac{400ac}{\left(\sqrt{16a^2-b^2+25c^2}\right)^3}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{cb}(a,b,c) &=& 25c\cdot\left(-\dfrac{1}{2}\right)\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}-1}\cdot (-2b) \\\\&=& 25bc\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}-1} \\\\&=& \dfrac{25bc}{\left(16a^2-b^2+25c^2\right)^{\frac{3}{2}}} \\\\&=& \dfrac{25bc}{\left(\sqrt{16a^2-b^2+25c^2}\right)^3}\end{array}

Vorgehen: Kettenregel
    \begin{array}{rcl} f'_{cc}(a,b,c) &=& 25\cdot\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}}+25c\cdot\left(-\dfrac{1}{2}\right)\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}-1}\cdot 50c \\\\&=& 25\left(16a^2-b^2+25c^2\right)^{-\frac{1}{2}}-625c^2\left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \left[25\left(16a^2-b^2+25c^2\right)-625c^2\right] \left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \left[400a^2-25b^2+625c^2-625c^2\right] \left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \left[400a^2-25b^2\right] \left(16a^2-b^2+25c^2\right)^{-\frac{3}{2}} \\\\&=& \dfrac{400a^2-25b^2}{\left(16a^2-b^2+25c^2\right)^{\frac{3}{2}}} \\\\&=& \dfrac{400a^2-25b^2}{\left(\sqrt{16a^2-b^2+25c^2}\right)^3}\end{array}

Vorgehen: Produktregel, Kettenregel


Bemerkung: Wenn Sie die partiellen Ableitungen 2. Ordnung lieber mit der Quotientenregel (statt der Produktregel) berechnen möchte, können Sie das natürlich auch tun. Die Kettenregel braucht man in jedem Fall.