TeX source:
\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}