d/(dh)(({V}(h))((3h^2+4)^{3/2}-8))

解

\frac{d}{d h} ((V (h)) ((3 h^{2} + 4)^{\frac{3}{2}} - 8))

\frac{d}{d h} (V (h)) ((3 h^{2} + 4)^{\frac{3}{2}} - 8) + 9 h (3 h^{2} + 4)^{\frac{1}{2}} V (h)

解答ステップ

\frac{d}{d h} ((V (h)) ((3 h^{2} + 4)^{\frac{3}{2}} - 8))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = V (h), g = (3 h^{2} + 4)^{\frac{3}{2}} - 8

= \frac{d}{d h} (V (h)) ((3 h^{2} + 4)^{\frac{3}{2}} - 8) + \frac{d}{d h} ((3 h^{2} + 4)^{\frac{3}{2}} - 8) V (h)

\frac{d}{d h} ((3 h^{2} + 4)^{\frac{3}{2}} - 8) = 9 h (3 h^{2} + 4)^{\frac{1}{2}}

= \frac{d}{d h} (V (h)) ((3 h^{2} + 4)^{\frac{3}{2}} - 8) + 9 h (3 h^{2} + 4)^{\frac{1}{2}} V (h)