(\partial)/(\partial z)(zsin(xy^2+2z))

解

\frac{\partial}{\partial z} (z sin (x y^{2} + 2 z))

sin (x y^{2} + 2 z) + 2 z cos (x y^{2} + 2 z)

解答ステップ

\frac{\partial}{\partial z} (z sin (x y^{2} + 2 z))

x, y

を定数として扱う

積の法則を適用:

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

f = z, g = sin (x y^{2} + 2 z)

= \frac{\partial}{\partial z} (z) sin (x y^{2} + 2 z) + \frac{\partial}{\partial z} (sin (x y^{2} + 2 z)) z

\frac{\partial}{\partial z} (z) = 1

\frac{\partial}{\partial z} (sin (x y^{2} + 2 z)) = cos (x y^{2} + 2 z) \cdot 2

= 1 \cdot sin (x y^{2} + 2 z) + cos (x y^{2} + 2 z) \cdot 2 z

簡素化

= sin (x y^{2} + 2 z) + 2 z cos (x y^{2} + 2 z)