(\partial)/(\partial x)(cos(ye^{xy})sin(x))

解

\frac{\partial}{\partial x} (cos (y e^{x y}) sin (x))

- e^{y x} y^{2} sin (e^{y x} y) sin (x) + cos (e^{y x} y) cos (x)

解答ステップ

\frac{\partial}{\partial x} (cos (y e^{x y}) sin (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 = cos (y e^{x y}), g = sin (x)

= \frac{\partial}{\partial x} (cos (y e^{x y})) sin (x) + \frac{\partial}{\partial x} (sin (x)) cos (y e^{x y})

\frac{\partial}{\partial x} (cos (y e^{x y})) = - sin (y e^{x y}) y^{2} e^{y x}

\frac{\partial}{\partial x} (sin (x)) = cos (x)

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

簡素化

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