(\partial)/(\partial y)(e^ysin(xy))

解

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

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

解答ステップ

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

x

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 = e^{y}, g = sin (x y)

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

\frac{\partial}{\partial y} (e^{y}) = e^{y}

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

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