(\partial)/(\partial y)(-2xysin(x^2y))

解

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

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

解答ステップ

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

x

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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