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

解

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

- 2 x sin (x^{2} y) - 2 x^{3} y cos (x^{2} y)

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

積の法則を適用:

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

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

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

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

簡素化

- (2 x sin (x^{2} y) + cos (x^{2} y) \cdot 2 y x x^{2}) : - 2 x sin (x^{2} y) - 2 x^{3} y cos (x^{2} y)

= - 2 x sin (x^{2} y) - 2 x^{3} y cos (x^{2} y)