(\partial)/(\partial x)(-3x^2y^3sin(x^3y^3))

解

\frac{\partial}{\partial x} (- 3 x^{2} y^{3} sin (x^{3} y^{3}))

- 3 y^{3} (2 x sin (x^{3} y^{3}) + 3 x^{4} y^{3} cos (x^{3} y^{3}))

解答ステップ

\frac{\partial}{\partial x} (- 3 x^{2} y^{3} sin (x^{3} y^{3}))

y

を定数として扱う

定数を除去:

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

= - 3 y^{3} \frac{\partial}{\partial x} (x^{2} sin (x^{3} y^{3}))

積の法則を適用:

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

f = x^{2}, g = sin (x^{3} y^{3})

= - 3 y^{3} (\frac{\partial}{\partial x} (x^{2}) sin (x^{3} y^{3}) + \frac{\partial}{\partial x} (sin (x^{3} y^{3})) x^{2})

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

\frac{\partial}{\partial x} (sin (x^{3} y^{3})) = cos (x^{3} y^{3}) \cdot 3 y^{3} x^{2}

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

cos (x^{3} y^{3}) \cdot 3 y^{3} x^{2} x^{2} = 3 x^{4} y^{3} cos (x^{3} y^{3})

= - 3 y^{3} (2 x sin (x^{3} y^{3}) + 3 x^{4} y^{3} cos (x^{3} y^{3}))