(\partial)/(\partial x)(xy^2cos(2+x^2y))

解

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

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

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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