derivative of x-sin^2(x-4xye^{xy^2})

解

\frac{d}{d x} (x - sin^{2} (x) - 4 x y e^{x y^{2}})

1 - sin (2 x) - 4 (e^{x y^{2}} y + x (e^{x y^{2}} \frac{d y}{d x} + e^{x y^{2}} (y^{2} + 2 x y \frac{d y}{d x}) y))

解答ステップ

\frac{d}{d x} (x - sin^{2} (x) - 4 x y e^{x y^{2}})

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d x}{d x} - \frac{d}{d x} (sin^{2} (x)) - \frac{d}{d x} (4 x y e^{x y^{2}})

\frac{d x}{d x} = 1

\frac{d}{d x} (sin^{2} (x)) = sin (2 x)

\frac{d}{d x} (4 x y e^{x y^{2}}) = 4 (e^{x y^{2}} y + x (e^{x y^{2}} \frac{d y}{d x} + e^{x y^{2}} (y^{2} + 2 x y \frac{d y}{d x}) y))

= 1 - sin (2 x) - 4 (e^{x y^{2}} y + x (e^{x y^{2}} \frac{d y}{d x} + e^{x y^{2}} (y^{2} + 2 x y \frac{d y}{d x}) y))