derivative of 2xe^{2y}-xcos(xy+2y)

解

\frac{d}{d x} (2 x e^{2 y} - x cos (x y) + 2 y)

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

解答ステップ

\frac{d}{d x} (2 x e^{2 y} - x cos (x y) + 2 y)

和/差の法則を適用:

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

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

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

\frac{d}{d x} (x cos (x y)) = cos (x y) - x sin (x y) (y + x \frac{d y}{d x})

\frac{d}{d x} (2 y) = 2 \frac{d y}{d x}

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

簡素化

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

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