derivative of e^ycos(xy)

解

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

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

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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