derivative of e^{{g}(x}sin(x))

解

\frac{d}{d x} (e^{g (x)} sin (x))

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

解答ステップ

\frac{d}{d x} (e^{g (x)} sin (x))

積の法則を適用:

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

f = e^{g (x)}, g = sin (x)

= \frac{d}{d x} (e^{g (x)}) sin (x) + \frac{d}{d x} (sin (x)) e^{g (x)}

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

\frac{d}{d x} (sin (x)) = cos (x)

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