derivative of-e^{cos(x}sin(x))

解

\frac{d}{d x} (- e^{c o s (x)} sin (x))

e^{c o s (x)} sin^{2} (x) - e^{c o s (x)} cos (x)

解答ステップ

\frac{d}{d x} (- e^{c o s (x)} sin (x))

定数を除去:

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

= - \frac{d}{d x} (e^{c o s (x)} sin (x))

積の法則を適用:

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

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

= - (\frac{d}{d x} (e^{c o s (x)}) sin (x) + \frac{d}{d x} (sin (x)) e^{c o s (x)})

\frac{d}{d x} (e^{c o s (x)}) = - e^{c o s (x)} sin (x)

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

= - ((- e^{c o s (x)} sin (x)) sin (x) + cos (x) e^{c o s (x)})

簡素化

- ((- e^{c o s (x)} sin (x)) sin (x) + cos (x) e^{c o s (x)}) : e^{c o s (x)} sin^{2} (x) - e^{c o s (x)} cos (x)

= e^{c o s (x)} sin^{2} (x) - e^{c o s (x)} cos (x)