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

解

\frac{d}{d x} ((cos (x))^{s i n (x)})

cos^{s i n (x)} (x) (cos (x) ln (cos (x)) - tan (x) sin (x))

解答ステップ

\frac{d}{d x} ((cos (x))^{s i n (x)})

指数の規則を適用する:

a^{b} = e^{b l n (a)}

(cos (x))^{s i n (x)} = e^{s i n (x) l n (c o s (x))}

= \frac{d}{d x} (e^{s i n (x) l n (c o s (x))})

連鎖法則を適用:

e^{s i n (x) l n (c o s (x))} \frac{d}{d x} (sin (x) ln (cos (x)))

= e^{s i n (x) l n (c o s (x))} \frac{d}{d x} (sin (x) ln (cos (x)))

\frac{d}{d x} (sin (x) ln (cos (x))) = cos (x) ln (cos (x)) - tan (x) sin (x)

= e^{s i n (x) l n (c o s (x))} (cos (x) ln (cos (x)) - tan (x) sin (x))

e^{s i n (x) l n (c o s (x))} = cos^{s i n (x)} (x)

= cos^{s i n (x)} (x) (cos (x) ln (cos (x)) - tan (x) sin (x))