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

解

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

sin^{c o s (x)} (x) (- sin (x) ln (sin (x)) + cot (x) cos (x))

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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