derivative (sin(x))^x

解

導関数

(sin (x))^{x}

sin^{x} (x) (ln (sin (x)) + x cot (x))

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

= sin^{x} (x) (ln (sin (x)) + x cot (x))