derivative of (sin(5x)^{ln(x)})

解

\frac{d}{d x} ((sin (5 x))^{l n (x)})

x^{l n (s i n (5 x))} (\frac{ln ( sin ( 5 x ) )}{x} + 5 cot (5 x) ln (x))

解答ステップ

\frac{d}{d x} ((sin (5 x))^{l n (x)})

指数の規則を適用する:

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

(sin (5 x))^{l n (x)} = e^{l n (x) l n (s i n (5 x))}

= \frac{d}{d x} (e^{l n (x) l n (s i n (5 x))})

連鎖法則を適用:

e^{l n (x) l n (s i n (5 x))} \frac{d}{d x} (ln (x) ln (sin (5 x)))

= e^{l n (x) l n (s i n (5 x))} \frac{d}{d x} (ln (x) ln (sin (5 x)))

\frac{d}{d x} (ln (x) ln (sin (5 x))) = \frac{ln ( sin ( 5 x ) )}{x} + 5 cot (5 x) ln (x)

= e^{l n (x) l n (s i n (5 x))} (\frac{ln ( sin ( 5 x ) )}{x} + 5 cot (5 x) ln (x))

e^{l n (x) l n (s i n (5 x))} = x^{l n (s i n (5 x))}

= x^{l n (s i n (5 x))} (\frac{ln ( sin ( 5 x ) )}{x} + 5 cot (5 x) ln (x))