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

解答

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

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

求解步骤

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

使用指数法则:

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

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

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

使用连锁律:

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

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

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

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

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

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