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

解

\frac{d}{d x} ((sin (x))^{t a n (x)})

sin^{t a n (x)} (x) (1 + sec^{2} (x) ln (sin (x)))

解答ステップ

\frac{d}{d x} ((sin (x))^{t a n (x)})

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

\frac{d}{d x} (tan (x) ln (sin (x))) = 1 + sec^{2} (x) ln (sin (x))

= e^{t a n (x) l n (s i n (x))} (1 + sec^{2} (x) ln (sin (x)))

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

= sin^{t a n (x)} (x) (1 + sec^{2} (x) ln (sin (x)))