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

解

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

tan^{c o s (x)} (x) (- sin (x) ln (tan (x)) + 2 csc (2 x) cos (x))

解答ステップ

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

指数の規則を適用する:

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

(tan (x))^{c o s (x)} = e^{c o s (x) l n (t a n (x))}

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

連鎖法則を適用:

e^{c o s (x) l n (t a n (x))} \frac{d}{d x} (cos (x) ln (tan (x)))

= e^{c o s (x) l n (t a n (x))} \frac{d}{d x} (cos (x) ln (tan (x)))

\frac{d}{d x} (cos (x) ln (tan (x))) = - sin (x) ln (tan (x)) + 2 csc (2 x) cos (x)

= e^{c o s (x) l n (t a n (x))} (- sin (x) ln (tan (x)) + 2 csc (2 x) cos (x))

e^{c o s (x) l n (t a n (x))} = tan^{c o s (x)} (x)

= tan^{c o s (x)} (x) (- sin (x) ln (tan (x)) + 2 csc (2 x) cos (x))