derivative of tan^x(x)

解

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

tan^{x} (x) (ln (tan (x)) + 2 x csc (2 x))

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

= e^{x l n (t a n (x))} (ln (tan (x)) + 2 x csc (2 x))

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

= tan^{x} (x) (ln (tan (x)) + 2 x csc (2 x))