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

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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