derivative of tan^{x^2}(x)

解

\frac{d}{d x} (tan^{x^{2}} (x))

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

解答ステップ

\frac{d}{d x} (tan^{x^{2}} (x))

指数の規則を適用する:

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

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

= \frac{d}{d x} (e^{x^{2} l n (t a n (x))})

連鎖法則を適用:

e^{x^{2} l n (t a n (x))} \frac{d}{d x} (x^{2} ln (tan (x)))

= e^{x^{2} l n (t a n (x))} \frac{d}{d x} (x^{2} ln (tan (x)))

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

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

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

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