derivative of (sin^2(x+1)^x)

解

\frac{d}{d x} ((sin^{2} (x) + 1)^{x})

(ln (sin^{2} (x) + 1) + \frac{x sin ( 2 x )}{sin ^{2} ( x ) + 1}) (sin^{2} (x) + 1)^{x}

解答ステップ

\frac{d}{d x} ((sin^{2} (x) + 1)^{x})

指数の規則を適用する:

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

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

= \frac{d}{d x} (e^{x l n (s i n^{2} (x) + 1)})

連鎖法則を適用:

e^{x l n (s i n^{2} (x) + 1)} \frac{d}{d x} (x ln (sin^{2} (x) + 1))

= e^{x l n (s i n^{2} (x) + 1)} \frac{d}{d x} (x ln (sin^{2} (x) + 1))

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

= e^{x l n (s i n^{2} (x) + 1)} (ln (sin^{2} (x) + 1) + \frac{x sin ( 2 x )}{sin ^{2} ( x ) + 1})

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

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