derivative of (x+1^{x^2+x^3})

解

\frac{d}{d x} ((x + 1)^{x^{2} + x^{3}})

(ln (x + 1) (2 x + 3 x^{2}) + x^{2}) (x + 1)^{x^{2} + x^{3}}

解答ステップ

\frac{d}{d x} ((x + 1)^{x^{2} + x^{3}})

指数の規則を適用する:

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

(x + 1)^{x^{2} + x^{3}} = e^{(x^{2} + x^{3}) l n (x + 1)}

= \frac{d}{d x} (e^{(x^{2} + x^{3}) l n (x + 1)})

連鎖法則を適用:

e^{(x^{2} + x^{3}) l n (x + 1)} \frac{d}{d x} ((x^{2} + x^{3}) ln (x + 1))

= e^{(x^{2} + x^{3}) l n (x + 1)} \frac{d}{d x} ((x^{2} + x^{3}) ln (x + 1))

\frac{d}{d x} ((x^{2} + x^{3}) ln (x + 1)) = ln (x + 1) (2 x + 3 x^{2}) + x^{2}

= e^{(x^{2} + x^{3}) l n (x + 1)} (ln (x + 1) (2 x + 3 x^{2}) + x^{2})

e^{(x^{2} + x^{3}) l n (x + 1)} = (x + 1)^{x^{2} + x^{3}}

= (ln (x + 1) (2 x + 3 x^{2}) + x^{2}) (x + 1)^{x^{2} + x^{3}}