derivative f(x)=x^3arcsin(x^2e^x)

解

導関数

f (x) = x^{3} arcsin (x^{2} e^{x})

3 x^{2} arcsin (e^{x} x^{2}) + \frac{x ^{3} ( 2 e ^{x} x + e ^{x} x ^{2} )}{1 - e ^{2 x} x ^{4}}

解答ステップ

\frac{d}{d x} (x^{3} arcsin (x^{2} e^{x}))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{3}, g = arcsin (x^{2} e^{x})

= \frac{d}{d x} (x^{3}) arcsin (x^{2} e^{x}) + \frac{d}{d x} (arcsin (x^{2} e^{x})) x^{3}

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

\frac{d}{d x} (arcsin (x^{2} e^{x})) = \frac{2 e ^{x} x + e ^{x} x ^{2}}{1 - e ^{2 x} x ^{4}}

= 3 x^{2} arcsin (x^{2} e^{x}) + \frac{2 e ^{x} x + e ^{x} x ^{2}}{1 - e ^{2 x} x ^{4}} x^{3}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 3 x^{2} arcsin (e^{x} x^{2}) + \frac{x ^{3} ( 2 e ^{x} x + e ^{x} x ^{2} )}{1 - e ^{2 x} x ^{4}}