derivative of (1+sqrt(x)(x^3))

解

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

\frac{x ^{\frac{5}{2}}}{2} + 3 x^{2} (1 + x)

解答ステップ

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

積の法則を適用:

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

f = 1 + x, g = x^{3}

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

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

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

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

\frac{1}{2 x} x^{3} = \frac{x ^{\frac{5}{2}}}{2}

= \frac{x ^{\frac{5}{2}}}{2} + 3 x^{2} (1 + x)