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

解

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

2 ln (1 + x) (1 + x) + 1 + x

解答ステップ

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

積の法則を適用:

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

f = (1 + x)^{2}, g = ln (1 + x)

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

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

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

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

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

= 2 ln (1 + x) (1 + x) + 1 + x