derivative of ln(x^2(3x^2+6))

解

\frac{d}{d x} (ln (x^{2}) (3 x^{2} + 6))

\frac{2 ( 3 x ^{2} + 6 )}{x} + 12 x ln (x)

解答ステップ

\frac{d}{d x} (ln (x^{2}) (3 x^{2} + 6))

積の法則を適用:

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

f = ln (x^{2}), g = 3 x^{2} + 6

= \frac{d}{d x} (ln (x^{2})) (3 x^{2} + 6) + \frac{d}{d x} (3 x^{2} + 6) ln (x^{2})

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

\frac{d}{d x} (3 x^{2} + 6) = 6 x

= \frac{2}{x} (3 x^{2} + 6) + 6 x ln (x^{2})

簡素化

\frac{2}{x} (3 x^{2} + 6) + 6 x ln (x^{2}) : \frac{2 ( 3 x ^{2} + 6 )}{x} + 12 x ln (x)

= \frac{2 ( 3 x ^{2} + 6 )}{x} + 12 x ln (x)