derivative of ln(sqrt(x)(x^2-5))

解

\frac{d}{d x} (ln (x) (x^{2} - 5))

\frac{x ^{2} - 5}{2 x} + x ln (x)

解答ステップ

\frac{d}{d x} (ln (x) (x^{2} - 5))

積の法則を適用:

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

f = ln (x), g = x^{2} - 5

= \frac{d}{d x} (ln (x)) (x^{2} - 5) + \frac{d}{d x} (x^{2} - 5) ln (x)

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

\frac{d}{d x} (x^{2} - 5) = 2 x

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

簡素化

\frac{1}{2 x} (x^{2} - 5) + 2 x ln (x) : \frac{x ^{2} - 5}{2 x} + x ln (x)

= \frac{x ^{2} - 5}{2 x} + x ln (x)