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

解答

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

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

求解步骤

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

使用微分乘法定则:

(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)

化简

\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)

\int 8 3 x + (x + 1) e^{x^{2} + 2 x} d x

\frac{d}{d x} (1 - x^{2} arccos (x))

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