derivative of (ln(x)(arctan(x)))

解答

\frac{d}{d x} ((ln (x)) (arctan (x)))

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

求解步骤

\frac{d}{d x} ((ln (x)) (arctan (x)))

使用微分乘法定则:

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

f = ln (x), g = arctan (x)

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

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

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

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

化简

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

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

n = 1 \sum \infty \frac{2 ^{n}}{n 3 ^{n}}

\int \frac{24}{1 + 16 x ^{2}} d x

n = 0 \sum \infty (\frac{1}{2})^{n} x^{n}

\int \frac{2}{1 - 2 x} d x

\int \frac{1}{( y + 1 )} d y