derivative of \sqrt[3]{x}ln(x)

解答

\frac{d}{d x} (3 x ln (x))

\frac{ln ( x )}{3 x ^{\frac{2}{3}}} + \frac{1}{x ^{\frac{2}{3}}}

求解步骤

\frac{d}{d x} (3 x ln (x))

使用微分乘法定则:

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

f = 3 x, g = ln (x)

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

\frac{d}{d x} (3 x) = \frac{1}{3 x ^{\frac{2}{3}}}

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

= \frac{1}{3 x ^{\frac{2}{3}}} ln (x) + \frac{1}{x} 3 x

化简

\frac{1}{3 x ^{\frac{2}{3}}} ln (x) + \frac{1}{x} 3 x : \frac{ln ( x )}{3 x ^{\frac{2}{3}}} + \frac{1}{x ^{\frac{2}{3}}}

= \frac{ln ( x )}{3 x ^{\frac{2}{3}}} + \frac{1}{x ^{\frac{2}{3}}}

\int \frac{2}{X} d X

f (x) = 5 e^{5 x}

\int \frac{x ^{3}}{x ^{3} - 1} d x

f (x) = \frac{3 x ^{2} - 9}{x}

\int 2 x y^{2} + 3 x^{2} d x