derivative y=ln(x)sqrt(x)

解答

导数

y = ln (x) x

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

求解步骤

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

使用微分乘法定则:

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

f = ln (x), g = x

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

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

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

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

化简

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

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

x \to 5 lim (\frac{5}{( x - 5 ) ^{2}})

\int \frac{x + 3}{x ^{4} + 9 x ^{2}} d x

x \to 0 + lim (x^{7} ln (x))

x \to 0 + lim (x ln (x^{7}))

\int x x - 7 d x