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Popular >

derivative of y=ln(x)sqrt(x)

Solution

derivative of

y = ln (x) x

Solution

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

Solution steps

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

Apply the Product Rule:

(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)

Simplify

\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}

Popular Examples

integral from 0 to 2pi of sin(x)+cos(x)

\int_{0}^{2 π} sin (x) + cos (x) d x

limit as x approaches 5 of 5/((x-5)^2)

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

integral of (x+3)/(x^4+9x^2)

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

derivative of sin(xln(6x))

\frac{d}{d x} (sin (x) ln (6 x))

limit as x approaches 0+of x^7ln(x)

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

Frequently Asked Questions (FAQ)

What is the derivative of y=ln(x)sqrt(x) ?
The derivative of y=ln(x)sqrt(x) is 1/(sqrt(x))+(ln(x))/(2sqrt(x))
What is the first derivative of y=ln(x)sqrt(x) ?
The first derivative of y=ln(x)sqrt(x) is 1/(sqrt(x))+(ln(x))/(2sqrt(x))