# Product rule of differentiation Calculator

## Get detailed solutions to your math problems with our Product rule of differentiation step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here!

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### Difficult Problems

1

Solved example of product rule of differentiation

$\frac{d}{dx}\left(\sqrt{x}\ln\left(x\right)\right)$
2

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\sqrt{x}$ and $g=\ln\left(x\right)$

$\frac{d}{dx}\left(\sqrt{x}\right)\ln\left(x\right)+\sqrt{x}\frac{d}{dx}\left(\ln\left(x\right)\right)$
3

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)+\sqrt{x}\frac{d}{dx}\left(\ln\left(x\right)\right)$
4

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)+\frac{\sqrt{x}}{x}$
5

Simplify the fraction by $x$

$\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)+x^{-\frac{1}{2}}$
6

Factoring by $x^{-\frac{1}{2}}$

$x^{-\frac{1}{2}}\left(1+\frac{1}{2}\ln\left(x\right)\right)$
7

Using the power rule of logarithms

$x^{-\frac{1}{2}}\left(1+\ln\left(\sqrt{x}\right)\right)$
8

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{1+\ln\left(\sqrt{x}\right)}{\sqrt{x}}$

$\frac{1+\ln\left(\sqrt{x}\right)}{\sqrt{x}}$