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Product rule of differentiation Calculator

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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 $\displaystyle f'(x)=\frac{a'}{a}$

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

The derivative of the linear function is equal to $1$

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

Simplify the fraction by $x$

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

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

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

Combine $\frac{\frac{1}{2}\ln\left(x\right)}{\sqrt{x}}+x^{-\frac{1}{2}}$ in a single fraction

$\frac{\frac{1}{2}\ln\left(x\right)+x^{-\frac{1}{2}}\sqrt{x}}{\sqrt{x}}$
9

When multiplying exponents with same base we can add the exponents

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

Final Answer

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

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