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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 $g=\ln\left(x\right)$

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

Subtract the values $\frac{1}{2}$ and $-1$

$x^{-\frac{1}{2}}$
3

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

$\frac{1}{2}x^{-\frac{1}{2}}\ln\left(x\right)+\sqrt{x}\cdot\frac{d}{dx}\left(\ln\left(x\right)\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

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

Any expression multiplied by $1$ is equal to itself

$\sqrt{x}$
5

Multiplying the fraction and term

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

Add the values $\frac{1}{2}$ and $-1$

$x^{-\frac{1}{2}}$
6

Simplify the fraction by $x$

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

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