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Step-by-step Solution

Find the derivative using the product rule $\frac{d}{dx}\left(\sqrt{x}\ln\left(x\right)\right)$

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Answer

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

Step-by-step explanation

Problem to solve:

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

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

$\frac{d}{dx}\left(\sqrt{x}\right)\ln\left(x\right)+\sqrt{x}\cdot\frac{d}{dx}\left(\ln\left(x\right)\right)$
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The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

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

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Answer

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

Main topic:

Product rule of differentiation

Used formulas:

2. See formulas

Time to solve it:

~ 0.78 seconds