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Step-by-step Solution

Derive the function $\ln\left(\sqrt{x}\sqrt{x-3}\right)$ with respect to x

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Answer

$\frac{1}{\sqrt{x}\sqrt{x-3}}\left(\frac{1}{2}x^{-\frac{1}{2}}\sqrt{x-3}+\frac{1}{2}\sqrt{x}\left(x-3\right)^{-\frac{1}{2}}\right)$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\ln\left(\sqrt{x}\sqrt{x-3}\right)\right)$
1

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

$\frac{1}{\sqrt{x}\sqrt{x-3}}\cdot\frac{d}{dx}\left(\sqrt{x}\sqrt{x-3}\right)$
2

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\sqrt{x}$ and $g=\sqrt{x-3}$

$\frac{1}{\sqrt{x}\sqrt{x-3}}\left(\sqrt{x-3}\cdot\frac{d}{dx}\left(\sqrt{x}\right)+\sqrt{x}\cdot\frac{d}{dx}\left(\sqrt{x-3}\right)\right)$

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Answer

$\frac{1}{\sqrt{x}\sqrt{x-3}}\left(\frac{1}{2}x^{-\frac{1}{2}}\sqrt{x-3}+\frac{1}{2}\sqrt{x}\left(x-3\right)^{-\frac{1}{2}}\right)$
$\frac{d}{dx}\left(\ln\left(\sqrt{x}\sqrt{x-3}\right)\right)$

Main topic:

Differential calculus

Used formulas:

6. See formulas

Time to solve it:

~ 0.6 seconds