Step-by-step Solution

Derive the function $\ln\left(\sqrt{xe^{2x}}\right)$ with respect to x

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$\frac{1}{x}$

Step-by-step explanation

Problem to solve:

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

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

Applying the power of a power property

$\frac{1}{\sqrt{x}e^x}\cdot\frac{d}{dx}\left(\sqrt{x}e^x\right)$

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