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

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Basic Derivatives

· Derivative of the natural logarithm

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

$\frac{d}{dx}\left(\ln\left(x\right)\right)=\frac{1}{x}\cdot\frac{d}{dx}\left(x\right)$
· Product rule for derivatives

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=[a]$ and $g=[b]$

$\frac{d}{dx}\left(ab\right)=b\frac{d}{dx}\left(a\right)+a\frac{d}{dx}\left(b\right)$
· Power rule for derivatives

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

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

Main topic:

Differential calculus

Related formulas:

3. See formulas

Time to solve it:

~ 0.09 seconds