# Step-by-step Solution

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### Videos

$\ln\left(x^{\left(x^2\right)}\right)=\ln\left(\ln\left(\ln\left(\ln\left(\ln\left(\ln\left(\ln\left(\ln\left(\ln\left(x+2x\ln\left(x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)$

## Step-by-step explanation

Problem to solve:

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\ln\left(x\right)$ and $g=x^2$

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

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

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

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

Product rule

~ 0.81 seconds

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