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To derive the function $x^x$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation
$y=x^x$
3
Apply natural logarithm to both sides of the equality
$\ln\left(y\right)=\ln\left(x^x\right)$
4
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
$\ln\left(y\right)=x\ln\left(x\right)$
5
Derive both sides of the equality with respect to $x$
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 $\displaystyle f'(x)=\frac{a'}{a}$
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 $\displaystyle f'(x)=\frac{a'}{a}$