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Solved example of logarithmic differentiation
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
Apply natural logarithm to both sides of the equality
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
Derive both sides of the equality with respect to $x$
Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=\ln\left(x\right)$
Intermediate steps
The derivative of the linear function is equal to $1$
Any expression multiplied by $1$ is equal to itself
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 $\displaystyle f'(x)=\frac{a'}{a}$
Simplify the fraction $\frac{x}{x}$ by $x$
Isolate $y'$
Divide fractions $\frac{\ln\left(x\right)+1}{\frac{1}{y}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Substitute $y$ for the original function: $x^x$
The derivative of the function results in
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