Here, we show you a step-by-step solved example of logarithmic differentiation. This solution was automatically generated by our smart calculator:
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
Apply logarithm properties to both sides of the equality
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
Apply logarithm properties to both sides of the equality
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 $g=\ln\left(x\right)$
The derivative of the linear function is equal to $1$
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 $\displaystyle f'(x)=\frac{a'}{a}$
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 linear function is equal to $1$
The derivative of the linear function is equal to $1$
Multiply the fraction by the term $x$
Any expression multiplied by $1$ is equal to itself
Simplify the fraction $\frac{x}{x}$ by $x$
Multiply the fraction by the term $x$
Multiply both sides of the equation by $y$
Substitute $y$ for the original function: $x^x$
The derivative of the function results in
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