Final Answer
Step-by-step explanation
Problem to solve:
Choose the solving method
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 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 $g=\ln\left(x\right)$
Any expression multiplied by $1$ is equal to itself
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 $\displaystyle f'(x)=\frac{a'}{a}$
Any expression multiplied by $1$ is equal to itself
Any expression multiplied by $1$ is equal to itself
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 $\displaystyle f'(x)=\frac{a'}{a}$
Any expression multiplied by $1$ is equal to itself
Any expression multiplied by $1$ is equal to itself
Any expression multiplied by $1$ is equal to itself
The derivative of the linear function is equal to $1$
Simplify the fraction $\frac{x}{x}$ by $x$
Divide fractions $\frac{\ln\left(x\right)+1}{\frac{1}{y}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Divide both sides of the equation by $\frac{1}{y}$
Substitute $y$ for the original function: $x^x$
The derivative of the function results in