** Final Answer

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** Step-by-step Solution **

Problem to solve:

** Specify the solving method

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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

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Apply natural logarithm to both sides of the equality

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Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

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Derive both sides of the equality with respect to $x$

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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 $1$

Any expression multiplied by $1$ is equal to itself

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The derivative of the linear function is equal to $1$

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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 $1$

Any expression multiplied by $1$ is equal to itself

The derivative of the linear function is equal to $1$

Any expression multiplied by $1$ is equal to itself

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The derivative of the linear function is equal to $1$

The derivative of the linear function is equal to $1$

Any expression multiplied by $1$ is equal to itself

The derivative of the linear function is equal to $1$

Any expression multiplied by $1$ is equal to itself

The derivative of the linear function is equal to $1$

Any expression multiplied by $1$ is equal to itself

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The derivative of the linear function is equal to $1$

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Multiply the fraction and term

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Simplify the fraction $\frac{x}{x}$ by $x$

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Multiply both sides of the equation by $y$

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Substitute $y$ for the original function: $x^x$

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The derivative of the function results in

** Final Answer

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