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Step-by-step Solution

Find the derivative using logarithmic differentiation method $\frac{d}{dx}\left(x^x\right)$

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Solution

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Final Answer

$x^x\left(\ln\left(x\right)+1\right)$

Step-by-step Solution

Problem to solve:

$\frac{d}{dx}\left(x^x\right)$

Solving method

1

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

$y=x^x$
2

Apply logarithm to both sides of the equality

$\ln\left(y\right)=\ln\left(x^x\right)$
3

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

$\ln\left(y\right)=x\ln\left(x\right)$
4

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

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(x\ln\left(x\right)\right)$
5

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=\ln\left(x\right)$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(x\right)\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=1\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$

Any expression multiplied by $1$ is equal to itself

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$
6

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

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$
7

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=1\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$

Any expression multiplied by $1$ is equal to itself

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$

$1y^{\prime}\left(\frac{1}{y}\right)=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$

Any expression multiplied by $1$ is equal to itself

$y^{\prime}\frac{1}{y}=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$
8

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

$y^{\prime}\frac{1}{y}=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$
9

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 $\displaystyle f'(x)=\frac{a'}{a}$

$y^{\prime}\frac{1}{y}=\ln\left(x\right)+x\frac{1}{x}\frac{d}{dx}\left(x\right)$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=1\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$

Any expression multiplied by $1$ is equal to itself

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$

$1y^{\prime}\left(\frac{1}{y}\right)=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$

Any expression multiplied by $1$ is equal to itself

$y^{\prime}\frac{1}{y}=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$

$y^{\prime}\frac{1}{y}=\ln\left(x\right)+1x\frac{1}{x}$

Any expression multiplied by $1$ is equal to itself

$y^{\prime}\frac{1}{y}=\ln\left(x\right)+\frac{x}{x}$
10

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

$y^{\prime}\frac{1}{y}=\ln\left(x\right)+\frac{x}{x}$
11

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

$y^{\prime}\frac{1}{y}=\ln\left(x\right)+1$

$y^{\prime}=\frac{\ln\left(x\right)+1}{\frac{1}{y}}$

Divide fractions $\frac{\ln\left(x\right)+1}{\frac{1}{y}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

$y^{\prime}=y\left(\ln\left(x\right)+1\right)$
12

Isolate $y'$

$y^{\prime}=y\left(\ln\left(x\right)+1\right)$
13

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

$y^{\prime}=x^x\left(\ln\left(x\right)+1\right)$
14

The derivative of the function results in

$x^x\left(\ln\left(x\right)+1\right)$

Final Answer

$x^x\left(\ln\left(x\right)+1\right)$

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Main topic:

Logarithmic differentiation

Related Formulas:

3. See formulas

Time to solve it:

~ 0.03 s

Related topics:

Logarithmic differentiationProperties of LogarithmsProduct rule of differentiationBasic Differentiation RulesFactorization

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