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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 explanation

Problem to solve:

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

Choose the 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 $g=\ln\left(x\right)$

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

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

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

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

Divide both sides of the equation by $\frac{1}{y}$

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

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

Similar Problems

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

Main topic:

Logarithmic differentiation

Related formulas:

3. See formulas

Time to solve it:

~ 0.16 s (SnapXam)

Related topics:

Logarithmic differentiationProperties of logarithmsProduct rule of differentiationBasic Differentiation RulesFactorization

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