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Exercise

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

Step-by-step Solution

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 natural 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 $g=\ln\left(x\right)$

Make diagrams like this with our diagram generator
$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(x\right)\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

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

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

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

Multiply the fraction by the term $x$

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

Multiply both sides of the equation by $y$

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

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

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

The derivative of the function results in

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

Final answer to the exercise

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

Try other ways to solve this exercise

  • Choose an option
  • Find the derivative using the definition
  • Find the derivative using the product rule
  • Find the derivative using the quotient rule
  • Find the derivative using logarithmic differentiation
  • Find the derivative
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
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