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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

$x^x\left(\ln\left(x\right)+1\right)$
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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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a
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f
g
m
n
u
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w
x
y
z
.
(◻)
+
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×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Tips on how to improve your answer:

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

Main topic:

Logarithmic differentiation

Related Formulas:

3. See formulas

Time to solve it:

~ 0.31 s

Related topics:

Logarithmic differentiationProperties of LogarithmsProduct Rule of differentiationBasic Differentiation RulesFactorization

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