Calculators Topics Solving Methods Go Premium
ENG • ESP
Topics

Step-by-step Solution

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

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

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

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Solution

Formulas

Videos

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

Find the derivative

$\frac{d}{dx}\left(\cos\left(x\right)\right)\cdot \csc\left(x\right)$

Find the derivative

$\frac{d}{dx}\left(x\cdot \left(x^2+1\right)\right)$

Find the derivative

$\frac{d}{dx}\left(x\cdot \tan\left(y\right)\right)$

Find the derivative

$\frac{d}{dx}\left(\left(2-x\right)^3\left(1-x^2\right)^2\right)$

Find the derivative

$\frac{d}{dx}\left(4\sin\left(x\right)\cdot \cos\left(x\right)\right)$

Find the derivative

$\frac{d}{dx}\left(5\cdot e^{7x}\cdot \cos\left(9x\right)\right)$

Find the derivative

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

Main topic:

Logarithmic differentiation

Related formulas:

3. See formulas

Time to solve it:

~ 0.03 s (SnapXam)

Related topics:

Logarithmic differentiationProperties of logarithmsProduct rule of differentiationBasic Differentiation RulesFactorization

Supercharge your math learning

By signing up, you agree that you're 13 years old or older, and also agree to Snapxam's Terms of Service and Privacy Policy.

Already have an account? Login here
Forgot password?

Supercharge your math learning



Register here if you don't have an account.
Forgot password?