👉 Try now NerdPal! Our new math app on iOS and Android

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

Step-by-step Solution

Go!
Math mode
Text mode
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
Dx
|◻|
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

Final Answer

$2x^{\left(\ln\left(x\right)-1\right)}\ln\left(x\right)$
Got another answer? Verify it here!

Step-by-step Solution

Specify the solving method

1

To derive the function $x^{\ln\left(x\right)}$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=x^{\ln\left(x\right)}$
2

Apply natural logarithm to both sides of the equality

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

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

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

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

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

When multiplying two powers that have the same base ($\ln\left(x\right)$), you can add the exponents

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(\ln\left(x\right)^2\right)$
6

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=2\frac{d}{dx}\left(\ln\left(x\right)\right)\ln\left(x\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=2\left(\frac{1}{x}\right)\frac{d}{dx}\left(x\right)\ln\left(x\right)$

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

$1\left(\frac{1}{y}\right)$

Any expression multiplied by $1$ is equal to itself

$\frac{1}{y}$
8

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

$\frac{y^{\prime}}{y}=2\left(\frac{1}{x}\right)\frac{d}{dx}\left(x\right)\ln\left(x\right)$

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

$1\left(\frac{1}{y}\right)$

Any expression multiplied by $1$ is equal to itself

$\frac{1}{y}$

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

$2\cdot 1\left(\frac{1}{x}\right)\ln\left(x\right)$

Any expression multiplied by $1$ is equal to itself

$2\left(\frac{1}{x}\right)\ln\left(x\right)$
9

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

$\frac{y^{\prime}}{y}=2\left(\frac{1}{x}\right)\ln\left(x\right)$
10

Multiply the fraction and term

$\frac{y^{\prime}}{y}=\frac{2\ln\left(x\right)}{x}$
11

Multiply both sides of the equation by $y$

$y^{\prime}=\frac{2y\ln\left(x\right)}{x}$
12

Substitute $y$ for the original function: $x^{\ln\left(x\right)}$

$y^{\prime}=\frac{2x^{\ln\left(x\right)}\ln\left(x\right)}{x}$
13

Simplify the fraction $\frac{2x^{\ln\left(x\right)}\ln\left(x\right)}{x}$ by $x$

$y^{\prime}=2x^{\left(\ln\left(x\right)-1\right)}\ln\left(x\right)$
14

The derivative of the function results in

$2x^{\left(\ln\left(x\right)-1\right)}\ln\left(x\right)$

Final Answer

$2x^{\left(\ln\left(x\right)-1\right)}\ln\left(x\right)$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of x^lnx using the product ruleFind derivative of x^lnx using the quotient ruleFind derivative of x^lnx using logarithmic differentiationFind derivative of x^lnx using the definition

Give us your feedback!

Function Plot

Plotting: $2x^{\left(\ln\left(x\right)-1\right)}\ln\left(x\right)$

SnapXam A2
Answer Assistant

beta
Got a different answer? Verify it!

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
Dx
|◻|
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

How to improve your answer:

Main Topic: Logarithmic Differentiation

The logarithmic derivative of a function f(x) is defined by the formula f'(x)/f(x).

Used Formulas

3. See formulas

Your Math & Physics Tutor. Powered by AI

Available 24/7, 365.

Complete step-by-step math solutions. No ads.

Includes multiple solving methods.

Support for more than 100 math topics.

Premium access on our iOS and Android apps as well.

20% discount on online tutoring.

Choose your Subscription plan:
Have a promo code?
Pay $39.97 USD securely with your payment method.
Please hold while your payment is being processed.
Create an Account