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

 Calculators  Topics Step Checker  Book a tutor Go Premium  Topics

 Tap to take a pic of the problem

Precalculus Calculator

Get detailed solutions to your math problems with our Precalculus step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

Go!

Symbolic 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

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

 Example

 Solved Problems

 Difficult Problems

1

Here, we show you a step-by-step solved example of logarithmic differentiation. This solution was automatically generated by our smart calculator:

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

2

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

$y=x^x$

3

Apply natural logarithm to both sides of the equality

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

Apply logarithm properties to both sides of the equality

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

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

Apply logarithm properties to both sides of the equality

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

5

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

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

6

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=

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

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

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

Any expression multiplied by $1$ is equal to itself

$\ln\left(x\right)$

7

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

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

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 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 $\displaystyle f'(x)=\frac{a'}{a}$

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

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

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

Any expression multiplied by $1$ is equal to itself

$\ln\left(x\right)$

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

$\frac{1}{y}$

9

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

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

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

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

Any expression multiplied by $1$ is equal to itself

$\ln\left(x\right)$

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 $1$

$1x\frac{1}{x}$

Any expression multiplied by $1$ is equal to itself

$x\frac{1}{x}$

10

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

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

11

Multiply the fraction and term

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

12

Simplify the fraction

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

13

Multiply both sides of the equation by $y$

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

14

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

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

15

The derivative of the function results in

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

 Final answer to the problem

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

Are you struggling with math?

Access detailed step by step solutions to thousands of problems, growing every day!

 Popular problems

Most popular problems solved with this calculator: