Try now NerdPal! Our new app on iOS and Android
Calculators Topics Solving Methods Step Checker
Book solutions
Algebra Baldor
Go Premium
ENG • ESP
Topics
  1. calculators
  2. Precalculus

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!
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

Solved example of logarithmic differentiation

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

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

4

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

$\ln\left(y\right)=x\ln\left(x\right)$
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 $f=x$ and $g=\ln\left(x\right)$

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

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

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

Any expression multiplied by $1$ is equal to itself

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\ln\left(x\right)+x\frac{d}{dx}\left(\ln\left(x\right)\right)$
7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the fraction $\frac{x}{x}$ by $x$

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

Isolate $y'$

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

Divide fractions $\frac{\ln\left(x\right)+1}{\frac{1}{y}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

$y^{\prime}=y\left(\ln\left(x\right)+1\right)$
13

Isolate $y'$

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

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

$y^{\prime}=x^x\left(\ln\left(x\right)+1\right)$
15

The derivative of the function results in

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

Final Answer

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

Struggling with math?

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

Related Calculators

Synthetic division of polynomials Calculator
Polynomial long division Calculator
Properties of Logarithms Calculator

Popular problems

$5x+3y=-5$ 3 views
$6x=0$ 2 views
$-2x=-10$ 2 views
$4x=-20$ 2 views
$-2x^2\cdot y=-6$ 2 views
$7+x=-9$ 2 views
$2y\cdot x=2$ 2 views
$z^3=56$