1. calculators
  2. Product Rule Of Differentiation

Product Rule of differentiation Calculator

Get detailed solutions to your math problems with our Product Rule of differentiation 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
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

1

Solved example of product rule of differentiation

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=3x+2$ and $g=x^2-1$

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

The derivative of a sum of two or more functions is the sum of the derivatives of each function

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

The derivative of the constant function ($2$) is equal to zero

$\left(\frac{d}{dx}\left(3x\right)+0\right)\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2-1\right)$

$x+0=x$, where $x$ is any expression

$\frac{d}{dx}\left(3x\right)\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2-1\right)$
4

The derivative of the constant function ($2$) is equal to zero

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

The derivative of a function multiplied by a constant ($3$) is equal to the constant times the derivative of the function

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

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

$3\left(x^2-1\right)$
5

The derivative of the linear function times a constant, is equal to the constant

$3\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2-1\right)$
6

The derivative of a sum of two or more functions is the sum of the derivatives of each function

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

The derivative of the constant function ($2$) is equal to zero

$\left(\frac{d}{dx}\left(3x\right)+0\right)\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2-1\right)$

$x+0=x$, where $x$ is any expression

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

$3\left(x^2-1\right)+\left(3x+2\right)\left(\frac{d}{dx}\left(x^2\right)+0\right)$

$x+0=x$, where $x$ is any expression

$3\left(x^2-1\right)+\left(3x+2\right)\frac{d}{dx}\left(x^2\right)$
7

The derivative of the constant function ($-1$) is equal to zero

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

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

$3\left(x^2-1\right)+2x^{\left(2-1\right)}\left(3x+2\right)$

Subtract the values $2$ and $-1$

$3\left(x^2-1\right)+2x\left(3x+2\right)$
8

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

$3\left(x^2-1\right)+2x\left(3x+2\right)$

Multiply the single term $3$ by each term of the polynomial $\left(x^2-1\right)$

$3x^2-1\cdot 3+2x\left(3x+2\right)$

Multiply $-1$ times $3$

$3x^2-3+2x\left(3x+2\right)$

Solve the product $2x\left(3x+2\right)$

$3x^2-3+x\left(2\cdot 3x+2\cdot 2\right)$

Multiply the single term $x$ by each term of the polynomial $\left(2\cdot 3x+2\cdot 2\right)$

$3x^2-3+2\cdot 3x\cdot x+2\cdot 2x$

Multiply $2$ times $3$

$3x^2-3+6x\cdot x+2\cdot 2x$

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

$3x^2-3+6x^2+2\cdot 2x$

Multiply $2$ times $2$

$3x^2-3+6x^2+4x$

Combining like terms $3x^2$ and $6x^2$

$9x^2-3+4x$
9

Simplifying

$9x^2-3+4x$

Final Answer

$9x^2-3+4x$

Struggling with math?

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