PRODUCT RULE OF DERIVATIVE

Product rule is one of the techniques in  derivative that is applied to differentiate product of two functions. 

Let U and V be the two functions given in the form UV.

Then, the product rule can be used to find the derivative of UV as shown below. 

(UV)'  =  UV' + VU'

Example : 1

Differentiate with respect to x :

(2x)(3x3 + 7)

Solution :

The given function is the product of two functions. So, we can use product rule to find the derivative. 

Product Rule : 

(UV)' =  UV' +VU'

Here, 

U  =  2x

U'  =  2

V  =  3x3 + 7

V'  =  3(3x2) + 0

V'  =  9x2

Derivative of the given function : 

=  (2x)(9x2) + (3x3 + 7)(2)

=  18x+ 6x3 + 14

=  24x3 + 14

Example 2 :

Differentiate with respect to x :

(x2 - 1)(x2 + 1)

Solution :

The given function is a rational function. So, we can use quotient rule to find the derivative. 

Product Rule : 

(UV)' =  UV' +VU'

Here, 

U  =  x2 - 1

U'  =  2x - 0

U'  =  2x

V  =  x2 + 1

V'  =  2x + 0

V'  =  2x

Derivative of the given function : 

=  (x2 - 1)(2x) + (x2 + 1)(2x)

=  2x3 - 2x - 2x3 + 2x

=  4x3

Example 3 :

Differentiate with respect to x :

x2ex

Solution :

The given function is a rational function. So, we can use quotient rule to find the derivative. 

Product Rule : 

(UV)' =  UV' +VU'

Here, 

U  =  x2

U'  =  2x

V  =  ex

V'  =  ex

Derivative of the given function : 

=  x2ex + ex(2x)

=  xex (x + 2)

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