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, 

Derivative of the given function : 

=  (2x)(9x²) + (3x³ + 7)(2)

=  18x³ + 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, 

Derivative of the given function : 

=  (x² - 1)(2x) + (x² + 1)(2x)

=  2x³ - 2x - 2x³ + 2x

=  4x³

Example 3 :

Differentiate with respect to x :

x²e^x

Solution :

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

Product Rule : 

(UV)' =  UV' +VU'

Here, 

Derivative of the given function : 

=  x²e^x + e^x(2x)

=  xe^x (x + 2)

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