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)
= 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,
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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