QUOTIENT RULE OF DERIVATIVE

Quotient rule is one of the techniques in  derivative that is applied to differentiate rational functions.

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

Then, the quotient rule can be used to find the derivative of U/V as shown below. 

Example : 1

Differentiate with respect to x :

2x / (3x³ + 7)

Solution :

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

Quotient Rule : 

(U/V)' =  [VU' - UV'] / V²

Here, 

Derivative of the given function : 

=  [(3x³ + 7)(2) - (2x)(9x²)] / (3x³ + 7)²

=  [6x³ + 14 - 18x³] / (3x³ + 7)²

=  [14 - 12x³] / (3x³ + 7)²

=  7(2 - 6x³) / (3x³ + 7)²

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. 

Quotient Rule : 

(U/V)' =  [VU' - UV'] / V²

Here, 

Derivative of the given function : 

=  [(x² + 1)(2x) - (x² - 1)(2x)] / (x² + 1)²

=  [(2x³ + 2x) - (2x³ - 2x)] / (x² + 1)²

=  [2x³ + 2x - 2x³ + 2x] / (x² + 1)²

=  4x / (x² + 1)²

Example 3 :

Differentiate with respect to x :

x² / e^x

Solution :

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

Quotient Rule : 

(U/V)' =  [VU' - UV'] / V²

Here, 

Derivative of the given function : 

=  [e^x(2x) - x²e^x] / (e^x)²

=  xe^x[2 - x] / (e^x)²

=  x(2 - x) / e^x

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