QUOTIENT RULE

About "Quotient rule"

Quotient rule is one of the techniques in  derivative that we apply when we have rational functions. 

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

Then, the quotient rule to find the derivative of U/V is given below. 

Quotient rule - Examples

Example 1

Differentiate (x² - 1)/ (x² + 1) with respect to x

Solution:

In this problem we have two function which are dividing.So we have to use the quotient rule

(U/V)' =  [VU' - UV'] /V²

let y = (x² - 1)/ (x² + 1)

 u = x² - 1                  v = x² + 1

 u' = 2x - 0                 v' = 2x + 0

 u' = 2x                      v' = 2x

        

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

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

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

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

                   = 4x /(x² + 1)²



Example 2

Differentiate (x² - sin x)/ (cos x + log x) with respect to x

Solution:

In this problem we have two function which are dividing.So we have to use the quotient rule

(U/V)' =  [VU' - UV'] /V²

let y = (x² - sin x)/ (cos x + log x)

 u = x² - sin x                 v = cos x + log x

 u' = 2x - cos x               v' = -sin x + (1/x)

                                         v' = (-x sin x + 1)/x

y'= [(cos x+log x)(2x-cos x)-(x²-sin x)((-x sin x + 1)/x)]/(cos x + log x)²


Example 3

Differentiate (sin x + x cos x )/ (x sin x - cos x)  with respect to x

Solution:

In this problem we have two functions are dividing so we have to use the quotient rule to solve this problem.

(U/V)' =  [VU' - UV'] /V²

Let y  =  (sin x + x cos x )/ (x sin x - cos x)

 u = sin x + x cos x                                        

 u' = cos x + x(-sin x) + cos x (1)

 u' = cos x - x sin x + cos x   

 u' = 2 cos x - x sin x

 v = x sin x - cos x

 v' = x (cos x) + sin x (1) - (-sin x)

 v' = x cos x + sin x + sin x

 v' = x cos x + 2 sin x

so y' = [(x sin x - cos x)(2 cos x - x sin x)]-[(sin x + x cos x)(x cos x + 2             sin x)]/(x sin x - cos x)²

      = [(2x sin x cos x -2 cos²x - (x sin x)² + x sin x cos x )]-[(x sin x cos           x + (x cos x)² + 2 sin² x + 2x sin x cos x )]/(x sin x - cos x) ²

      = [x sin 2x -2 cos²x - x² sin² x + x sin x cos x -x sin x cos x - x²                cos² x - 2 sin² x - x sin 2x]/(x sin x - cos x) ²

      = [- x²(sin² x +  cos² x) - 2 (sin² x + cos²x)]/(x sin x - cos x)²

      = [- x²(1) - 2 (1)]/(x sin x - cos x)²

      = [-x² - 2]/(x sin x - cos x)²

      = -(x²+2)/(x sin x - cos x)²

After having gone through the stuff given above, we hope that the students would have understood the stuff given above. 

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