## 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

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