QUOTIENT RULE OF DERIVATIVE
About "Quotient rule of derivative"
Quotient rule of derivative :
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 of derivative
Quotient rule of derivative - 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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