The quotient rule is a method used in calculus for finding the derivative of a function that is formed as the division of two differentiable functions. If there is a function f(x) that is the division of two functions, u(x) and v(x), where v(x) is not zero, the quotient rule is used to find the derivative of f(x).
Quotient Rule Formula
The formula for the quotient rule is expressed as: the derivative of f(x), denoted as f'(x), equals [the derivative of u(x) times v(x) minus u(x) times the derivative of v(x)] divided by [v(x) squared]. In notation, this is written as f'(x) = [u'(x)v(x) – u(x)v'(x)] / [v(x)^2].
Example 1
Find the derivative of f(x) = x^2 / (x + 1).
Let u(x) = x^2 and v(x) = x + 1. The derivatives u'(x) and v'(x) are 2x and 1, respectively.
Calculation: f'(x) = [(2x)(x + 1) – (x^2)(1)] / (x + 1)^2 = [2x^2 + 2x – x^2] / (x + 1)^2 = [x^2 + 2x] / (x + 1)^2.
Example 2
Find the derivative of f(x) = sin(x) / x^2.
Let u(x) = sin(x) and v(x) = x^2. The derivatives u'(x) and v'(x) are cos(x) and 2x, respectively.
Calculation: f'(x) = [cos(x)x^2 – sin(x)2x] / x^4 = [x^2cos(x) – 2xsin(x)] / x^4.
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