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How to Differentiate Functions Using the Quotient Rule

The quotient rule is the rule that tells you how to take the derivative of a function that is a ratio of two differentiable functions.

Formula

The Quotient Rule

(u v) = uv uv v2

where u = u(x) and v = v(x)

Note! Sometimes you get lucky, and it’s possible to simplify the answer. Most of the time that’s not possible though, and you can just leave the fraction as it is.

Example 1

Differentiate the expression 2x + 1 ex

Here, u = 2x + 1 and v = ex. You get u = 2 and v = ex, which gives you

(2x + 1 ex ) = (2x + 1) ex (2x + 1) (ex) (ex) 2 = 2 ex (2x + 1) ex (ex) 2 = 2ex 2xex ex (ex) 2 = ex 2xex (ex) 2 = ex(1 2x) (ex) 2 = 1 2x ex .

(2x + 1 ex ) = (2x + 1) ex (2x + 1) (ex) (ex) 2 = 2 ex (2x + 1) ex (ex) 2 = 2ex 2xex ex (ex) 2 = ex 2xex (ex) 2 = ex(1 2x) (ex) 2 = 1 2x ex

Example 2

Differentiate the expression 3x3 2x2 + 7 x 1

Here, you have u = 3x3 2x2 + 7 and v = x 1. That means u = 9x2 2 and v = 1, and the derivative is

= (3x3 2x2 + 7 x 1 ) = 1 (x 1)2 ( (3x3 2x2 + 7) (x 1) (3x3 2x2 + 7) (x 1) ) = (9x2 4x) (x 1) (3x3 2x2 + 7) 1 (x 1)2 = 9x3 9x2 4x2 + 4x 3x3 + 2x2 7 (x 1)2 = 6x3 11x2 + 4x 7 (x 1)2

(3x3 2x2 + 7 x 1 ) = (3x3 2x2 + 7) (x 1) (3x3 2x2 + 7) (x 1) (x 1)2 = (9x2 4x) (x 1) (3x3 2x2 + 7) 1 (x 1)2 = 9x3 9x2 4x2 + 4x 3x3 + 2x2 7 (x 1)2 = 6x3 11x2 + 4x 7 (x 1)2 .

As x = 1 is not a root of the numerator, you can’t simplify the expression.

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