proof of quotient rule

Let F(x)=f(x)/g(x). Then

F(x) = limh0F(x+h)-F(x)h=limh0f(x+h)g(x+h)-f(x)g(x)h
= limh0f(x+h)g(x)-f(x)g(x+h)hg(x+h)g(x)

Like the product ruleMathworldPlanetmath, the key to this proof is subtracting and adding the same quantity. We separate f and g in the above expression by subtracting and adding the term f(x)g(x) in the numerator.

F(x) = limh0f(x+h)g(x)-f(x)g(x)+f(x)g(x)-f(x)g(x+h)hg(x+h)g(x)
= limh0g(x)f(x+h)-f(x)h-f(x)g(x+h)-g(x)hg(x+h)g(x)
= limh0g(x)limh0f(x+h)-f(x)h-limh0f(x)limh0g(x+h)-g(x)hlimh0g(x+h)limh0g(x)
= g(x)f(x)-f(x)g(x)[g(x)]2
Title proof of quotient rule
Canonical name ProofOfQuotientRule
Date of creation 2013-03-22 12:38:58
Last modified on 2013-03-22 12:38:58
Owner drini (3)
Last modified by drini (3)
Numerical id 5
Author drini (3)
Entry type Proof
Classification msc 26A06