# Homework Help: Proof of the Quotient Rule

1. Dec 28, 2009

### Jimmy84

1. The problem statement, all variables and given/known data
if p(x) = f(x)/g(x)

Prove that

p'(x) = g(x) f '(x) - f(x) g '(x) / g(x)ˆ2

2. Relevant equations

3. The attempt at a solution

The proof goes like this in my book

p(x + h) - p(x) / h = [ f(x+h)/ g(x+h) - f(x) / g(x) ] / h

= f(x + h) g(x) - f(x) g(x + h) / h g(x) g(x + h)

I dont understand why did g(x) and g(x + h) appeared in the numerator on the last part of the proof? Since g(x) and g(x +h) were already multiplyed by h in the denominator.

Thanks in advance.

2. Dec 28, 2009

### futurebird

It's just subtracting fractions:

a/b-c/d = (ad-cb)/d

a=f(x+h)
b=g(x+h)

c=f(x)
d=g(x)

3. Dec 28, 2009

### futurebird

By the way... don't leave out the "lim h-->0" when you write out the proof!

4. Dec 28, 2009

### futurebird

Here is a good write up of the proof with reasons for each step.

http://people.hofstra.edu/Stefan_waner/RealWorld/proofs/quotientruleproof.html [Broken]

Last edited by a moderator: May 4, 2017
5. Dec 28, 2009

### Phrak

Alternatively,

f(x)/g(x) = f(x)[g(x)]-1.

Solve by applying the product rule, and the power rule.

6. Dec 28, 2009

### Jimmy84

thanks a lot ...

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