Understanding the Quotient Rule in Differentiation: A Proof and Explanation

[Jimmy84]

Homework Statement

if p(x) = f(x)/g(x)

Prove that

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

Homework Equations

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 don't 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.

 

[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)

 

[futurebird]

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

 

[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

 

[Phrak]

Alternatively,

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

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

 

[Jimmy84]

thanks a lot ...