Chain Rule: Prove the Quotient

Date: 05/30/2001 at 09:47:45
From: David Liu
Subject: Proving the quotient rule using the product and chain rule

I've been asked to prove the quotient where u/v is uv^-1 using the 
product rule and the chain rule for (dv^-1/dx

I get this far:  V^-1*(du/dx)+u(dv^-1/dx)

but I don't know how to use the chain rule to to find dv^-1/dx and 
then convert it all into the quotient rule. Could you help by giving 
me a step-by-step guide to the answer?

Thank you!

Date: 05/30/2001 at 13:06:35
From: Doctor Peterson
Subject: Re: Proving the quotient rule using the product and chain 
rule

Hi, David.

You applied the product rule to u * v^-1 and got

    d/dx (u * v^-1) = v^-1 * du/dx + u * d(v^-1)/dx

Now you have to apply the chain rule to find d(v^-1)/dx. You might 
find this clearer if you define a new variable

    w = v^-1

The chain rule says

    dw/dx  = dw/dv * dv/dx

Since dw/dv = -v^-2, this gives

    d(v^-1)/dx = -v^-2 * dv/dx

You should be able to finish the work. If not, write back and show 
again how far you got.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/