Homework Help: Need help with quotient rule

1. Jan 11, 2012

rustynail

1. The problem statement, all variables and given/known data

I want to prove that if $$y = \frac{u}{v}$$

then $$\frac{dy}{dx} = \frac{ v \frac{du}{dx} - u \frac{dv}{dx} }{v²}$$

u and v are functions of x.

2. The attempt at a solution

$$y = uv^{-1}$$

$$y + dy = ( u + du ) ( v + dv )^{-1}$$

then I suppose I could use Newton's Binomial to develop

$$( v + dv )^{-1}$$

but I don't know how to use the formula

$$(a+b)^{n} = \sum_{k=0}^{n} \dbinom{n}{k} a^{n-k} b^k$$

with a negative exponent. I'm familiar with binomial coefficients but that negative exponent is leaving me without a clue.

Any help would be very much appreciated, thank you!

Last edited: Jan 11, 2012
2. Jan 11, 2012

Char. Limit

That seems like a rather awkward way to do it. Are you allowed to use the product rule in your proof?

3. Jan 11, 2012

rustynail

I am learning by myself for now, so I'm pretty much allowed to use anything.
How would you do it using the product rule?

4. Jan 11, 2012

Char. Limit

Well, if you can use the product rule, it becomes a lot easier. Since the product rule is:

(uv)' = u v' + u' v

Just say (u * 1/v)' = u (1/v)' + u' (1/v)

and solve from there.

5. Jan 11, 2012

rustynail

Oooh this is indeed much simpler. Thank you!