# 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!

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