How to Use the Quotient Rule to Find Derivatives of Functions

[rustynail]

Homework Statement

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!

 

[Char. Limit]

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

 

[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?

 

[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.

 

[rustynail]

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.

Oooh this is indeed much simpler. Thank you!