# Quotient Rule for Derivatives

1. Jun 28, 2013

### duncanrager

I am a little confused about when to use the quotient rule. When you have one function over another function, and are taking the derivative, are you required to use this technique? I thought you were, but then I was watching this video on Khan Academy...

...and the instructor, Sal, did not use the quotient rule. You don't even need to watch the video, the screenshot at the beginning shows exactly what he did. Using L'Hopital's rule, he simply took the derivative of the function in the numerator using the power rule, and did the same to the denominator.

Take the limit as x approaches infinity of: (8x-5)/-6x

He couldn't evaluate the limit by replacement because you get infinity over negative infinity... so he used L'Hopital's rule and got 8/-6.

Why didn't he have to use the quotient rule? How is this problem different from any others?

Thanks so much,

Colin

2. Jun 28, 2013

### E'lir Kramer

There are two different concepts.

The quotient rule is how to find the derivative of a function by decomposing it into the quotient of two smaller functions, f and g, each which you independently know how to differentiate. The quotient rule yields a formula for a derivative.

L'Hopital's rule is a technique for finding the limit of a quotient of two functions under certain conditions. L'Hopital's rule yields the limit of the quotient of two functions.

Does that clear it up for you? He was just trying to find a particular limit, so he used L'Hopital's rule. He was not trying to find a formula for the derivative of the function f(x) such that f(x) = (8x-5)/-6x.

Edit: The key thing here is see that the limit of a function is a completely separate concept from the derivative of a function. It just so happens that the definition of the derivative of a function includes limits in it.

Last edited: Jun 28, 2013
3. Jun 28, 2013

### CAF123

No, not necessarily. Some simplification beforehand may allow you to reexpress the quotient into a form that can be integrated by simply using the power rule, for example.

Can you state l'Hospital's Rule and when it is used?

4. Jun 28, 2013

### duncanrager

Yes that clears things up, thanks guys.

Is this right?: When one function is being divided by another (i.e. f(x)/g(x)), and you can't simplify it any further, and you want to take the derivative, then you use the quotient rule.

If, on the other hand, in the same situation, you want to find the limit, then you use L'Hopital's rule and can take the individual derivatives independent of the other, not using the quotient rule.

To answer your question CAF123, I think L'Hopital's Rule states that the limit of a function is equal to the limit of that function's derivative. And so if you evaluate the limit of a function and get 0/0 or inf/inf, then you can use L'Hopital's rule as justification to take the derivative of each function (because I think it needs to be in f(x)/g(x) form) and get an equivalent result.

Do I have all that correct?

5. Jun 28, 2013

### CAF123

Yes, unless you want to find the derivative by first principles. Note also you may equally use the product rule here, writing f/g = fg-1.
Yes, but only on an indeterminate form.

I think I know what you mean (given by what you write below) but the above is not quite right. Take f = const. as counter example.

Yes