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[MUSIC] Given what we've done so far we can differentiate a bunch of functions.

We can differentiate sums and differences and products.

But what about quotients. Given a fraction I'd like to be able to

differentiate that fraction. I like to be able to differentiate a

really complicated looking function like f(x)=2x+1/x^2+1, for instance.

But we're stuck immediately because we don't have anyways to differentiate

quotients, until now. Here's the The quotient rule so to state

this really precisely let's suppose I got two functions f and g and then I define a

new function that I'm just going to call h for now.

H(x) is this quotient f(x) over g(x). Now I also want to make sure that the

denominator isn't 0 at the point a so it makes sense to evaluate this function at

the point a. And I want ot assume that f and g are

differential at the point a. And I'm trying to understand how h

changes so I'm going to need to know how f and g change when the input wiggles a

bit. Alright so given all this set up, then I

can tell you what the derivative of the quotient is, the derivative of the

quotient at a is the denominator at a times the derivative of the numerator at

a. Minus the numerator at a times the

derivative of the denominator at a, all divided by the denominator at a^2.

Let's use the quotient rule to differentiate the function that we saw

earlier. So, the function we were thinking about

is f(x)=2x+1/ x^2+1. I want to calculate the derivative of

that, with respect to x. Now the derivative of this quotient is

given to us by the quotient rule. It's just the denominator times the

derivative of the numerator minus the numerator times the derivative of the

denominator. That's all divided by the denominator

squared. Now, I've calculated the derivative of

this quotient in terms of the derivatives of the numerator and denominator.

So we can simplify this further, X^2+1 times the derivative of this sum is

the sum of the derivatives. It's the derivative of 2x + the

derivative of 1-2x+1 times at again, the derivative of a sum, so the derivative of

x^2, with respect to x plus the derivative of 1.

And it's all divided by the denominator, the original denominator squared.

I can keep going, I've got x^2+1 times what's the

derivative of 2x? It's just 2. What's the derivative of this constant?

zero, minus 2x+1 times, what's the derivative of x^2? It's 2x,

and what's the derivative of 1? It's the derivative of a constant zero, all

divided by x^2+1^2. So, this is the derivative of the

original function we're considering, there's no more differentiation to be

done and we did it using the quotient rule.

We've done a ton of work on differentiation so far, we differentiate

sums, differences, products, now quotients.

What sorts of functions can we differentiate using all of these rules?

Well, here's one big collection. If you've got a polynomial divided by

polynomial, these things are called rational functions.

Sort of an analogy with rational numbers which are integers over integers.

A polynomial over a polynomial is by analogy, being called a rational

function. Now, since this is just a quotient of two

things you can differentiate, you can differentiate these rational functions.

This is a huge class of functions that you can now differentiate.

[MUSIC] I encourage you to practice with the quotient rule.

With some practice, you'll be able to differentiate any rational function that

we can throw at you. [MUSIC]

[MUSIC]