[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. 
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