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

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We can differentiate sums and differences 
and products. 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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a. 
Minus the numerator at a times the 

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

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

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earlier. 
So, the function we were thinking about 

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

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

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

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

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denominator. 
That's all divided by the denominator 

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squared. 
Now, I've calculated the derivative of 

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

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

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

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

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

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

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

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

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

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

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

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

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done and we did it using the quotient 
rule. 

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

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sums, differences, products, now 
quotients. 

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

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

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polynomial, these things are called 
rational functions. 

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

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

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

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

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

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

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

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we can throw at you. 
[MUSIC] 

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[MUSIC]