[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]