Differentiation: Principles, Techniques and Applications

00:01 So, things are getting a little bit more exciting now.

00:05 We're going to start to look at three new rules.

00:08 We'll look at the chain rule, the product rule, and the quotient rule.

00:12 And we'll talk about when to use them, and what kind of functions we need to look at in order to use those rules.

00:18 Now, remember, previously, you've learned how to differentiate from first principles, we've learned how to differentiate the faster way, and we've also looked at the applications of differentiation.

00:27 So we've looked at finding the gradient of a tangent, the gradient of a normal, and also move to finding equations of tangents and normals.

00:36 So let's start looking at our three new rules.

00:39 The principles here are the three different types of functions that we'll be looking at.

00:45 The first type of function is called a function of a function.

00:48 I'll explain what that means better as we go through numerical examples.

00:52 But what that really is, is a function inside of another function.

00:56 And you can have a chain of different functions.

00:58 And we also refer to this as the chain rule as well.

01:02 The second type of function that we'll be looking at is just a product of two functions.

01:06 So that's fairly straightforward to spot.

01:08 You just look at two functions that are multiplying together.

01:11 And then we'll talk about what rule to use to differentiate it.

01:15 And the last type, which is predictable is going to be two functions which are dividing each other.

01:19 so you have one on top of the other.

01:21 And then we'll look at the rules that we need to differentiate a function, which is a quotient.

01:26 We call it one above the other or dividing it.

01:30 The techniques for each one for a function of a function is the chain rule, because it's a function within a function.

01:37 When you have the product of two functions, we just call it the product rule.

01:40 And when you have two functions dividing, we call it the quotient rule.

01:45 So we'll learn how to apply them.

01:46 And we'll also look at the derivatives and the proofs for them.

01:50 So you're fully convinced that it's the right thing to do.

01:53 And the applications once again, we're differentiating.

01:55 So we're going to be finding the gradient of course, with the gradient, we can do lots of other things, we can find the equations of tangents and lines, and we'll move on defining stationary points of curves as well.

02:07 So let's start and look at our first problem.