11/12/2022 0 Comments Limit chain rule![]() ![]() When I show this to students for the first time, they freak out, because what we had to do before, I had the chain rule, was I'd have to foil this thing four times. You can already see the value of the chain rule, I like this one a lot. We'll do these together, then I'm going to telling you to go off and pause the video and try some on your own, but let's start off with something like this. The way to get good at this one is to find lots of derivatives, here's your assignment, take the derivatives of a bunch of problems. For the most part, for now at least, we're going to use Newton's notation with the primes. You can think of it as y as a function of u, and then u is some function of g of x. It looks something like this, if I have the function little y as my function, we can say the derivative of y, so y is now my function over here, dy dx becomes dy du, du dx. We're going to see it in its other form, so in leib, but it's notation. ![]() This is a little prime by the way, it's Newton's notation. ![]() ![]() I'll write this out a bunch of times as we go through the first one. It is important that we keep the inside the same, multiplied by the derivative of the outside function. This is the one that we're going to use the most. It's pretty common to see sine of 2x or sine of cosine or e to the whatever. The reason why is a lot of stuff, a lot of functions are composed with each other. We're just going to state the rule, and then I just want you to get good at using it. You can work this out using limit definition, it's a bit of a pain, but this is the idea. But here it is, it is the derivative of the outside function, and we keep the inside function the same, multiplied, so times the derivative of the inside function. You could see it as d, dx of f composed with g of x, they might use a little circle notation, either one is fine. Well, here it comes, folks, here's the chain rule, the derivative of this composition, which you'll see a couple of ways. I have one function and I compose it with another. Let's define some big function, capital F to be the composition of f of g of x. I promise you'll have it memorized by the time we get past this, you just need it to do almost everything in this class, but the idea now is we want to study the functions that are compositions. The chain rule is one of the most important differentiation rules we're going to have in this class, we're going to use it over and over again. Welcome to our lecture on the chain rule. ![]()
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