 Hello again, and welcome to another screencast where we're gonna see a couple of examples of the chain rule in action. And the focus of this screencast is on exponential functions and using the chain rule. So let's dive in, and we're gonna look at two examples, the first of which is differentiating y equals 2 to the tangent x power. So first of all, let's remind ourselves why we should even use the chain rule in the first place and not something else. Well, this isn't gonna be a product rule situation because I don't have anything applied together. This is 2 to the tangent x power, not 2 times tangent x. And certainly not a quotient rule question. And furthermore, you can see this is a composite. If I take this function and just run an input through it, let's say if I were to calculate y of 1, what would I do? Well, I have to do two things in stages. I would have to first of all calculate tangent of 1, whatever that is. So I'd have one thing to do first, and then I'd have to take the output of that and run it through the 2 to the x function second. So I can see that this y function is composed of two functions, literally composed of two functions that are linked together. The first function that I encountered in this process was tangent of x right there. And then the second function I encountered was 2 to the x. So that does two things for me. It convinces me that this is really a composite function and it tells me what the f and g are in the definition of the chain rule. Let's just write those down really quick. The function called g is the quote unquote inside function here, the first link in the chain, and that was tangent of x. And the f function, which is the outside function here, is 2 to the x power. So in composing this function, y, I do g first and then I do f second. And that's going to tell me how to put together the chain rule. Let's do that now. The chain rule itself is quite quick and quite simple once I know what the parts are and that's why this is important. So the chain rule would say that y prime would be the derivative of f with g put into it times the derivative of g. So now let's just go through and think about what those derivatives are. The derivative of f right here, we saw this is an exponential rule that we haven't seen in a while but f prime of x is 2 to the x times the natural log of 2. And g, I'll put this over here, g prime of x, we saw in an earlier section is secant squared of x. So now, all I have to do is put some pieces together here and I have f prime of g of x. That's 2 to the, and I'll leave a space there where there was an x before times log 2. And now what goes in here is the original function g of x and that was tangent of x. So 2 to the tangent x times the natural log of 2 times all of this times the derivative of g which we said was secant squared x. There was no real simplification I can do on this so I'm just going to leave it right here. So the answer here is 2 to the tangent x times the natural log of 2 times secant squared of x. And again, this isn't so bad once we know that the original function is a composite and we know what the first and second or inside and outside functions are. Now let's look at another similar example where I'm differentiating a function but it has e as its base. And this is a very important class of examples because exponential functions with e as the base, super, super common in most applications. So let's play with this. Again, I believe this is a composite function. Let's try to remind ourselves what the inside and outside functions are. So if I were to take a number and run it through here like y of 1, let's say, that would be e to the 1 cubed minus 1 plus 1. The important thing about this is knowing what comes first and what comes second. It looks like the first function I encounter here, the g of x or inside function, is the exponent x cubed minus x plus 1. And the f function, the outside or second function, was e to the x. I'm going to erase my little sketch writing here. And now if I have that composition, I'm going to be using the chain rule so I'll need the derivative of each of these. So the derivative of g of x, very simple in this case, since g of x is a polynomial, is 3x squared minus 1. And f prime of x, the derivative of e to the x is just itself. Okay, so this is g prime, this is f prime. Now we have enough information to put together the derivative of y that we're after. Okay, derivative of y is, I'll just write it down, f prime of g of x times g prime of x and now we'll just use the info we have in the blue upstairs here. f prime is e to the x. And in place of the x, I'll put a, just a blank. And what goes into that blank, what gets plugged in for x is g. Okay, the original g function that's here, that's x cubed minus x plus 1. And then I need to multiply all that by g prime of x and we have that right up here. That's 3, put a parenthesis around that, 3x squared minus 1. So just to clean that up slightly, the answer here is 3x squared minus 1, that entire group times e to the x cubed minus x plus 1. Okay, so just a word about this example here, a couple of words. Whenever you see a function that's of the form e to the something, e to some power, that's not just a straight variable, e to a function power. This happens quite a lot. You're going to be using the chain rule. Second of all, notice that the derivative, although the derivative of e to the x is just itself, the derivative of e to a function power is not just itself. Okay, we have to also multiply by the derivative of the x bonus. This is a very important example. Study this and master it and you'll have plenty of opportunities to practice further. Thank you for watching.