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Applying the chain rule graphically 3 (old)

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Consider the functions f and g with the graphs shown below. If capital G of x is equal to g of f of x whole thing squared, what is the value of G prime, capital G prime, of 5? And I encourage you to now pause this video and try to solve it on your own. So let's try to think through this somewhat complicated-looking function definition right over here. So we have capital G of x. And actually, let me write it this way. Let me write it this way, I'll do it in yellow. We have capital G of x is equal to this quantity squared. What we're squaring is g of f of x. g of f of x is what we're squaring. Or another way to write G of x, If h of x were to be equal to x squared, we could write G of x is equal to h of this business, h of g of f of x. Let me just copy and paste that so I don't have to keep switching colors. So copy and paste, there we go. So this is another way of writing G of x, where whatever g of f of x, we input then to h of x, which is really just squaring it. So there's a couple of ways that we can write out the derivative of capital G with respect to x. And you could imagine this is going to involve the chain rule. But I like to write it out, just to clarify in my head what's going on and to make sure that it actually makes some sense. So one thing that we could write, we could write the derivative of G with respect-- I'll mix notations a little bit-- but I'll write the derivative of G of x with respect to x is equal to the derivative of this whole thing. So let me copy and paste it, copy and paste. It's equal to this derivative of this whole thing with respect to what's inside of that whole thing. So if you wanted to treat g of f of x as a variable, so with respect to that. So copy and paste. So it's going to be the derivative of this whole thing with respect to g of f of x times the derivative of g of f of x with respect to f of x, with respect to-- I'll just copy and paste this part, whoops-- with respect to f of x. And I like to write this out. It feels good. It looks like these are rational expressions with differentials. It's really a notation more than to be taken literally. But it feels good, or at least in my mind it's a little bit more intuitive why all of this works out. So with respect to f of x times the derivative of-- and I'm using non-standard notation here, but it helps me really conceptualize this-- times the derivative of f of x with respect to x. Or another way we could write this is G prime of x is equal to h prime of g of f of x, h prime of-- actually, let me do it here-- h prime of this. So copy and paste, h prime of that, times g prime of f of x, times g prime of this. So copy and then paste. So times g prime of that. Put some parentheses there. Times f prime of x. And I like writing it this way, because you notice if these were-- and once again, this is more notation, but it gives a sense of what's going on. If you did view these as fractions, that would cancel with that. That would cancel with that. You're taking the derivative of everything with respect to x, which is exactly what you wanted to do. And let me put some parentheses here so it makes a little bit clearer what's going on. But this thing, in my brain, I like to translate that as, well, that's just h prime of g of f of x. This is g prime of f of x. This is f prime of x. And going from this to try to answer your question, the question that they're asking us actually isn't too bad. So we want to know, what's G prime of 5? So everywhere we see an x, let's change it to a 5. So we're going to say, we need to figure out what G prime of 5 is. G prime of 5 is equal to-- and actually, let me just copy and paste this whole thing. So copy and paste. And so let me, everywhere where I see an x, I'm going to replace it with a 5. So let me get rid of that. Let me get rid of that. And let me get rid of that. And so I have a 5, a 5, and a 5. So what is f of 5? f of 5 is equal to negative 1. So this right over here simplifies to negative 1. This right over here simplifies to negative 1. And what's f prime of negative 5? Well, that's the slope of the tangent line at this point right over here. And we see that the derivative, or the slope, of the tangent line here is 0. So this right over here is going to be equal to 0. Now that's really interesting. So we could keep trying to, well, what's g of negative 1? What's g prime of negative 1? You could see g of negative 1, g of negative 1 we see is negative 1. g prime of negative 1 is the slope here, which is also negative 1. Then we could calculate h prime of these values, et cetera, et cetera. But we don't even have to do that. Because this is the product of three things, and one of these things right over here is a 0. So 0 times anything times anything is going to be equal to 0. Another way of thinking about it is, f of x is isn't changing when x is equal to 5. If f of x isn't changing when x is equal to 5, then the input into the g isn't going to be changing. So the g function isn't going to be-- in the composition g of f of x-- isn't going to be changing. And so h of g of f of x isn't going to be changing. So g of x isn't going to be changing. And so the derivative of capital G of x at x equals 5 is going to be equal to 0.