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Current time:0:00Total duration:7:08

Applying the chain rule graphically 3 (old)

Video transcript

consider the functions F and G with the graph 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 five 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 this somewhat complicated looking function definition right over here so we have we have capital G of X and actually let me do it 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 and what we're squaring is G of F of 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 if H of X would be equal to x squared we could write G of X G of X is equal to H of 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 so copy and paste there we go so this is another way of writing G of X or whatever we g of f of X we can put that into H of X which is really just squaring it 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 can 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 that the cat the derivative of G with respect let me I'll kind of mix notations a little bit but all right the derivative of G of X with respect to X with respect to X is equal to is equal to the derivative of this whole thing this whole thinks let me copy and paste it copy and paste it's equal to this derivative of this whole thing with respect to with respect to what's inside of that whole thing so what 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 x times the derivative of 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'd like to write this out just to it feels good it looks like it's not these aren't these are kind of rational expressions with differentials it's really a notation more than to be taken literally but it feels good Y or it's 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 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 with respect to with respect to X or another way we can write this is G prime G prime of X is equal to H prime of G of f of X H prime of X let me do it here H prime of this H prime of this so copy and paste H prime of that x times G prime 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 times F prime of x times F prime of X and when you write it you 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 it a little bit clearer what's going on but this thing in my brain I like to translate there 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 them this actually isn't too bad so we want to know what's G prime of five so everywhere we see an X everywhere we see an X let's change it to a five so we're gonna say we need to figure out what G prime of five 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 and let me everywhere where I see an X I'm gonna replace it with a five 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 zero so this right over here is going to be equal to zero now that's really interesting so we can keep trying to try to well it'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 etc etc 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 is going to be equal to another way of thinking about it is f of X 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 G that 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