WEBVTT
Kind: captions
Language: en
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a chain rule which is what we need to do when we have a function described as the
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composition of two functions each of which we know how to take the derivative
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of so for example maybe I have the sin of 3x squared plus 2 and I want to think
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of this as one function with another function inside it so here
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f u is the sign u and g of x is 3x squared plus 2 which is u
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when i put them together I want to somehow relate the derivative here i
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take the derivative of the sine with respect to u and that will be the
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cosine and if i take the derivative of 3x squared with respect X that will be
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6x and I want to somehow put these together and really what the chain rule
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says is that these two derivatives just multiply but we need to so the
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derivative will be just of the sin 3x plus two will be just this
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derivative times that
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but I made up u so i have to rewrite you in terms of X and us 3x squared plus
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2 so this is really derivative I want 6x times usually write the 6x in front so
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it's the derivative of the outer function times the derivative of the
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inner function
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why would this be so so let's think about it's not just an arbitrary rule
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that i wrote down i'm gonna erase this example and why would we expect this to
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work this way all so we want we want to take the derivative of the composition
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like this and what is it
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so let's think about what the derivative is in the first place i have let's just
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think about what does G prime of X represent well I think of g is some kind
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of a transformational object i put in number like X and outcomes another
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number 3 x square to that's what she does it transforms an X into a 3x
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squared plus 2 and what the derivative is telling me is that if i move ex a
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little bit in one direction or another
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then the amount that comes out will be moved by about 6 x the derivative
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multiplies my input error and ads on about of wiggle that we x 66 x times
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let's put this wiggle as H exact times H so it tells me that my wiggles on the
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input are going to be magnified by a fat
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after of 6x on the output
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ok so now what is the derivative of F do well the derivative of F says let's call
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this number
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you said that if I take a number you and I feed it into the f machine that's not
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like dropping the f-bomb that's a different thing I feed you into the f
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machine outcomes sign of you but the derivative tells me that any amount of
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wiggling that i have on you or any small amount of whittling that i have on you
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will be magnified by a factor of the cosine so if i have a so now if i hook
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these two things together then we see that the amount of Wiggles are going to
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multiply that is if i take an x and move it a little bit and then I feed
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it into G then that will also move a little bit and that little bit will be
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controlled exactly by the derivative 6x and then if i take that output
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and move it and put it into the f machine but it's off a little bit
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because I moved it it will come out factored by the cosine so that means
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that this derivative will be 6x which is what I get out of you out of removing
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the X by the g machine times the amount that i get by moving f but you was of
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course 3x squared plus 2
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more formally we can write this by saying that the derivative of this
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composition is the derivative of the outside function plug-in where it is
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times the derivative of the inside if we write this in the other
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notation in the likeness notation here we're on thinking you is g of x and y is
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f of g of x f you then the amount that why changes when i move ex a little bit
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is this same as the amount that y changes when I move x a little bit
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that's F prime u times the amount that you change when i move x a little bit
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works just like fractions i can just cancel the du's this is strictly
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speaking not true but it works that way if we do the proof which i'm not doing
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in this and you'll see that actually we can carefully adjust these so that they
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they do cancel so it works this is the chain rule as but so we have the chain
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rule here and why it works