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Current time:0:00Total duration:3:34

what I want to do in this video is think about how we can take the derivative of an expression that can be viewed as the product not of two functions but of three functions and we're going to do it using what we know the product rule and the way we can think about it is we can view this as the product first of two functions of this function here and then that function over there and then separately take the derivative so if we just take if we just view the standard product rule it tells us that the derivative of this thing will be equal to the derivative of f of X the derivative of f of X let me close it with the white bracket times the rest of the function so times G of X let me close it with the times G of X times H of X times G of X times H of X times H of X plus plus just f of X plus f of X plus f of X times the derivative of this thing times the derivative with respect to X of G of x times H of X G of x times H of X times H of X let me write that a little bit neater times H of X but what is this thing right over here going to be equal well we can you apply the product rule again so here I'm just focusing on this part right over here the derivative of this is just going to be G prime of X G prime of x times H of X times H of X plus plus G of X plus G of X times the derivative of H times H prime of X H prime of X so everything that we had the derivative of G of x times H of X is this stuff right over here and so we're going to multiply that times f of X so let's rewrite all of this stuff here so this first term right over here we can rewrite so all this is going to be equal to F prime of X that's that right over there times G of X times H of X times H of X plus plus and now we're going to distribute this f of X so it's f of X times this plus f of X times this so f of X times this is f of X times G prime of X the derivative of G G prime of x times H of X times H of X and then finally we do that in a white color and then finally F of x times this is just f of X is just f of X times G of x times G of X times H prime of X H prime of X and this was a pretty neat result this is essentially we can view this as a product rule where we have three where we can have our expression viewed as a product of three functions now we have three terms in each of these terms we take a derivative of one of the functions and not the other two here we took the derivative of F here we took the derivative of G here we took the derivative of H and you could imagine if you had the product of four functions here you would have four terms in each of them you'd be taking a derivative of one of the functions if you had n if you add n functions here then you would have n terms here and in each of them you would take the derivative of one of the function so this is kind of a neat result