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AP.CALC:

FUN‑3 (EU)

, FUN‑3.B (LO)

, FUN‑3.B.1 (EK)

the following table lists the values of functions F and H and of their derivatives F Prime and H prime for X is equal to 3 so this is telling us when X is equal to 3 the value of the function is 6 f of 3 is 6 you could view it that way H of 3 is 0 F prime of 3 is 6 and H prime of 3 is 4 and now they want us to evaluate the derivative with respect to X of the product of f of X and H of X when X is equal to 3 so one way you could view this is if we viewed some function if we viewed some function G of X G of X as being equal to the product of f of X and H of X this expression is the derivative of G of X so we could write G prime of X is equal to the derivative with respect to X of f of X times H of X which is what we see right here which is what we want to evaluate at x equals 3 so we essentially want to evaluate G prime of 3 this is what they're asking us to do well to do that let's go first up here let's just think about what it's doing they're asking us to take the derivative with respect to X of the product of two functions that we have some information about well if we're taking the praat the derivative of the product of two functions you could imagine that the product rule could prove useful here so I'm just going to restate the product rule this is going to be equal to the derivative of the first function f prime of x times the second function no dirt not taking its derivative plus the first function not taking its derivative f of X times the derivative of the second function H prime of X so if you're trying to find G prime of 3 well that's just going to be F prime of 3 times H of 3 plus F of 3 times H prime of 3 and lucky for us they give us what all of these things evaluate to F prime of 3 right over here they tell us F prime when X is equal to 3 is equal to 6 so this right over here is 6 H of 3 they give us that 2h of 3 when X is 3 the value of our function is zero so this is zero so this first term is you can just get six times zero which is going to be zero but we'll get to that now F of three F of three well the function when X is equal to 3 y or the func F of 3 is equal to 6 so that is 6 and then finally the H prime evaluated 3 H prime of X when X is equal to 3 H prime of X is equal to 4 or you could say this is H prime of 3 so this is 4 and so there you have it this is going to evaluate to 6 times 0 which that's all just going to be 0 plus 6 times 4 which is going to be equal to 24 and we're done

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