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# Worked example: Product rule with mixed implicit & explicit

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.B (LO)
,
FUN‑3.B.1 (EK)

## Video transcript

let F be a function such that F of negative 1 is 3 and F prime of negative 1 is equal to 5 let G be the function G of X is equal to 1 over X let capital F function to find it as the product of those other two functions what is capital F prime of negative 1 well we can just apply the product rule here let me just rewrite let me just essentially state the product rule capital F prime of X is going to be equal to since capital f of X is the product of these two functions we apply the product rule this is going to be F prime of X times G of X plus plus f of X times G prime of X and so if we want to evaluate this at F at negative 1 capital F prime at negative 1 is equal to F prime of negative 1 times G of negative 1 plus function f evaluated at negative 1 times the derivative of G evaluated at negative 1 now let's see if we can figure these things out so do they tell us this anywhere can we figure this out F prime of negative 1 well they tell us right if your F prime of negative 1 is equal to 5 so this is equal to 5 now let's actually stick with F what is F of negative 1 well they tell that to us right over here F of negative 1 is equal to 3 so F of negative 1 is equal to 3 now G G of negative 1 and G prime of negative 1 they don't give it to us explicitly here but we could figure it out we can we know that well if G of X is equal to this G of negative 1 is equal to 1 over negative 1 which is equal to negative 1 so this is equal to negative 1 and then last but not least if we want to find G prime of negative 1 we just have to take the derivative of this so G prime of X actually let me just rewrite G of X G of X 1 over X is just the same thing as X to the negative 1 so we're going to use the power rule to figure out G prime of X is equal to bring that exponent out front negative one times X to the and then decrement the exponent negative two power so G prime of negative 1 of negative one is equal to negative one times negative one to the negative two power and that's just the same thing as negative one over negative one squared this is one so this is just all going to evaluate to negative one so this is negative one and so we have five times negative one which is negative 5 plus three times negative one which is negative three which is equal to negative eight so f prime of negative one is equal to negative eight