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# Derivatives of inverse functions

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

## Video transcript

so let's say I have two functions that are the inverse of each other so I have f of X and then I also have G of X which is equal to the inverse of f of X and f of X would be the inverse of G of X as well if the notion of an inverse function is completely unfamiliar to you I encourage you to review inverse functions on Khan Academy now one of the properties of inverse functions are that if I were to take G of f of X G of f of X where I could say the F inverse of f of X that this is just going to be equal to X and it comes straight out of what an inverse of a function is if this is X right over here the function f would map to some value f of X so that's f of X right over there and then the function G or F inverse if you input f of X into it it would take you back it would take you back to X so that would be F inverse or we're saying G is the same thing as f inverse so all of that so far is a review of inverse functions but now we're going to apply a little bit of calculus to it using the chain rule and we're gonna get a pretty interesting result what I want to do is take the derivative of both sides of this equation right over here so let's apply the derivative operator DDX on the left hand side d DX on the right hand side and what are we going to get well on the left hand side we would apply the chain rule so this is going to be the derivative of G with respect to f of X so that's going to be G prime of f of X G prime of f of X times the derivative of f of X with respect to X so times F prime of X and then that is going to be equal to what well the derivative with respect to X of X that's just equal to 1 and this is where we get our interesting result all we did so far is we used something we knew about inverse functions and we'd use the chain rule to take the derivative of the left hand side but if you divide both sides by G prime of f of X what are you going to get you're going to get a relationship between the derivative function and the derivative of its inverse so you get f prime of X is going to be equal to one over all of this business 1 over G prime of f of X G prime of f of X and this is really neat because if you know something about the derivative of function you can then start to figure out things about the derivative of its inverse and we can actually see this is true with some classic functions so let's say that f of X is equal to e to the X and so G of X would be equal to the inverse of F so f inverse which is what's the inverse of e to the X well one way to think about it is if you have Y is equal to e to the X if you want the inverse you can swap the variables and then solve for y again so you'd get X is equal to e to the Y you take the natural log of both sides you get natural log of X is equal to Y so the inverse of e to the X is natural log of X and once again that's all review of inverse functions if that's unfamiliar review it on Khan Academy so if G of X is going to be equal to the natural log of X now let's see if this holds true for these two functions well what does f prime of X going to be well this is one of those amazing results in calculus one of these neat things about the number e is that if the derivative of e to the X is e to the X and in other videos we also saw that the derivative of the natural log of X is 1 over X so let's see if this holds out so we should get a result f prime of X e to the X should be equal to 1 over G prime of f of X so G prime of f of X so G prime is 1 over our f of X and f of X is e to the X 1 over e to the X is this indeed true yes it is 1 over 1 over e to the X is just going to be e to the X so it all checks out and you could do the other way because these are inverse of each other you could say G prime of X is going to be equal to 1 over F prime of G of X because they're inverses of each other and actually what's really neat about this is that you could actually use this to get a sense of what the derivative of an inverse function is even going to be