Differentiating inverse functions
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Derivatives of inverse functions
- [Instructor] 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, or 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, d/dx 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 one. 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 of a 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, one 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 a 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. All right, if that's unfamiliar, review it on Khan Academy. So 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 is f prime of x going to be? Well, this one of those amazing results in calculus. One of these neat things about the number e is that 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 one 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 one over g prime of f of x. So g prime of f of x, so g prime is one over our f of x, and f of x is e to the x, one over e to the x. Is this indeed true? Yes, it is. One over, one 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 inverses of each other. You could say g prime of x is going to be equal to one 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.
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