Derivatives of inverse functions (video) | Khan Academy
The basic idea of this inverse function theorem was discovered by John Nash . TX and V and let Djf denote they'th covariant derivative of a section ƒ of V. Put . Even in the Banach space case there is a large difference in what it means for Hence the integral of a vector function may be obtained by ordinary integra-. In addition, the chapter on differential equations and the section on numerical integra- tion are largely Derivatives of the Trigonometric Functions. Inverse functions. .. In our example, ∆y = 3 - 1 = 2, the difference between the .. The special case r = 1 is called the unit circle; its equation is x2 + y2 = 1. Inverse Functions and Their Derivatives. Inverses of . The whole subject of calculus is built on the relation between u and f. The question .. The ratio of those changes equals w., which is the x Integrals. The Idea of the Integral.
Derivatives of inverse functions: from equation
The left hand side used the chain rule. You're going to get F prime of H of X. F prime of H of X times H prime of X comes straight out of the chain rule is equal to, is equal to the derivative of X is just going to be equal to one. And then you derive, you divide both sides by F prime of H of X and you get our original property there. So now with that out of the way let's just actually apply this.
So, we want to evaluate H prime of Or sorry, H prime of negative Is going to be equal to one over F prime of H of negative H of negative Now had they given us H of negative But they didn't give it to us explicitly, we have to remember that F and H are inverses of each other.Derivative of Inverse Functions Examples & Practice Problems - Calculus
So F of negative two is negative Well, H is gonna go from the other way around. If you input negative 14 into H you're going to get negative two.
So H of negative 14 well, this is going to be equal to negative two. Once again, they are inverses of each other. So H of negative 14 is equal to negative negative two. And once again, I just swapped these two around. That's what the inverse function will do. If you're wrapping from if F goes from negative two to negative 14 H is going to go from negative 14 back to negative two.
So now we want to evaluate F prime of negative two. Well, let's figure out what F prime of X is.
So, F prime of X is equal to remember the power rule, so three times one half is three halves times X to the three minus one power which is the second power. Plus the derivative of three X with respect to X which that's just going to be three.
And you could do that, it's just the power rule. But this was X to the first power, one times three, X to the zero power, but X to the zero is just one so you're just left with three. And derivative of a concept that's just gonna be zero.
So that's F prime of X. So F prime of, F prime of negative two is going to be three halves times negative two squared is four, positive four.
So, this is going to be equal to two times three plus three. 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.
Derivatives of Inverse Functions - Mathematics LibreTexts
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.
Derivatives of inverse functions
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.
Calculus II - Arc Length
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.