# What is the derivative of an inverse function?

## What is the derivative of an inverse function?

The Derivative of an Inverse Function. (f−1)′(a)=pq. f′(f−1(a))=qp.

## How do you find the inverse of a function in calculus?

Finding the Inverse of a Function

- First, replace f(x) with y .
- Replace every x with a y and replace every y with an x .
- Solve the equation from Step 2 for y .
- Replace y with f−1(x) f − 1 ( x ) .
- Verify your work by checking that (f∘f−1)(x)=x ( f ∘ f − 1 ) ( x ) = x and (f−1∘f)(x)=x ( f − 1 ∘ f ) ( x ) = x are both true.

**Is the derivative of the inverse the inverse of the derivative?**

This means that the derivative of the inverse function is the reciprocal of the derivative of the function itself, evaluated at the value of the inverse function.

### What is implicit differentiation?

: the process of finding the derivative of a dependent variable in an implicit function by differentiating each term separately, by expressing the derivative of the dependent variable as a symbol, and by solving the resulting expression for the symbol.

### Do inverse functions have the same derivative?

Functions f and g are inverses if f(g(x))=x=g(f(x)). For every pair of such functions, the derivatives f’ and g’ have a special relationship. Learn about this relationship and see how it applies to 𝑒ˣ and ln(x) (which are inverse functions!).

**What is the derivative with respect to the inverse sine of X?**

The derivative with respect to X of the inverse sine of X is equal to one over the square root of one minus X squared, so let me just make that very clear. If you were to take the derivative with respect to X of both sides of this, you get dy,dx is equal to this on the right-hand side.

#### How does Sal evaluate the derivative of the inverse of G?

Given a table of values of g, its inverse h, and its derivative g’, Sal evaluates the derivative of the inverse, h’, at a given x-value. This is the currently selected item. – [Voiceover] Let G and H be inverse functions.

#### How do you interpret the derivative of a graph?

Another common interpretation is that the derivative gives us the slope of the line tangent to the function’s graph at that point. Learn how we define the derivative using limits.

**What is the derivative of G with respect to X?**

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.