# Derivative of an inverse question

• 3.141592654
In summary, the problem is asking to find the derivative of the inverse function h(x) of f(x)=(x^3)+x, denoted as h'(2), using the equation (F^{-1})'(x)=\frac{1}{F'(F^{-1}x)}. After realizing that h(2)=1, it is then necessary to find f'(x) and f'(1) to solve for h'(2) using the equation.
3.141592654

## Homework Statement

f(x)=(x^3)+x. If h(x) is the inverse of f(x), find h'(2).

## Homework Equations

(F$$^{-1}$$)'(x)=$$\frac{1}{F'(F^{-1}x)}$$

## The Attempt at a Solution

I want to find h'(2)=(F$$^{-1}$$)'(2)=$$\frac{1}{F'(F^{-1}(2))}$$

I know f'(x)=3(x^2)+1, so I just need to find h(2), but I don't know how to solve f(x)=(x^3)+x for its inverse. Is it possible to solve for x and then switch x and y with this type of function?

It might be difficult finding the inverse function, but it's easy to see that f(1)=2, so h(2)=1.

Since h is the inverse of f, h(f(x)) = x
Differentiating, you get h'(f(x))*f'(x) = 1, so h'(f(x)) = 1/f'(x).

Can you work in grief's comment that f(1) = 2 to find h'(2)? You will also need to find f'(x), and from that f'(1).

## What is the derivative of an inverse function?

The derivative of an inverse function is the reciprocal of the derivative of the original function.

## How do you find the derivative of an inverse function?

To find the derivative of an inverse function, you can use the formula (f^-1)'(x) = 1 / f'(f^-1(x)).

## What is the difference between the derivative of an inverse function and the derivative of a regular function?

The derivative of an inverse function is the reciprocal of the derivative of the original function, while the derivative of a regular function is simply the slope of the tangent line at a given point.

## Can you find the derivative of any inverse function?

Yes, the derivative of an inverse function can be found for any function as long as the function is one-to-one and differentiable.

## Why is the derivative of an inverse function important?

The derivative of an inverse function is important because it allows us to find the rate of change of the original function at a given point, and it also helps us find the slopes of tangent lines on inverse functions, which are useful in many applications such as optimization and curve sketching.

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