## Derivatives of Inverse Functions (at inverse points)

### To understand the concept of derivatives of inverse functions at inverse points, we should first discuss what an inverse function is

To understand the concept of derivatives of inverse functions at inverse points, we should first discuss what an inverse function is. An inverse function essentially undoes the effect of the original function. If we have a function f(x), its inverse is denoted as f^(-1)(x), and it swaps the roles of the inputs and outputs of the original function. In other words, if f(x) takes an input x and gives an output y, then f^(-1)(y) gives an input x that produces the output y.

Now, let’s talk about the derivative of a function. The derivative of a function f(x) at a specific point x=a gives us the instantaneous rate of change of the function at that point. It tells us how fast or slow the function is changing at that specific point.

When it comes to the derivative of inverse functions, we can say that the derivative of the inverse function at a certain point is the reciprocal of the derivative of the original function at the corresponding inverse point.

To illustrate this, let’s say we have a function f(x) and its inverse function f^(-1)(x). If we take a point (a, b) on the graph of f(x), it means that f(a) = b. Now, if we take the corresponding point (b, a) on the graph of f^(-1)(x), it means that f^(-1)(b) = a.

According to the derivative concept, the derivative of f(x) at x=a is given by f'(a), and the derivative of f^(-1)(x) at x=b is given by f’^(-1)(b). The relationship between the derivatives is as follows:

f’^(-1)(b) = 1 / f'(a)

This result highlights the inverse relationship between the derivatives of a function and its inverse function. The derivative of the inverse function at a certain point is the reciprocal of the derivative of the original function at the corresponding inverse point.

It is important to note that this concept only applies when both the function and its inverse function are differentiable at the specific points of interest.

So, when differentiating inverse functions at inverse points, remember to find the derivative of the original function at the point where it intersects with the inverse function’s graph, and take the reciprocal of that to find the derivative of the inverse function at the respective point.

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