Understanding the Derivative of the Inverse Function and Its Calculation

(f^-1(x))’ –> derivative of inverse function

To find the derivative of the inverse function, we first need to understand what the inverse function represents

To find the derivative of the inverse function, we first need to understand what the inverse function represents. The inverse of a function f(x) is denoted as f^-1(x) and represents a function that undoes the action of the original function.

Now, let’s find the derivative of the inverse function (f^-1(x))’.

To clarify, let’s use a new notation to avoid confusion. Let y = f(x) be the original function, and let x = f^-1(y) be its inverse.

We want to find the derivative dy/dx of the inverse function. This can be computed using the chain rule of differentiation, which states that the derivative of the composite of two functions is equal to the product of their derivatives.

The chain rule can be expressed as follows:
(dy/dx) = (dy/dy) * (dy/dx)

Since x = f^-1(y), we can rewrite the equation as:
(dy/dx) = (1/dy/dx) * (dy/dy)

Notice that we used the fact that dx/dy = 1/(dy/dx) since the derivatives of x and y are inverses of each other.

Now, let’s solve for (dy/dx). Since we are finding the derivative of the inverse function, we want to express (dy/dx) in terms of x. To do this, we can rewrite it as (dx/dy).

(dy/dx) = 1 / (dx/dy)

Finally, we have the derivative of the inverse function:
(f^-1(x))’ = 1 / (df(x)/dx)

To summarize, to find the derivative of the inverse function (f^-1(x)), you take the reciprocal of df(x)/dx, the derivative of the original function f(x). Remember to use appropriate notation to avoid confusion when working with inverse functions and their derivatives.

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