Inverse Derivative Calculator

What is an Inverse Derivative?

The inverse derivative is used to find the derivative of the inverse of a specific function. For a function ( f(x) ), the derivative of its inverse, ( f^{-1}(x) ), can be calculated using the formula:

( (f^(-1)(x))' = 1 / f'(f^(-1)(x)) )

This formula comes from the relationship ( f(f^(-1)(x)) = x ). By differentiating both sides with respect to ( x ), we derive:

( f'(f^(-1)(x)) * (f^(-1)(x))' = 1 )

Rearranging for ( (f^(-1)(x))' ), we get:

( (f^(-1)(x))' = 1 / f'(f^(-1)(x)) )

This idea is especially handy in calculus for understanding how quickly an inverse function changes at a certain point.

Features of the Inverse Derivative Calculator

• Detailed Steps: Enter a function and an ( x )-value to see a comprehensive step-by-step solution.
• Example Functions: Try the calculator with built-in functions like ( f(x) = x^2 + 1 ), ( f(x) = e^x ), or ( f(x) = ln(x) ).
• Graphical Visualization: The calculator displays both the function and its inverse derivative graphically.

How to Use the Inverse Derivative Calculator

1. Enter a Function: Input the function ( f(x) ) for which you want to calculate the inverse derivative. For instance: x^2 + 1 or e^x.
2. Specify an ( x )-Value: Enter the point at which you want to find the derivative of the inverse function.
3. Click Calculate: Check the result along with a detailed explanation of the calculation.
4. Explore Preloaded Examples: Use the dropdown menu to try out example functions and see how the calculator operates.

Example Walkthrough

Let’s say you want to find the inverse derivative of ( f(x) = x^2 + 1 ) at ( x = 2 ):

1. The derivative of ( f(x) ) is:

( f'(x) = 2 * x )

1. Evaluate ( f'(2) ):

( f'(2) = 2 * 2 = 4 )

1. Using the formula for the inverse derivative:

( (f^(-1)(x))' = 1 / f'(f^(-1)(x)) )

At ( x = 2 ), the inverse derivative is:

( (f^(-1)(2))' = 1 / 4 = 0.25 )

Key Benefits of Using This Calculator

• Quickly find the inverse derivative of complicated functions.
• Visualise the function and its inverse derivative on an interactive graph.
• Grasp the process through step-by-step solutions.