Inverse Derivative Calculator

Category: Calculus
- December 06, 2024
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What is an Inverse Derivative?

The inverse derivative helps calculate the derivative of the inverse of a given function. For a function ( f(x) ), the derivative of its inverse, ( f^{-1}(x) ), is determined using the formula:

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

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

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

Solving for ( (f^(-1)(x))' ), we obtain:

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

This concept is particularly useful in calculus for analyzing how fast an inverse function changes at a specific point.

Features of the Inverse Derivative Calculator

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

How to Use the Inverse Derivative Calculator

  1. Enter a Function: Input the function ( f(x) ) whose inverse derivative you want to calculate. For example: x^2 + 1 or e^x.
  2. Specify an ( x )-Value: Enter the point where you want to calculate the derivative of the inverse function.
  3. Click Calculate: View 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 works.

Example Walkthrough

Suppose you want to compute 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 compute the inverse derivative of complex functions.
  • Visualize the function and its inverse derivative on an interactive graph.
  • Understand the process through step-by-step solutions.