Calculus How To

Inverse Function: Definition, Derivative

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Types of Functions >

What is an Inverse Function?

An inverse function is a function that undoes another function; you can think of a function and its inverse as being opposite of each other.

The slopes of inverse linear functions are multiplicative inverses of each other. For example, a linear function that has a slope of 4 has an inverse function with a slope of 14.

Finding Derivatives for Inverse Functions

The multiplicative inverse fact above means that you can find the derivative of inverse functions by using a little geometry. Or, you could find the derivative of inverse functions by finding the inverse function for the derivative and then using the usual rules of differentiation to differentiate the inverse function.

However, there is a third option: using the formula to find the derivative of inverse functions.

Example problem: Find the derivative of the inverse function for the following function:

Part One: Find the inverse function (non-formula)

Step 1: Swap f(x) for y:

Step 2: Switch x and y:

Step 3: Solve for y, using algebra:

  • x2 = y-3
  • y = x2 + 3

Solution: y = x2 + 3.

Note: You could find the derivative of the inverse function at this point, using the usual rules for differentiation. Continue with the Steps if you want to use the formula.

Part Two: Using the Formula for the Derivative of Inverse Functions

Step 4: Calculate the derivative for the original function. Use the chain rule for this example problem.

Step 5: Insert your answer from Step 4 into the derivative of inverse functions formula:

Step 6: Replace the “x” from your answer in Step 5 with the function from Step 3:

Note that the square and square root will cancel, so will the 3s, leaving 2x as the derivative of the function.

That’s it!

Tip: In order for the derivative of the inverse function to work, the function must be differentiable at f-1(x) and f′(f-1(x)) is not equal to zero.
Warning: If you flip the graph of this sample function, you only get half of the parabola. Therefore, this particular inverse only holds for x>0.
derivative of inverse functions

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Stephanie Glen. "Inverse Function: Definition, Derivative" From Calculus for the rest of us!

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