How to Find the Derivative of Inverse Functions

TL;DR

The derivative of an inverse function can be found using three methods: the formula (1 over the derivative of the original function), implicit differentiation, or directly differentiating the inverse function after finding it. Always ensure you're using values that correspond to the correct functions when calculating derivatives to avoid errors.

Transcript

now in this video we're going to focus on the derivative of inverse functions perhaps you have seen this formula in your textbook the derivative of the inverse function with respect to x is one over the derivative so looking at this formula it might not always make sense you might see it and wonder what exactly does this mean so here's a question f... Read More

Key Insights

• 🫡 The derivative of the inverse function is equal to one over the derivative of the original function with respect to y.
• ❓ There are multiple methods to find the derivative of the inverse function: using the formula, implicit differentiation, and finding the inverse function first.
• 😒 It is important to use the correct values when plugging into the derivative formula, depending on whether the point corresponds to the original function or the inverse function.
• ☺️ The slope of the tangent line for the inverse function can be found by plugging the corresponding x value into the derivative formula.
• 🫥 The slope of the tangent line becomes approximately equal to the slope of the secant line as the two points on the secant line approach the point of tangency.
• ❓ The derivative of the inverse function can be found by finding the inverse function first and then differentiating it.
• ❓ It is important to be aware of the difference between the inverse function and the original function when finding the derivative of the inverse function.

Questions & Answers

Q: What is the formula for finding the derivative of the inverse function?

The formula is one over the derivative of the original function with respect to y.

Q: How can you find the derivative of the inverse function using implicit differentiation?

Switch the x and y variables in the original function, differentiate both sides with respect to x, and solve for the derivative of the inverse function.

Q: Can you find the derivative of the inverse function without finding the original function?

Yes, you can find the inverse function first and then differentiate it to find the derivative of the inverse function.

Q: How do you know which values to use when plugging into the derivative formula?

For the inverse function, use the x value of the original function if finding the derivative in terms of y, and use the y value of the inverse function if finding the derivative in terms of x.

Summary & Key Takeaways

• The derivative of the inverse function is equal to one over the derivative of the original function.

• There are three methods to find the derivative of the inverse function: using the formula, implicit differentiation, and finding the inverse function first.

• It is important to use the correct values when plugging into the derivative formula, depending on whether the point corresponds to the original function or the inverse function.