Implicit Differentiation and Inverse Functions Discussion

This discussion invites you to explore how implicit differentiation can be used to find derivatives of functions whose inverse’s you already know a derivative of. For example, you know the derivative of sin x, so you can (it turns out) use implicit differentiation to find the derivative of arcsin x aka sin^-1x. Try solving (or asking questions about, commenting on, etc.) the following problems to start seeing how this idea works. We’ll pull it all together through this discussion and/or in class on Friday (October 16).

How about finding a derivative function for arcsin x (aka sin^-1x)? How would you set it up as an implicit differentiation problem?

Can you find a derivative function for arctan x (i.e., tan^-1x)?

Can you find a derivative function for ln x (remember that ln x is the inverse of e^x)?

Noting that the nth root of x, also written x^1/n, is the inverse of xⁿ, can you show that that power rule for derivatives also applies to roots?

Can you come up with a general rule for derivatives of inverse functions, i.e., a general rule for finding dy/dx if y = f^-1(x) and you know what f′(x) is?