Overview
Functions are a powerful tool in mathematics: each describes a rule or process that takes any valid input to one and only one output. One of the natural questions that arises regarding any function is “can the rule be undone (or reversed)?” Indeed, this is connected to the concept of a function’s inverse. While not every function has an inverse, for those that do, knowing the inverse can be valuable in a wide range of settings.
Among the most important inverse functions in mathematics are the natural logarithm function, \(\ln(x)\) (which is the inverse of the exponential function, \(e^x\)) and the inverse trigonometric functions, such as \(\arcsin(x)\) and \(\arctan(x)\).
In this section of the text, we explore how we can use the relationship between a function and its inverse to determine the derivative formula for the inverse function and, along the way, learn a key geometric connection between a function’s derivative and the derivative of its inverse.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

Recall and know the definition of an inverse function.

Understand the horizontal line test and what it tells us about a function.

Know the basic facts about inverse functions that are stated in our text and in the online tutorial you will read below.

Understand and recognize that writing “\(y = f(x)\)” and “\(x = f^{1}(y)\)” say the exact same thing.

For a simple function \(y=f(x)\), find the function \(g(x)\) for which \(f(g(x))=x\) by solving \(y=f(x)\) for \(x\).

Know the key relationship between \(y = e^x\) and \(y = \ln(x)\).

Calculate basic values of the natural logarithm, arcsine, and arctangent functions without a calculator. (For example, \(\ln(e^5)\) and \(\arcsin(1/2)\).)
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Understand how the derivative of an inverse function is related to the derivative of the original function.

Know and use derivative rules for \(\ln(x)\), \(\arcsin(x)\), and \(\arctan(x)\).

Be able to use the relationship between a function and its inverse to develop the inverse function’s derivative rule (in particular: be able to do this “from scratch” for \(y=\ln x\), \(y=\arcsin x\), and \(y=\arctan x\))

Differentiate a function involving logarithmic functions, \(\arcsin\) and \(\arctan\) functions, and for which the derivative involves a combination of chain, product and quotient rules.
To prepare for class

Watch the following video to review some general facts about inverse functions (video created at CCSL):

Read and study the introduction and the first subsection of Section 2.6 (stop before subsection 2.6.2), as well as this online tutorial as a refresher on inverse functions.

Do the Preview Activity for section 2.6 (on WeBWorK if required by your teacher).

Read the second subsection of Section 2.6 (stop just before Activity 2.6.2).

Watch the following videos about the derivatives of inverse functions (the first and last videos were created at CCSL):

Do some experimentation with the following interactive applet: Derivatives of Inverse Functions