# Differentiation: Composite, Implicit, and Inverse Function – Unit 3

This is a re-post and update of the third in a series of posts from last year. It contains links to posts on this blog about the differentiation of composite, implicit, and inverse functions for your reference in planning. Other updated post on the 2019 CED will come throughout the year, hopefully, a few weeks…

# Teaching and Learning Theorems

Theorems are carefully worded statements about mathematical facts that have been proved to be true. Important (and some not so important) ideas in calculus and all of mathematics are summarized as theorems. When you come across a theorem you need to understand it; the author of your textbook would not have included it and the…

# Inverses Graphically and Numerically

In this final post in this series on inverses we consider the graphical and numerical concepts related to the derivative of the inverse and look at an important formula. To make the notation a little less messy, let’s let g(x) = f -1(x). Then we know that f (g(x))= x.  Differentiating this implicitly gives Great formula, but one…

# The Calculus of Inverses

Today we will consider computing the derivative of the inverse of a function. This is pretty standard and is in all the textbooks. The usual suspects are the inverse trigonometric functions. So let’s start with  and then rewrite this as . Differentiating this gives Since we would like this in terms of x we can…

# The Range of the Inverse

The last two post discussed inverse functions and some concerns about them. We continue that today be considering that fact that sometimes the inverse of a function is not a function, and what can be done in that case. Since the square of both 3 and –3 is 9. Which number should you get when…

# Writing Inverses

In my last post I identified two “problems” related to inverses. The first of these is that there may be no string of operations, no algebra or arithmetic, that tells us how to evaluate the inverse function. For simple functions you can find the inverse function by switching the x and the y and then…

# Inverses

The next few posts will concern functions and their inverses. Today we will just get into the basics which, hopefully, students know from their work prior to calculus. Of course they will have forgotten some of this and claim they never learned it. (Which may be correct, but one hopes they were taught it.) This…