(A new question of the week) We are often asked to help a student understand a solution to a problem, obtained from a book or a website, that is not fully explained there. Here, we’ll look at a rather odd demonstration that a function satisfies a differential equation, both figuring out what the author did, …
In response to a recent request for information about implicit differentiation (hi, Brian!), let’s take a look at that topic. It happens to be distantly related to Friday’s topic, which was about implicitly defined curves. We’ll start with a thorough explanation, and then look at several specific examples, capping it off with a weird one.
(A new question of the week) An interesting question we received in mid-January concerned two implicit derivative problems with an unusual feature: the derivative we are seeking disappears! How do you track down such elusive quarry? Each case is a little different.
(A new question of the week) Often the hardest part of solving a problem is interpreting what it means. Math is precise; human language can be ambiguous, and assumptions can be hidden. Today, we look at a multi-variable calculus problem that looked enough like a classic single-variable maximization problem to fool the reader into not …
To close out this series on the definition of the derivative, I want to look at a few questions about alternative versions of the definition, primarily the “symmetric difference quotient”. We’ll see that this leads to a slightly different result, not always equivalent to the original, and we’ll observe some associated ways that calculators can …
(New questions of the week) We’ve had a number of brief discussions recently, which feel too small on their own for a post; but several happen to be dealing with similar types of issues. These four questions, all from July, involve limits or derivatives at edges or holes in the domain of a function. Let’s …
Last week we looked at the meaning of the derivative. In doing so, we mostly used the notation “”, but mentioned another in passing. Here I want to look at our answers to several questions about the different notations you will see. Several different notations We’ll start with this question from 2013: Differences in Differentiation …
To start a series of posts on differentiation (one of the basic concepts studied in calculus), I’d like to look at a number of answers we’ve given to the basic question, What is a derivative? This includes questions about the meaning of the concept and its definition, as well as examples. Something all these questions …