# Derivatives

## Implicit Differentiation: What to Do When It’s “Wrong”

(A new question of the week) Having just discussed the Chain Rule and the Product and Quotient Rules, a recent question about implicit differentiation (which we covered in depth two years ago) fits in nicely. This raises an important issue: when you get an apparently wrong answer, you may just have done something wise that …

## How to Think About the Product and Quotient Rules

Last time, we considered the Chain Rule for derivatives. This time, we’ll look at the product and quotient rules, focusing on how to keep the formulas straight, and make them easier to apply. We’ll look primarily at the quotient rule to start with, and then examine the product rule at the end.

## How to Think About the Chain Rule

Having recently helped some students (in person) with the rules of differentiation, I’m reminded to do so here, starting with the chain rule. It is easy to make this topic look harder than it really is; the two main ways to state the rule are often confusing, and different approaches fit different problems. We’ll try …

## More On Mixing Trig Functions

I’ve had several occasions in face-to-face tutoring lately to refer to a past post on mixing (that is, composition) of trig and inverse trig functions. Several recent questions have touched directly or indirectly on this same general idea and extended it, so I thought I’d post them.

## A Surprising Route to a Differential Equation

(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, …

## Implicit Differentiation: Explanation, Examples, and a Surprise

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.

## The Case of the Disappearing Derivative

(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.

## Maximum Volume of a Box: Two Interpretations

(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 …

## The Symmetric Derivative

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 …

## Limits and Derivatives on the Edge

(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 …