# Seeing the Chain Rule

Being a believer in the Rule of Four, I have been trying for years to find a good visual (graphical) illustration of why or how the Chain Rule for derivatives works. This very simple example is the best I could come up with. Consider the function  . (See figure 1. A tangent segment at  is drawn.)…

# The Chain Rule

Most of the function students are faced with in beginning calculus are compositions of the Elementary Functions. The Chain Rule allows you to differentiate composite functions easily. The posts listed below are ways to introduce and use the Chain Rule. Experimenting with a CAS – Chain Rule  Using a CAS to discover the Chain Rule…

# The Chain Rule

Most of the function students are faced with in beginning calculus are compositions of the Elementary Functions. The Chain Rule allows you to differentiate composite functions easily. The posted listed below are ways to introduce and then use the Chain Rule. Experimenting with a CAS – Chain Rule  Using a CAS to discover the Chain…

# Good Question 10 – The Cone Problem

Today’s good question is an optimization problem, but its real point is choosing how to do the computation. As such it relates to MPAC 3a and 3b: “Students can  … select appropriate mathematical strategies [and] sequence algebraic/computational processes logically.” The algebra required to solve this questions can be quite daunting, unless you get clever. Here’s the…

# Power Rule Implies Chain Rule

Having developed the Product Rule  and the Power Rule  for derivatives in your class, you can explore similar rules for the product of more than two functions and suddenly the Chain Rule will appear. For three functions use the associative property of multiplication with the rule above: So expanding with a slight change in notation: For…

# Foreshadowing the Chain Rule

I assigned another very easy but good problem this week. It was simple enough, but it gave a hint of things to come. Use the Product Rule to find the derivative of . Since we have not yet discussed the Chain Rule, the Product Rule was the only way to go.  And likewise for higher…

# The Chain Rule

Except for the simplest functions, a procedure known as the Chain Rule is very helpful and often necessary to find derivatives. You can start with an example such as finding the derivative of  .  Most students will expand the binomial to get and differentiate the result to get . They will try the same approach…