Basics Calculus – The Chain Rule (Grade A)

Summary:

The document titled “The Chain Rule” explains the concept and application of the chain rule in differentiation. The chain rule is a special rule that differentiates a function from another. The document introduces the topic and emphasizes the importance of practice in mastering the techniques. It covers the definition of a function, states the chain rule, and demonstrates how to differentiate such functions using examples, including trigonometric and exponential functions. The document also introduces a simple technique for directly differentiating functions of functions. Overall, it serves as a guide for understanding and applying the chain rule in calculus.

Excerpt:

Basics Calculus – The Chain Rule

A special rule, the chain rule, exists to differentiate a function from another. This unit illustrates this rule.

To master the techniques explained here you must undertake plenty of practice exercises so that they become second nature.

After reading this text, and/or viewing the video tutorial on this topic, you should be able to:
• explain what is meant by a function of a function
• state the chain rule
• differentiate a function from a function

Contents
1. Introduction 2
2. A function of a function 2
3. The chain rule 2
4. Some examples involving trigonometric functions 4
5. A simple technique for differentiating directly 5

1. Introduction
In this unit, we learn how to differentiate a ‘function of a function’. We first explain what this term means and then learn about the Chain Rule, which is the technique used to perform the differentiation.
2. A function of a function
Consider the expression cos x2. Immediately we note that this differs from the straightforward cosine function, cos x. We are finding the cosine of x2, not simply the cosine of x. We call such an expression a ‘function of a function’.

Suppose, in general, that we have two functions, f(x) and g(x). Then y = f(g(x))

is a function of a function. In our case, the function f is the cosine function, and the function g is the square function. We could identify them more mathematically by saying that
f(x) = cos x g(x) = x 2
so that
f(g(x)) = f(x2) = cos x2