Chapter Two: Section Four The Chain Rule

Chapter Two: Section Four Up to this point all of our derivative rules have considered functions of x in their definitions. What do we do if our function has a more complex argument? For example, consider the function y = (3x) 2.  If we think about the power rule we might be tempted to say that the derivative looks like 2(3 x )  If we expand the function to 9x 2 first and apply the constant product rule we get 18 x as the derivative.

Chapter Two: Section Four So, which of these derivatives is correct? Using the NDERIV function on your calculator (do you remember where it is found?) type the following two functions on your graphing calculator:

Chapter Two: Section Four Now, let’s look at the table of values that the calculator creates for us. Using my table set feature I asked the calculator to start at x = 0 and have a delta x of 1 as seen below:

Chapter Two: Section Four Your table of values should look like this: Where the values you are interested in are in the Y2 column. It looks like 18 x is the derivative of choice here.

Chapter Two: Section Four So, this means that the derivative of y = (3 x ) 2 is not simply 2(3 x ), but it is 2(3 x )*3. Make a guess about the derivative of each of the following functions and then check them with your calculator:  y = sin(2 x )  y = (4 x ) 5  y = (3x – 2) 2 [Hint: Check this algebraically first by expanding the binomial]  y = (sin x ) 2 Be prepared to discuss your findings in class.

Chapter Two: Section Four This chain rule, which relates the behavior of the derivative of a function to the idea of composition of functions, is a crucial tool for us. We will be working extensively with these kind of function compositions during the year. Ask some of your friends in Calculus BC, I think that they will verify that this is a pretty important derivative rule to look out for.