Differentiation using the Chain Rule

1
Differentiation using the Chain Rule
2
What you need to know

• This section will aim to teach the use of the
  chain rule in differentiation.
• This section extends the work done in the two
  main sections.
• IMPORTANT NOTE MAKE SURE YOU ARE COMFORTABLE
  WITH BOTH OF THE PREVIOUS SECTIONS, INTRODUCTION
  TO DIFFERENTIATION AND DIFFERENTIATION FROM
  FIRST PRINCIPLES, BEFORE PROCEEDING.

3
The Chain Rule Introduced

• From the exercise at the end of Differentiation
  from First Principles you should have
  established the following
• For yaxn, ?y naxn-1. ?x
• This rule is easy to implement when you are
  working with simple functions, but it is possible
  that you would need to find the derivative of a
  composite function, such as y(x3)3. In cases
  like this you would need to multiply out the
  brackets before you can proceed. This can get
  quite complicated.
• In addition, some composite functions are unable
  to be expanded in this way, for example those
  comprised of roots or negative powers etc.

4
The Chain Rule

• So, how can we deal with composite functions?
• The easiest way is to use the chain rule, which
  avoids this problem.
• The chain rule is as follows
• dy dy x du dx du
  dx
• The next slide will show how the chain rule is
  derived.

5
Deriving the Chain Rule

• If we have y f(x) where f is a composite
  function of x, we can say that we have yf(u),
  where u is a function of x, u(x), which is a
  component of the function f. As such, we have
  yf(u(x)). From this we can establish the
  following
• dy limit ?y limit ?y x ?u
  dx ?x?0 ?x ?x?0 ?u ?x
• This is because there is a corresponding change
  in the value of u, ?u each time there is a change
  in x, ?x.
• This means that as ?x?0, ?u also tends to 0.
• dy limit ?y x limit ?u
  dx ?u?0 ?u ?x?0 ?x
• This then leads us to the Chain Rule as shown on
  the previous slide.

6
The Chain Rule in Action

• So, how do we implement the Chain Rule?
• dy dy x du dx du dx
• Suppose we want to calculate the function
  y(x2)4
• First we need to decide what u stands for.
• In this case we decide to let yu4, where ux2.
• Then we calculate dy , which gives us 4u3
  du
• Then we calculate du , which gives us 1
  dx
• Can you see why this is?

7
The Chain Rule in Action 2

• Now that we have the two derivatives, we can use
  the Chain Rule
• dy dy x du 4u3 x 1 4u3
  dx du dx
• Now we can substitute u for the original function
  x1
• This gives us the derivative 4(x1)3
• Are you satisfied that you see how this works?

8
Exercise 3 (see Answers.ppt for worked answer)

• Try to find the derivative of the following
  composite functions
• y (4x-2)6
• y 1 1 5x
  2
• y ?(4x-3)