Unit 3 : Trigonometry and the Chain Rule
The chain rule is the most complex of the differentiation procedures.
It is also the most powerful: with it you will be able to
differentiate any algebraic function. By a trick of notation, it
looks innocuous enough:
Just cancel the du s. It should be stressed that this is a
trick of the notation; the chain rule is actually something that needs
to be proven.
The chain rule allows us to expand our previous, rather limited,
power rule. Let us say we want to differentiate
without multiplying the entire mess out. Let's introduce a ``dummy''
(a fancier term is ``auxiliary'')
We now have
We can also calculate
The chain rule then gives
A messy answer, true, but an answer we can get. Thus, our new and
improved power rule is
Note that this contains our old power rule as the special case u=x,
in which case dy/dx=1. In this unit's reading, the power rule is
extended to take care of any rational n. We'll extend it to all
real n eventually.
So that we have plenty of functions to differentiate, we also
introduce the derivatives of the trigonometric functions this unit:
The chain rule expands these to
For example, for
One final part of this unit is implicit differentiation: we want
dy/dx, but we don't have y expressed as a function
of x. For example, the equation that represents a circle of radius
a in the x-y plane is
where a is a constant. Implicit differentiation is simple: just
take derivatives of every term:
(In the above, a2 is a constant, so its
derivative is zero.) We can then
solve to get
This unit contains several new concepts. Now is the time to
master them. A few pointers to help you in doing so:
- Remember the du/dx tacked onto the end of the power
rule and the trig rules.
- When functions are deeply nested within each other, you aren't
through differentiating until you've gone all the way to the inside.
- When doing implicit differentiation, remember to differentiate
both sides of the equation.
After completing this unit you should be able to
Differentiate trig functions.
Use the chain rule to differentiate composite functions.
Use implicit differentiation to find derivatives of implicitly
- Read Simmons 3.3 - 3.5, 9.1, 9.2, 9.4
- Read World Web Math Pages
- Work lots of problems. The specific problems don't matter, but
you need to work enough of these things to be able to differentiate
in your sleep. Problems are located in Simmons:
- 3.3 : #1-6
- 3.4 : #1-8
- 3.5 : #1-12
- 9.2 : #1-14, 19-43 (chose a few)
- 9.4 : #1-8
- 9.5 : #4-12
- Take the Practice Unit
Test, Xdvi or PDF.
- Ask your instructor for a unit test.
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Last Modified July 31, 1998