Preface Here are my online notes for my Calculus I course that I teach here at Lamar University. Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn Calculus I or needing a refresher in some of the early topics in calculus. I’ve tried to make these notes as self contained as possible and so all the information needed to read through them is either from an Algebra or Trig class or contained in other sections of the notes. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn calculus I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. You will need to find one of your fellow class mates to see if there is something in these notes that wasn’t covered in class.

2. Because I want these notes to provide some more examples for you to read through, I don’t always work the same problems in class as those given in the notes. Likewise, even if I do work some of the problems in here I may work fewer problems in class than are presented here.

3. Sometimes questions in class will lead down paths that are not covered here. I try to anticipate as many of the questions as possible when writing these up, but the reality is that I can’t anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that I’ve not included here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.

4. This is somewhat related to the previous three items, but is important enough to merit its own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.

Logarithmic Differentiation There is one last topic to discuss in this section. Taking the derivatives of some complicated functions can be simplified by using logarithms. This is called logarithmic differentiation. It’s easiest to see how this works in an example. Example 1 Differentiate the function.

( )

5

21 10 2xy

x x=

− +

Solution Differentiating this function could be done with a product rule and a quotient rule. However, that would be a fairly messy process. We can simplify things somewhat by taking logarithms of both sides.

( )

5

2ln ln

1 10 2xy

x x

= − +

Of course, this isn’t really simpler. What we need to do is use the properties of logarithms to expand the right side as follows.

( ) ( )( )( ) ( ) ( )

5 2

5 2

ln ln ln 1 10 2

ln ln ln 1 10 ln 2

y x x x

y x x x

= − − +

= − − − +

This doesn’t look all that simple. However, the differentiation process will be simpler. What we need to do at this point is differentiate both sides with respect to x. Note that this is really implicit differentiation.

( ) ( )

( )

12 24

152 2

2

1 2 25 10 21 10 2

5 101 10 2

x xy xy x x x

y xy x x x

−+′ −

= − −−

+

′= + −

− +

To finish the problem all that we need to do is multiply both sides by y and the plug in for y since we do know what that is.

( )

2

5

22

5 101 10 2

5 101 10 21 10 2

xy yx x x

x xx x xx x

′ = + − − +

= + − − + − +

Depending upon the person, doing this would probably be slightly easier than doing both the product and quotient rule. The answer is almost definitely simpler than what we would have gotten using the product and quotient rule. So, as the first example has shown we can use logarithmic differentiation to avoid using the product rule and/or quotient rule. We can also use logarithmic differentiation to differentiation functions in the form.

( )( ) ( )g xy f x=

Let’s take a quick look at a simple example of this. Example 2 Differentiate xy x= Solution We’ve seen two functions similar to this at this point.

( ) ( )1 lnn n x xd dx nx a a adx dx

−= =

Neither of these two will work here since both require either the base or the exponent to be a constant. In this case both the base and the exponent are variables and so we have no way to differentiate this function using only known rules from previous sections. With logarithmic differentiation we can do this however. First take the logarithm of both sides as we did in the first example and use the logarithm properties to simplify things a little.

ln lnln ln

xy xy x x

==

Differentiate both sides using implicit differentiation.

1ln ln 1y x x xy x′ = + = +

As with the first example multiply by y and substitute back in for y.

( )( )1 ln

1 lnx

y y x

x x

′ = +

= +

Let’s take a look at a more complicated example of this.

Example 3 Differentiate ( ) ( )cos1 3 xy x= − Solution Now, this looks much more complicated than the previous example, but is in fact only slightly more complicated. The process is pretty much identical so we first take the log of both sides and then simplify the right side. ( ) ( ) ( ) ( )cosln ln 1 3 cos ln 1 3xy x x x = − = −

Next, do some implicit differentiation.

( ) ( ) ( ) ( ) ( ) ( )3 3sin ln 1 3 cos sin ln 1 3 cos1 3 1 3

y x x x x x xy x x′ −

= − − + = − − −− −

Finally, solve for y′ and substitute back in for y.

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )cos

3sin ln 1 3 cos1 3

31 3 sin ln 1 3 cos1 3

x

y y x x xx

x x x xx

′ = − − + − = − − − + −

A messy answer but there it is. We’ll close this section out with a quick recap of all the various ways we’ve seen of differentiating functions with exponents. It is important to not get all of these confused.

( )

( )

( )

( ) ( )

1

0 This is a constant

Power Rule

ln Derivative of an exponential function

1 ln Logarithmic Differentiation

b

n n

x x

x x

d adxd x nxdxd a a adxd x x xdx

−

=

=

=

= +

It is sometimes easy to get these various functions confused and use the wrong rule for differentiation. Always remember that each rule has very specific rules for where the variable and constants must be. For example, the Power Rule requires that the base be a variable and the exponent be a constant, while the exponential function requires exactly the opposite. If you can keep straight all the rules you can’t go wrong with these.

Calculus I

© 2007 Paul Dawkins 5 http://tutorial.math.lamar.edu/terms.aspx