Calculus Formulas Essay

Submitted By NasVan
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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.

Differentiation Formulas
In the first section of this chapter we saw the definition of the derivative and we computed a couple of derivatives using the definition. As we saw in those examples there was a fair amount of work involved in computing the limits and the functions that we worked with were not terribly complicated. For more complex functions using the definition of the derivative would be an almost impossible task. Luckily for us we won’t have to use the definition terribly often. We will have to use it on occasion, however we have a large collection of formulas and properties that we can use to simplify our life considerably and will allow us to avoid using the definition whenever possible.
We will introduce most of these formulas over the course of the next several sections. We will start in this section with some of the basic properties and formulas. We will give the properties and formulas in this section in both “prime” notation and “fraction” notation.
Properties
1)

( f ( x ) ± g ( x ) )¢ = f ¢ ( x ) ± g ¢ ( x )

OR

d df dg
( f ( x ) ± g ( x ) ) = dx ± dx dx In other words, to differentiate a sum or difference all we need to do is differentiate the individual terms and then put them back together with the appropriate signs. Note as well that this property is not limited to two functions.
See the Proof of Various Derivative Formulas section of the Extras chapter to see the proof of this property. It’s a very simple proof using the definition of the derivative.
2)

( cf ( x ) )¢ = cf ¢ ( x )

OR

d df ( cf ( x ) ) = c dx , c is any number dx In other words, we can “factor” a multiplicative constant out of a derivative if we need to.
See the Proof of Various Derivative Formulas section of the Extras chapter to see the proof of this property.
Note that we have not included
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