On Friday, Quiz #5 happens. It covers all the derivative shortcuts to date: Power Rule, Product Rule, Quotient Rule, and Chain Rule, and all those rules about constants, addition, and subtraction. That was all in Sections 3.3, 3.4, and 3.5.

Study hard!

(My apologies to Sam Cooke. Look it up.)

Thursday, we talked about a way of taking derivatives of compositions of functions. Suppose where f and g are differentiable functions. Then the composed function F is differentiable and



That is we take the derivative of the "outer" function f and multiplying by the derivative of the "inner" function g. This is known as the Chain Rule.

For example, if we want to take the derivative of , we can think of the outer function as being "something to the 100th power." So the derivative should start off with "100 times the same something to the 99th power." We then multiply by the derivative of the inner function; that is, . So we have that . (In the special case where the outer function is a power function, the Chain Rule is often called the General Power Rule: .) It certainly beats writing 100 rows of Pascal's Triangle in this case!

We looked at another example or two of the Chain Rule. Remember what Prof. LaVarnway says: "Not a day goes by that you don't use the Chain Rule!"

On all of these the difficulty comes from (1) remembering the rules and (2) finishing the algebra in a satisfactory manner. We've been lax about algebra up until this point, but we will now tighten up for the following reason.

If you do your homework, you often gauge how well you do by how often you agree with the answer in the back of the book (assuming that the right answer is there, which it most assuredly is not sometimes). Many times, we have the correct answer, but we don't know it because the book's author has performed an algebraic trick or two to get the answer as compact as possible. Therefore, I'm going to ask that you work on the algebraic aspects of your work as much as possible.

We'll learn about higher order derivatives and their meaning on Friday and take a quiz. See you then.

Wednesday in Calc I, we worked out formulas for the derivatives of the six trigonometric functions.

This lecture was an exercise in how and when to work from first principles. The trigonometric functions are not algebraic, so we can't get anything new from using, say, the power rule. We had to pick one of the two base trigonometric functions (in this case, we picked ) and worked from the definition of the derivative. After a few minutes -- using identities we saw in section 1.8 and limits we saw in section 2.5 -- we quickly got our derivative.

But once we saw how to do this, we could quickly get a derivative for cos x.

For the other four trig functions, we noted that they are all defined as quotients of sines and/or cosines. Therefore, we didn't have to go back to the actual definition of the derivative to work out the needed formulas, but to the Quotient Rule. We did a full calculation for and noted the others would be the same. What we found was:



These formulas can be used in the midst of the other rules we already know, like the Product Rule and Quotient Rule.

On Thursday, we'll begin talking about the Chain Rule -- not a day will go by for the rest of your lives when you won't use the Chain Rule. See you then.

The solutions to Quiz #4 have been posted and are available from the Test and Quiz Solutions page.

It was one of those weekends ... you'll get your quizzes back on Wednesday.

Lab #1 has been posted on the Computer Labs page. It is due on Wednesday, October 10th as an e-mail attachment. Especially since this is your first Mathematica assignment ever, please don't put this off until the last minute. Come for help often. Feel free to talk to each other and help one another out in the lab, but make sure your work is all your own. In particular, please do not do the lab and give the completed file to your friend to just put their name on.

Here is Tom Lehrer at the same celebration as before singing about limits: "There's a Delta for Every Epsilon." Enjoy!

(I should mention that these are excerpted from a video available on YouTube. You can get the full video there.)

Today we talked more about derivatives. Now that we're getting to "shortcuts," we should be able to differentiate a lot of different functions.

We added three fairly simple rules to our toolbox.



The fun began when we looked at the derivatives of products and quotients. From very simple examples (e.g. ), we saw that the derivative of a product is not the product of the derivatives, and the derivative of a quotient is not the quotient of the derivatives. What we got instead was:



These can be tough to memorize, but the most difficult part of using them can be the algebra in simplifying.

We'll look Wednesday at derivatives of trigonometric functions. We'll have to go back to the definition of the derivative once more for this. You'll live. See you Wednesday.