A winter storm warning has been issued for the region including NYC, beginning tonight and extending to Thursday. It is not clear right now (as I write) how and how badly NYC will be affected. It’s a good idea to monitor the situation – check weather forecasts as the storm approaches tonight.

Also, if you have not already done so, it’s a good idea to sign up for CUNY Alerts. You can do that here.

For the notes from the class, see this post.

Homework:
• Make sure that your citytech email address is in the User information in WeBWorK. There are still a few students who have not done this. (It was supposed to be done at the start of the semester.)

• Review the use of the various derivative rules, particularly the Product Rule and the Quotient Rule

• Do the WeBWorK on Derivatives-ProductQuotient, and at least start the assignment on Higher Derivatives. Problem 8 in Derivatives-ProductQuotient is extra credit.

• Use the Quotient Rule to show that the derivative of $\cot(x)$ is $-\csc^{2}(x)$, and the derivative of $\sec(x)$ is $\tan(x)\sec(x)$ (remember that $\sec(x) = \frac{1}{\cos(x)$).

• Also do the following problems from the textbook:

 P. 94: 15, 17, 18, 19, 23, 25, 29, 31+32 (tangent line only), 34, 39

• There will be a Quiz on Wednesday: it will be on the Product Rule and/or Quotient Rule (and you also need to know the other rules then, of course.)

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Topics:

• Review of the derivative rules (so far)

• The Product Rule and the Quotient Rule

• Notation for higher derivatives

My slide show (with the corrected derivative for cosine) is here:

MAT1475DerivativeRules

Here is a video which uses the “LO dee HI” formula for the Quotient Rule:

Examples I worked:

Derivative of $f(x) = x^{3}e^{x}$ using the Product Rule:

$\displaystyle \frac{\textrm{d}}{\textrm{d}x}\left(x^{3}e^{x}\right) = x^{3}\frac{\textrm{d}}{\textrm{d}x}\left(e^{x}\right) + e^{x}\frac{\textrm{d}}{\textrm{d}x}\left(x^{3}\right)$

$ = x^{3}e^{x} +e^{x}3x^{2} = e^{x}\left(x^{3} + 3x^{2}\right)$

Derivative of a product of polynomials (two ways) – first by using the Product Rule, and then by  multiplying it out and taking the derivative. See Example 51 in section 2.4 in the textbook. (This is not the one I did, but it’s similar.)

Derivative of a rational function using the Quotient Rule.  A similar example is in the video above. (Also see Example 58 in section 2.4 in the textbook, which shows how sometimes you may be able to avoid using the Quotient Rule.)

Derivative of $\tan{x}$ using the Quotient Rule. See Example 55 in section 2.4 in the textbook.

Derivative of $f(x) = x\ln(x)$ using the Product Rule

Derivative of $g(x) = x\ln(x) – x$ For both this and the previous, see Example 53 in section 2.4 in the textbook.

Then we found the first, second, and third derivatives of $f(t) = -16t^{2} + 150$, our position function for the object falling from a height of 150 feet.

For the homework assignment please see this separate post.