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:
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.
