SUNY Geneseo Department of Mathematics
Math 221 02
Fall 2020
Prof. Doug Baldwin
This discussion consolidates your understanding of the product and quotient rules for derivatives. Our textbook describes these rules in “The Product Rule” and “The Quotient Rule” in section 3.3.
Respond to at least one of the following by class time on Monday, September 28.
Use the product rule to find the derivative of
\[f(t) = (t^2-2)(t+3)\]The textbook uses the fact that the derivative of x2 is not the product of the derivative of x with itself to show that derivatives of products must be more complicated than just the product of the derivatives. Can you do the reverse to check the product rule? In other words, if you use the product rule to calculate
\[\frac{d}{dx}(x \cdot x)\]do you get 2x, the derivative of x2?
As you no doubt know, the area of a rectangle of width w and height h is wh. Suppose a rectangle is growing in such a way that the change in width with respect to time is 2 inches per second, and the change in height with respect to time is 4 inches per second. How fast is the area changing at the moment the rectangle’s width is 10 inches and height is 15 inches? (Hint: width and height are both functions of time, although you don’t know exactly what those functions are.)
Use the quotient rule to find the derivatives of...
\[g(x) = \frac{2x^2 + 1}{x-1}\] \[g(s) = \frac{s^2}{s^3+1}\]Use the quotient rule (not the book’s extended power rule) to find the derivative of
\[f(x) = \frac{1}{x^2}\]Is your result consistent with what the power rule would have given if it worked for negative exponents? Equivalently, is your answer the same as what you would get using the book’s extended power rule?
These examples extend the product and quotient rules to more sophisticated or complicated applications, building on the examples explored in September 28’s class meeting. Respond to one or more of the following by Thursday, October 1.
Find the derivative of
\[h(t) = \frac{(t^2+2)(6t-5)}{t-2}\]Assume for the moment that the derivative of sin x is cos x, and the derivative of cos x is -sin x (both of these are true, and we’ll see why shortly).
What is the derivative of x2 sin x?
What is the derivative of tan x?
Can you derive differentiation formulas for other trigonometric functions?