Worksheet for Section 3.3

Secton 3.3 is about derivatives of products and quotients. Unlike the rules for sums and differences in Section 3.1, these rules to not resemble the corresponding rules for limits of products and quotients. To begin, here are the two rules:

• Product Rule: $$\begin{aligned}\frac{d}{dx}[f(x)g(x)] &=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]\\ &=f(x)g'(x)+g(x)f'(x)\end{aligned}$$
• Quotient Rule: $$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] =\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}$$, as long as $g(x)\neq 0$.

I will show you in class where these rules come from. It is important that you memorize these rules -- there are patterns you may notice, and I have even heard songs and rhymes that some people use. Here are some examples using these rules -- in each case, compute the indicated derivative:

$$<tex2html_comment_mark> \frac{d}{dx}[xe^x]\qquad \frac{d}{dx}[(x-... ...+3)]\qquad \frac{d}{dx}[3x\sin x - 2\cos x]\qquad \frac{d}{dx}[x^2e^x-2x\cos x]$$

$$\frac{d}{dx}\left[\frac{2x+1}{x^2+2}\right]\qquad \frac{d}{dx}\le... ...ft[\frac{\sin x}{\cos x}\right]\qquad \frac{d}{dx}\left[\frac{x^3-3x}{7}\right]$$

It is important to notice that you do not have to use the Quotient Rule when the denominator of the fraction is a constant -- you can simply use the Constant Multiple Rule from Section 2.2, which makes the computation much simpler (this is what is going on in the last example above -- try it with the Quotient Rule, too, to see the difference). Also, don't forget to simplify the expressions that each of these rules gives you.