Worksheet for Section 3.1

Secton 3.1 begins the search for rules that can be used in computing derivatives. Here are some of the basic rules for derivatives, the first of which comes directly from Section 2.3 and the last of which is a pattern that began to emerge in that section:

• Constant Rule: $$\frac{d}{dx}[c]=0$$
• Constant Multiple Rule: $$\frac{d}{dx}[c\cdot f(x)] =c\cdot f'(x)$$
• Sum and Difference Rules: $$\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)$$
• Power Rule: $$\frac{d}{dx}[x^n]=nx^{n-1}$$

All of these can be proven using the limit definition of the derivative of a function, and I will do that in class for a couple of these rules.

These rules provide, in particular, everything necessary to compute the derivative of any polynomial function. Here are some examples -- in each case, compute the derivative of the function using the rules:

$$f(x)=2x^3-3x^2+7x-4\qquad g(x)=\sqrt{5}x^6-\frac{\pi x^3}{2}+7.33$$

The rules can do more, though -- for example, as seen in Chapter 2, $sqrt{x}$ can be treated as a power of $x$, so that the Power Rule above applies; also, expressions such as $1/x^2$ can be treated as powers of $x$, and again the Power Rule applies. Here is an example which includes expressions like this -- again, compute the derivative:

$$h(x)=\frac{5}{x^3}+\frac{1}{3\sqrt{x}}-7x\sqrt{x}$$