3.3 Differentiation rules

Finding derivatives of functions by using the definition of the derivative can be a lengthy and, for certain functions, a rather challenging process. For example, previously we found that $\frac{d}{dx}\left(\sqrt{x}\right)=\frac{1}{2\sqrt{x}}$ by using a process that involved multiplying an expression by a conjugate prior to evaluating a limit. The process that we could use to evaluate $\frac{d}{dx}\left(\sqrt[3]{x}\right)$ using the definition, while similar, is more complicated. In this section, we develop rules for finding derivatives that allow us to bypass this process. We begin with the basics.

The basic rules

The functions $f\left(x\right)=c$ and $g\left(x\right)=x^n$ where $n$ is a positive integer are the building blocks from which all polynomials and rational functions are constructed. To find derivatives of polynomials and rational functions efficiently without resorting to the limit definition of the derivative, we must first develop formulas for differentiating these basic functions.

The constant rule

We first apply the limit definition of the derivative to find the derivative of the constant function, $f\left(x\right)=c.$ For this function, both $f\left(x\right)=c$ and $f\left(x+h\right)=c,$ so we obtain the following result:

The rule for differentiating constant functions is called the constant rule    . It states that the derivative of a constant function is zero; that is, since a constant function is a horizontal line, the slope, or the rate of change, of a constant function is $0.$ We restate this rule in the following theorem.

The power rule

We have shown that

At this point, you might see a pattern beginning to develop for derivatives of the form $\frac{d}{dx}\left(x^n\right).$ We continue our examination of derivative formulas by differentiating power functions of the form $f\left(x\right)=x^n$ where $n$ is a positive integer. We develop formulas for derivatives of this type of function in stages, beginning with positive integer powers. Before stating and proving the general rule for derivatives of functions of this form, we take a look at a specific case, $\frac{d}{dx}(x^3).$ As we go through this derivation, pay special attention to the portion of the expression in boldface, as the technique used in this case is essentially the same as the technique used to prove the general case.