第3章微分

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 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.

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.

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, note that the technique used in this case is essentially the same as the technique used to prove the general case.

As we shall see, the procedure for finding the derivative of the general form $f\left(x\right)=x^n$ is very similar. Although it is often unwise to draw general conclusions from specific examples, we note that when we differentiate $f\left(x\right)=x^3,$ the power on $x$ becomes the coefficient of $x^2$ in the derivative and the power on $x$ in the derivative decreases by 1. The following theorem states that the power rule holds for all positive integer powers of $x.$ We will eventually extend this result to negative integer powers. Later, we will see that this rule may also be extended first to rational powers of $x$ and then to arbitrary powers of $x.$ Be aware, however, that this rule does not apply to functions in which a constant is raised to a variable power, such as $f(x)=3^x.$