第3章微分

As we have seen, the derivative of a function at a given point gives us the rate of change or slope of the tangent line to the function at that point. If we differentiate a position function at a given time, we obtain the velocity at that time. It seems reasonable to conclude that knowing the derivative of the function at every point would produce valuable information about the behavior of the function. However, the process of finding the derivative at even a handful of values using the techniques of the preceding section would quickly become quite tedious. In this section we define the derivative function and learn a process for finding it.

The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows.

A function $f(x)$ is said to be differentiable at $a$ if $f^{\prime }(a)$ exists. More generally, a function is said to be differentiable on $S$ if it is differentiable at every point in an open set $S,$ and a differentiable function is one in which $f^{\prime }(x)$ exists on its domain.

In the next few examples we use Equation 3.9 to find the derivative of a function.

We use a variety of different notations to express the derivative of a function. In Example 3.12 we showed that if $f\left(x\right)=x^2-2x,$ then $f^{\prime }\left(x\right)=2x-2.$ If we had expressed this function in the form $y=x^2-2x,$ we could have expressed the derivative as $y^{\prime }=2x-2$ or $\frac{dy}{dx}=2x-2.$ We could have conveyed the same information by writing $\frac{d}{dx}\left(x^2-2x\right)=2x-2.$ Thus, for the function $y=f\left(x\right),$ each of the following notations represents the derivative of $f\left(x\right)\text{:}$

In place of $f^{\prime }\left(a\right)$ we may also use $\frac{dy}{dx}|\begin{array}{l}\\ {}_{x=a}\end{array}$ Use of the $\frac{dy}{dx}$ notation (called Leibniz notation) is quite common in engineering and physics. To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line. The slopes of these secant lines are often expressed in the form $\frac{\text{Δ}y}{\text{Δ}x}$ where $\text{Δ}y$ is the difference in the $y$ values corresponding to the difference in the $x$ values, which are expressed as $\text{Δ}x$ (Figure 3.11). Thus the derivative, which can be thought of as the instantaneous rate of change of $y$ with respect to $x,$ is expressed as

We have already discussed how to graph a function, so given the equation of a function or the equation of a derivative function, we could graph it. Given both, we would expect to see a correspondence between the graphs of these two functions, since $f^{\prime }(x)$ gives the rate of change of a function $f\left(x\right)$ (or slope of the tangent line to $f(x)).$

In Example 3.11 we found that for $f\left(x\right)=\sqrt{x},f^{\prime }(x)=1\text{/}2\sqrt{x}.$ If we graph these functions on the same axes, as in Figure 3.12, we can use the graphs to understand the relationship between these two functions. First, we notice that $f(x)$ is increasing over its entire domain, which means that the slopes of its tangent lines at all points are positive. Consequently, we expect $f^{\prime }\left(x\right)>0$ for all values of $x$ in its domain. Furthermore, as $x$ increases, the slopes of the tangent lines to $f(x)$ are decreasing and we expect to see a corresponding decrease in $f^{\prime }(x).$ We also observe that $f\prime \left(0\right)$ is undefined and that $\underset{x\to 0^{+}}{\text{lim}}{f}^{\prime }\left(x\right)=\text{+}\infty ,$ corresponding to a vertical tangent to $f(x)$ at $0.$

In Example 3.12 we found that for $f\left(x\right)=x^2-2x,f^{\prime }\left(x\right)=2x-2.$ The graphs of these functions are shown in Figure 3.13. Observe that $f(x)$ is decreasing for $x<1.$ For these same values of $x,f^{\prime }\left(x\right)<0.$ For values of $x>1,f(x)$ is increasing and $f^{\prime }\left(x\right)>0.$ Also, $f(x)$ has a horizontal tangent at $x=1$ and $f^{\prime }\left(1\right)=0.$