3.2 The derivative as a function  (Page 2/10)

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 $\text{Δ}x$ ( [link] ). Thus the derivative, which can be thought of as the instantaneous rate of change of $y$ with respect to $x,$ is expressed as

Graphing a derivative

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 [link] 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 [link] , 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 $f\left(0\right)$ 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 [link] 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 [link] . 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 $f^{\prime }\left(1\right)=0.$

Sketching a derivative using a function

Use the following graph of $f\left(x\right)$ to sketch a graph of $f^{\prime }\left(x\right).$

The solution is shown in the following graph. Observe that $f(x)$ is increasing and $f^{\prime }\left(x\right)>0$ on $\left(–2,3\right).$ Also, $f(x)$ is decreasing and $f^{\prime }\left(x\right)<0$ on $\left(\text{−}\infty ,\mathrm{-2}\right)$ and on $\left(3,\text{+}\infty \right).$ Also note that $f(x)$ has horizontal tangents at $–2$ and $3,$ and $f^{\prime }\left(\mathrm{-2}\right)=0$ and $f^{\prime }\left(3\right)=0.$

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Derivatives and continuity

Now that we can graph a derivative, let’s examine the behavior of the graphs. First, we consider the relationship between differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there; however, a function that is continuous at a point need not be differentiable at that point. In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons.