Section 6.2 Definition of the Derivative
Subsection 6.2.1 Limits of Secant Lines
In Section 3.1, I defined the derivative as the rate of change of a function. In Section 6.1, I connected that definition to the geometry of slopes of tangent lines and constructed a process by which algebra can approximate a tangent line by using secant lines, shown in Figure 6.1.4. Finally, since I have defined limits in Section 4.1, I can ask for the limit of that approximation process of secant lines.
I want to calculate the slope of the tangent line at a point \((a,f(a))\) on the graph of a function. I can take \(a\) as one point to define a secant line and let \(x\) be the other point (with \(x \neq a\)). Then the slope of the secant line is the difference in output (\(f(x) - f(a)\)) dividing by the difference in input (\(x-a\)).
\begin{equation*}
\frac{f(x) - f(a)}{x-a}
\end{equation*}
I said that the approximation should get closer and closer to the tangent line as \(x\) gets closer to \(a\text{.}\) Now I can ask for the limit as \(x \rightarrow a\text{.}\)
\begin{equation*}
\lim_{x \rightarrow a} \frac{f(x) - f(a)}{x-a}
\end{equation*}
This limit, if it exists, will be the slope of the tangent line. It is called the derivative and it calculates the rate of change of the function at \(x=a\text{.}\)
\begin{equation*}
f^\prime(a) = \frac{df}{dx} (a) = \lim_{x \rightarrow a}
\frac{f(x) - f(a)}{x-a}
\end{equation*}
There is a slight variation of this definition, which is useful for some calculations. If I define \(h = x-a\text{,}\) then I can write the limit in terms of \(h\) instead of \(x\text{.}\)
\begin{equation*}
f^\prime(a) = \frac{df}{dx} (a) = \lim_{h \rightarrow 0}
\frac{f(a+h) - f(a)}{h}
\end{equation*}
This second definition show that the derivative is an entirely new function. At each point \(x\) in the domain of \(f\text{,}\) I can ask for the slope of the tangent line. That gives a new function which measure the slope of the original.
\begin{equation*}
f^\prime(x) = \frac{df}{dx} = \lim_{h \rightarrow 0}
\frac{f(x+h) - f(x)}{h}
\end{equation*}
Subsection 6.2.2 Differential Operators
I can think of the derivative as an operation on functions: it takes a function and gives a new function which measures the rate of change of the previous function. This solves the velocity problem: if \(x(t)\) is a position function, then its derivative \(x^\prime(t)\) is the velocity function.
Leibniz notation is useful for thinking of derivatives as operators. If I seperate the notation slightly, I can write
\begin{equation*}
\frac{df}{dx} = \frac{d}{dx} f
\end{equation*}
With this notation, I think of \(\frac{d}{dx}\) as the operator: the thing that takes derivatives. Having notation for such an operator is extremely convenient.
Subsection 6.2.3 Differentiability
The limit defining the derivative may not always exist. If it does exist at a point \(a\) in the domain, then \(f\) is said to be differentiable at the point \(a\text{.}\) If it exists for all points in the domain of \(f\text{,}\) then \(f\) is called a differentiable function. Differentiability requires continuity: if a function makes a sudden jump, it doesn’t make sense to talk about a rate of change. A tangent line cannot be defined. Differentiability also requires a ‘smoothness’ condition. A function whose graph has sharp corners is not differentiable at the sharp corners, because it doesn’t make sense to define a tangent line at a sharp corner. The graph of the function must be smooth. Figure 6.2.1 shows how a jump or sharp corner makes the choice of a tangent line problematic.
Subsection 6.2.4 Interpretation
Now that I have formally defined the derivative and differentiability, I’ll try to review what this all means. The derivative has two major interpretation, one geometric and one applied.
- The derivative measures the slope of a tangent line to a function.
- The derivative measures the rate of change of a function.
If a function is differentiable on its domain, that means its derivative exists at all points in the domain. In the geometric interpretation, this means that the graph of the function has a well-defined tangent line at all points in its domain. In the applied interpretation, this means that the function has a well-defined rate of change at all points in its domain.
