4.1 Secant lines

This section introduces the derivative. Recall derivative measures change, change is what is of interest, so we are measuring what is of interest!

Here's a sketch of how we will get to the derivative:

Draw a picture of the secant line approximation to a function at a particular point (x).

The tangent line is the limit of the secant line as the "interval" ($\Delta x$) gets smaller.

The derivative is the slope of the tangent line at x.

The derivative of a function at the point x is defined as

$$\lim_{\Delta x\rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}.$$

In English, the derivative is the "rate of change" of the function.

Take a look at these ideas pictorially:

As $\Delta x$ gets smaller and smaller, how much does y change, and what about the slope, that is the ratio of $\frac{\Delta y} {\Delta x}$?

 

Table 1: Calculation of the slope of the secant line approximation to the function y = x² about the point x = 3.

x	y = x2	$\Delta x$	$x + \Delta x$	$y = f(x + \Delta x)$	$\Delta y$	$\frac{\Delta y} {\Delta x}$
3	9	4	7	49	40	10
3	9	3	6	36	27	9
3	9	2	5	25	16	8
3	9	1	4	16	7	7
3.00	9.00	0.50	3.50	12.25	3.25	6.50
3.0000	9.0000	0.2500	3.2500	10.5625	1.5625	6.2500
3.00	9.00	0.10	3.10	9.61	0.61	6.10
3.0000	9.0000	0.0100	3.0100	9.0601	0.0601	6.0100

Note, that the ratio tends to a limit!

Call this limit, $\frac{dy}{dx}$. Note the lower case d's rather than the $\Delta$'s.

Formally defined as

$$\lim_{\Delta x\rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}.$$

Do an example (once) ...

Take the function, f, where f(x) = x².

$$\lim_{\Delta x \rightarrow 0} \frac{f(3 + \Delta x) - f(3)}{\Delta x}.$$

$$\lim_{\Delta x \rightarrow 0} \frac{(3 + \Delta x)^2 - 3^2}{\Delta x}.$$

$$\lim_{\Delta x \rightarrow 0} \frac{9 + 6\,\Delta x + (\Delta x)^2 - 9}{\Delta x}.$$

$$\lim_{\Delta x \rightarrow 0} \frac{6\,\Delta x + (\Delta x)^2}{\Delta x}.$$

'

$$\lim_{\Delta x \rightarrow 0} (6 + \Delta x) = 6.$$

So, the answer is 6, just as the picture suggested.

Back to interpretation.

Consider a function that tells you how far you have gone after a particular amount of time. For example, think about flying from JFK to LHR; how far have you traveled in 4.5 hours etc.

Call x the time measured in hours, and y the distance you have traveled measured in kilometers (km). Then the function y = f(x) simply relates time flown to distance traveled.

Question: how do you interpret the derivative of f?

Interpretations come from looking at the units of measurement.

What are the units of the derivative?

The derivative is a ratio; change in y over change in x.

If y is measured in km, then a change in y must be measured in km too.

If x is measured in hours then a change in x must be measured in hours.

So the derivative has units km/hour, that is kilometers per hour.

So the derivative is simply speed or velocity - how fast you are going.

Challenge: what's the derivative of velocity with respect to time?