Posted by: Rebekah

"If the ax is dull,

And one does not sharpen the edge,

Then he must use more strength;

But wisdom brings success." Ecclesiastes 10:10

In algebra, finding the slope of a line is pretty straight forward. As long as you have two points on the line, you can find the slope by simply dividing the difference in y by that of x: you know, $\frac{\text{difference in}y}{\text{difference in}x}$

But what about curves? How can we find the slope of a curve ( like that in Figure 1)?

Well, this is where the scripture above comes in: unless we find a way to "sharpen" our algebra, finding the slope to any curve would consist of multiple calculations--all of which would probably be pretty tiresome.

Up to this point, we know a lot about lines, so lets try using what we know to solve this problem.

Figure 2

In Figure 1, lets draw two lines (see Figure 2 above): a green tangent through a greenish-reddish point, and a red secant through the greenish-reddish point and through a red point. If we wanted to find the slope at the greenish-reddish point, all we would have to do is find the slope of the green tangent line. But wait a minute...this is impossible! Unless we have TWO points to work with, it cannot be done.

So...

lets get the red point as close as possible to the greenish-reddish point, so much so that the distance between both points is almost zero, but never actually zero.

This means then, that we are left with the red secant line to work with. Remember though, both points are now extremely close to each other, so it is now necessary to zoom in (see Figure 3).

Figure 3

*very*zoomed in), lets consider the values presented to us in Figure 3.

As you can see, the x-axis is labeled x, the y-axis is labeled f(x), and the letter "a" is obviously an x value (consequently, f(a) is a y value). This means then, that the coordinates for the greenish-reddish point are (a, f(a)).

But what about the red point? Well remember, we want the red point to be extremely close to the greenish point, so that means that the value "da" is

*extremely*small--almost zero but never actually equal to zero. As a result, the coordinates for the red point are (a + da, f(a + da)).

**Important Note: We could actually call the small space between f(a) and f(a + da) "dy" (after all, it is in the y-axis), and the small space between a and a + da "dx".**

Now that we have both points, we can finally start looking for the slope of the line (remember, the slope of a line is the difference in y divided by the difference in x).

Slope of the line = $\frac{dy}{dx}=\frac{f(a+da)-f\left(a\right)}{a+da-a}$

If you simplify, you then get: $\frac{dy}{dx}=\frac{f(a+da)-f\left(a\right)}{da}$

Now, because the red point is theoretically VERY close to the greenish point (don't forget that the greenish point belongs to the green tangent line), this means that what we really found was an extremely close approximation to the slope of the green tangent line.

**Important Note: because da is as close to zero as possible (i.e. as small as possible), the common notation is**$\underset{da\to 0}{lim}\frac{f(a+da)-f\left(a\right)}{da}$

**This notation is read as "the limit as da approaches zero is..."**

So...where do derivatives come in?

The slope of the tangent line

__IS__the derivative! (;

But how will this help us find the slope of a curve?

**To be continued...**