The Definition of the DerivativeLESSON 3 OF 20

Deriving the Formula

The best way to understand the definition of the derivative is to start by looking at the simplest continuous function: a line.

You can determine the slope of a line by taking two points on that line and plugging them into the slope formula.

You can use the coordinates in reverse order and still get the same result. It doesn’t matter in which order you do the subtraction as long

as you’re consistent.

Calculating Slope Given Two Points on a Line

Suppose a line goes through two points.

First, you subtract the -coordinates.

Next, subtract the corresponding -coordinates.

Finally, divide the -coordinate difference by the -coordinate difference.

The result is the slope of the line, .

Suppose a line goes through the points and .

Looking at Slope Graphically

The slope measures the steepness of the line.

You probably remember your teachers referring to the slope as the “rise” over the “run.”

The rise is the difference between the -coordinates, and the run is the difference between the -coordinates. The slope is the ratio of the two.

Looking at Slope Graphically (cont.)

Instead of calling the -coordinates and , we’re going to call them and , where is the difference between the two -coordinates.

Instead of using and , we use and .

Don’t panic – only the notation has changed.

The Slope of a Curve

Suppose that we wanted to find the slope of a curve instead. Here the slope formula no longer works because the distance from one point to the other is along a curve, not a straight line.

But we could find the approximate slope if we took the slope of the line between the two points. This is called the secant line.The formula for the slope of the secant line is

This is called the Difference Quotient.

The Secant and the Tangent

As you can see, the farther apart the two points are, the less the slope of the line corresponds to the slope of the curve.

Conversely, the closer the two points are, the more accurate the approximation is.

The Secant and the Tangent (cont.)

There is one line, called the tangent line, that touches the curve at exactly one point. The slope of the tangent line is equal to the slope of the curve at exactly this point.

The object of using the above formula, therefore, is to shrink down to an infinitesimally small amount. If we could do that, then the difference between and would be a point.

The Definition of the Derivative

How do we perform this shrinking act?

By using the concept of limits!

We set up a limit during which approaches zero, like the following:

This is the definition of the derivative, and we call it .

Example 1

This means that and . If we can figure out , then we can find the slope.

Well, how did we find the value of ? We plugged into the equation . To find we plug into the equation, which now looks like this

Now plug this into the slope formula.

Next simplify by factoring out of the top.

Taking the limit as approaches , we get . Therefore, the slope of the curve at the point is .

Now we’ve found the slope of a curve at a certain point, and the notation looks like this: .

Find the slope of the curve at the point

Example 2

This means that and . If we can figure out , then we can find the slope.

This time,

Now plug this into the formula for the derivative.

Next simplify by factoring out of the top.

Taking the limit as approaches , we get . Therefore, the slope of the curve at the point is , or: .

Find the derivative of the equation at the point

Example 3

Follow the steps in the last two problems, but instead of using a number, use .

This means that

And .

Then the derivative is

Factor out of the top.

Now take the limit as goes to : you get . Therefore, .Find   the  slope  of   the   equation   𝑓 (𝑥 )=𝑥2   at   the   point  (𝑥1 ,𝑥12 ) .

Example 3 This example gives us a general formula for the

derivative of this curve. Now we can pick any point, plug it into the formula, and determine the slope at that point. For example, the derivative at the point is . At the point , the derivative is .Find   the  slope  of   the   equation   𝑓 (𝑥 )=𝑥2   at   the   point  (𝑥1 ,𝑥12 ) .

Differentiability

One of the important requirements for the differentiability of a function is that the function be continuous.

But, even if a function is continuous at a point, the function is not necessarily differentiable there.

If a function has a “sharp corner,” you can draw more than one tangent line at that point, and because the slopes of these tangent lines are not equal, the function is not differentiable there.

Another possible problem occurs when the tangent line is vertical (which can also occur at a cusp) because a vertical line has an infinite slope.

For example, if the derivative of a function is , it doesn’t have a derivative at .