The Derivative and the Tangent Line
(Lecture Notes for L-H-E Calculus 7e section 2.1)

Summary:  You already know how to find the slope of a line; calculus finds the slope of a curve at a point. The technique is to draw a secant line through that point and a nearby point and compute the slope of the secant line, Δy/Δx. Now slide the second point along the curve toward the first; obviously the secant line gets closer and closer to being a tangent line. In the limit, as the two points coincide, the secant line becomes the tangent line and its slope becomes the slope of the curve, written and called the derivative.

A. Slope of a Curve

One of the two main problems of calculus is to find the slope of a curve at a given point. Today we’ll solve this problem!

f(x) = –x²+9x–14 is graphed at right. We will solve the classic tangent line problem by finding the slope of this curve at some point, such as (6,4).

How to find the slope of the curve? It’s the same as the slope of a line that is tangent to the curve at that point. (A tangent line touches the curve at just one point.)

How to find the slope of the tangent line? Draw a secant line, which crosses through the curve at two points, (6,4) and another point nearby. Find the slope of that secant line. Then use a limit process to find the slope as you bring the second point closer to (6,4).

How to find the slope of the secant line? Use the normal formula for slope, Δy/Δx. The first point is at (6,4) and the second point is at (6+Δx, 4+Δy). But since y = f(x), you can also say that the y coordinate of the second point is f(6+Δx). Therefore the slope of the secant line is the difference quotient, as described on pages 95–97:

The above is the slope of any secant line, but we need the slope of the tangent line. To find it, take the limit as the two points come closer together. That is the same thing as taking the limit as the x distance between the points approaches 0. The slope of the tangent line is that limit:

In class we’ll evaluate this and find that the limit is –3. So the slope at x=6 is –3. We write f′(6) = –3 (“f-prime of 6 equals –3”) and say that the derivative of f at 6 is –3.

Obviously the slope is different at different points on the curve. Can we write an expression that will give the slope for any x? Yes, by using x instead of 6 in the difference quotient:

Evaluate the difference quotient and take the limit, and you get f′(x) = –2x+9. f′ (“f-prime”) is a new function. Since it is derived from the original function, it’s called the derivative. The derivative of a function at any x value gives the slope of the original function’s graph at that x value.

As you might expect, there are shortcut methods for finding the derivative. The rest of Chapter 2 goes into those methods. You can also use your TI-83 to check your calculation of the derivative, as another Web page explains.

B. Vocabulary and Notation

The function is called differentiable at a given point if the derivative exists at that point. It’s differentiable on an open interval if the derivative exists at every point on that interval.

Different authors use many symbols for the derivative of a function, as shown below. The first two or three are the most common.

C. Continuity

The derivative does not exist (the function is not differentiable) at any point where the function is not continuous. In other words, wherever the function is not continuous it’s also not differentiable.

But even if the function is continuous at a point, it still might not be differentiable there. These three conditions must be fulfilled for the function to be differentiable at a given point:

1. The function must be continuous there.
2. The graph must be smooth there, with no sharp bend.
3. The tangent line must be oblique (slanted) or horizontal, not vertical.

Wherever any of those conditions is violated, the derivative does not exist at that point. Turning it around, if the derivative does exist at a given point, then you know that all three of the above are true at that point.