# The Derivative and the Tangent Line

Copyright © 2002–2013 by Stan Brown, Oak Road Systems

Copyright © 2002–2013 by Stan Brown, Oak Road Systems

**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, Δ*y*/Δ*x*. 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**.

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, Δ*y*/Δ*x*. The first point
is at (6,4) and the second point is at
(6+Δ*x*, 4+Δ*y*). But since
y = f(x), you can also say that the y coordinate of the
second point is f(6+Δ*x*).
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; see Derivatives on TI-83/84.

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.

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:

- The function must be continuous there.
- The graph must be smooth there, with no sharp bend.
- 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.

This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an official statement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.

For updates and new info, go to http://www.tc3.edu/instruct/sbrown/calc/