- Topics At a Glance
**Derivative as a Limit of Slopes****Slope of a Line Between Two Points**- Slope of a Line Between Two Points on a Function
- Slope at One Point?
- Estimating Derivatives Given the Formula
- Estimating Derivatives from Tables
- Finding Derivatives Using Formulas
- Derivatives as an Instantaneous Rate of Change
- Average Rate of Change
- Instantaneous Rate of Change
- Units, Words, and Notation
- Tangent Lines
- How Tangent Lines Look
- When Tangent Lines Don't Exist
- Tangent Lines and Derivatives
- Tangent Line Approximation
- Finding Tangent Lines
- Using Tangent Lines to Approximate Function Values
- Differentiability and Continuity
- The Derivative Function
- Graphs of
*f*(*x*) and*f*' (*x*) - Theorems
- Rolle's Theorem
- The Mean Value Theorem
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem
- Appendix: Speed v. Velocity

The slope of a line between two given points is equal to

where "rise" and "run" mean the same things they did when we first learned about the slopes of lines.

Consider the line drawn here:

We have

{rise} = ({ending *y*-value}) - ({starting *y*-value}) = 8 - 4 = 4

and

{run} = ({ending *x*-value}) - ({starting *x*-value}) = 5 - 3 = 2.

This means the slope of the line is

When we calculate the slope this way, we're assuming that we start at the point (3, 4) and travel to the point (5, 8). If the equation of the line is given by the function *f*, we think of our starting point as (*a*, *f*(*a*)) and our ending point as (*b*, *f*(*b*)):

However, if we want to be contrary, we can instead start at the point (5, 8) and travel to the point (3, 4):

Now our starting point is (a, *f*(*a*)) = (5, 8) and our ending point is (*b*, *f*(*b*)) = (3, 4). We find

{rise} = ({ending *y*-value}) - ({starting *y*-value}) = 4 - 8 = -4

and

{run} = ({ending *x*-value}) - ({starting *x*-value}) = 3 - 5 = -2.

This means the slope of the line is

When we switched our starting and ending points in the example we changed the signs of both the rise and the run. These two sign changes canceled each other out, we got the same value of slope either way. The moral of the story is that when we're asked to find the slope of the line between two points, it doesn't matter which point we think of as the "start."

If we have two numbers *a* and *b* and a line *f*(*x*), we can say that "run" is the distance from *a* to *b* and "rise" is the distance from *f*(*a*) to *f*(*b*). This produces the formulas

{run} = *b* - *a*

{rise} = *f*(*b*) - *f*(*a*).

This will work regardless of which point is to the left.

Example 1

Find the slope of the line between the points (3, 15) and (5, 30). |

Exercise 1

In the picture, find the slope of the line between the two given points.

Exercise 2

In the picture, find the slope of the line between the two given points.

Exercise 3

In the picture, find the slope of the line between the two given points.

Exercise 4

Find the slope of the line between the two points (4, 7) and (6, 14).

Exercise 5

Find the slope of the line between the two points (5, 8) and (-1, -2).

Exercise 6

Find the slope of the line between the two points (-2, -3) and (-7, -1).