1. Definition of the Derivative
Lesson 3.4

2. Tangent Line Recall from geometry
Tangent is a line that touches the circle at only one point
Let us generalize the concept to functions
A tangent will just "touch" the line but not pass through it
Which of the above lines are tangent?

3. A Secant Line Crosses the curve twice The slope of the secant will be
x = a h

4. Tangent Line Now let h get smaller and smaller
x = a
Try this with Geogebra h
The slope of the tangent line

5. Tangent Line Try it out …given Evaluate
Determine the slope of the tangent line at x = 0
Evaluate

6. Tangent Line
Use the limit command on your calculator to determine the slope of for x = 1
Once you have the slope and the point
You can determine the equation of the line

7. The Derivative We will define the derivative of f(x) as Note
The derivative is the rate of change function for f(x)
The derivative is also a function of x
The limit must exist

8. Comparison Difference Quotient Slope of secant Average rate of change
Average velocity
Derivative
Slope of tangent
Instantaneous rate of change
Instantaneous velocity

9. Finding f'(x) from Definition
Strategy
Find f(x + h)
Find and simplify f(x + h) – f(x)
Divide by h to get
Let h → 0

10. Hint: rationalize the numerator
Try It Out
Use the strategy to find the derivatives of these functions
Hint: rationalize the numerator

11. Equation of the Tangent Line
We stated previously that once we determine the slope of the tangent
We can "cut to the chase" and state it as

12. Warning
Our definition of derivative included the phrase "if the limit exists"
Derivatives do not exist at "corners" or "sharp points" on the graph
The slope is different on each side of the point
The limit does not exist
f(x) = | x – 3 |

13. Assignment
Lesson 3.4
Page 210
Exercises 1 – 51 odd