1. 2-1: The Derivative Objectives: Explore the tangent line problem
Define the derivative
Discuss the relationship between differentiability and continuity

2. Important Idea
The tangent line problem is: in general, how do you find the slope of the tangent line at a point?

3. Analysis

4. Analysis
As , the slope of the secant linethe slope of the tangent line at x

5. Analysis
As , the slope of the secant linethe slope of the tangent line at x

6. Analysis
As , the slope of the secant linethe slope of the tangent line at x

7. Analysis
As , the slope of the secant linethe slope of the tangent line at x
is very, very close to 0

8. Analysis
Slope of secant line:
AP Exam: instead of

9. Analysis
The slope of the secant line becomes the slope of the tangent line:

10. Example
Find the slope of the line tangent to the following graph at the point (2,5):

11. Try This
Find the slope of the line tangent to the following graph at the point (3,17):
m=12

12. Try This
Find the slope of the line tangent to the following graph at the point (1,1):
Hint: rationalize the numerator

13. Try This
Estimate the slope of the line tangent to the following graph at the indicated point:

14. Try This
Estimate the slope of the line tangent to the following graph at the point (0,0)

15. Definition
The slope of the tangent line, if it exists, is:
sometimes:

16. Definition
The slope of the tangent line at a point is called the derivative of f(x) and the derivative is:
if the limit exists.

17. Important Idea
Differentiation at a point implies continuity at the point, however,…
continuity at a point does not imply differentiability at a point

18. f(x) is differentiable at (3,1) implies f(x) is continuous at (3,1)
Example
f(x)
f(x) is differentiable at (3,1) implies f(x) is continuous at (3,1)
(3,1)

19. Example
f(x) continuous at (3,1) does not imply f(x) is differentiable at (3,1)
f(x)
(3,1)

20. Definition An alternative form of the derivative at a point c is:
providing the limit exists

21. Example
Use the alternative form to find f’(x) at x=2:

22. Example
Use your knowledge of derivatives and algebra to find the equation of the line tangent to
at (2,8).

23. Try This
Use your knowledge of derivatives and algebra to find the equation of the line tangent to
at (-3,4).

24. Lesson Close
A derivative is…