2. Slope at Point of Tangency
Mr. Miehl

3. Objective To determine the slope of a function at a given point.
To determine the slope of a tangent line to a function at a given point.
Both are done the same way!

4. The Derivative is…
Computed by finding the limit of the difference quotient as ∆x approaches 0.
Used to find the slope of a function at a point.
Used to find the slope of the tangent line to a graph f (x), and is usually denoted f’(x).
Used to find the instantaneous rate of change of a function.

5. Limit Definition of the Derivative
Use the limit definition to find the derivative of:

6. Limit Definition of the Derivative
CAUTION:
Possible mistakes ahead!

7. Limit Definition of the Derivative

8. Limit Definition of the Derivative

9. Limit Definition of the Derivative
A formula for finding the
slope of the tangent line
of f (x) at a given point.

10. Slope of a Function
Find the slope of at

11. Slope of a Function

12. Slope of a Function

13. Slope of a Function
Find the slope of at

14. Slope of Tangent Line
Find the slope of the tangent line to the graph of at (–2, 16).

15. Slope of Tangent Line

16. Slope of Tangent Line
Find the slope of the tangent line to the graph of at (–2, 16).

17. Value of the Derivative
Find the value of the derivative of

18. Value of the Derivative

19. Value of the Derivative

20. Value of the Derivative

21. Value of the Derivative

22. Value of the Derivative
Find the value of the derivative of

23. Value of the Derivative
Find if

24. Value of the Derivative

25. Value of the Derivative

26. Value of the Derivative
Find if

27. Conclusion
The derivative is a formula used to find the slope of function or slope of the tangent line to a function.
To find the slope of a function or slope of the tangent line to a function, first, find the derivative and, second, plug the corresponding x-value into the derivative.