1. MTH 251 – Differential Calculus Chapter 3 – Differentiation Section 3.2 The Derivative as a Function Copyright © 2010 by Ron Wallace, all rights reserved.

2. Review – The Derivative at a Point The derivative was defined as the limit of the difference quotient. That is … If x 0 + h = z, then an alternate definition would be … Note that the result of this limit is a number. That is, the derivative at a specific value of x. Remember: x 0 refers to a specific value of x.

3. The Derivative as a Function If we do not specify a specific value of x (i.e. use x instead of x 0 ) we get a function called the derivative of f(x). That is, the derivative of f(x) is the function … OR f(x+h) x x+h h f(x) f(x+h) – f(x)

4. Derivative Notation All of the following can be used to designate the function that is the derivative of y = f(x) Reminder: The results of these will be a function.

5. Derivative at x = a Notation All of the following can be used to designate the derivative of y = f(x) at x = a Reminder: The results of these will be a number.

6. Examples … Determine the following derivatives … IMPORTANT! Memorize these 3 results.

7. Examples … Determine the following derivatives …

8. Sketching the Graph of f ’(x) using the Graph of f(x) Where is the derivative (i.e. slope) zero? Where is the derivative (i.e. slope) positive?  Large or small positive? Where is the derivative (i.e. slope) negative?  Large or small negative? Where is the derivative (i.e. slope) constant?  Function is a line segment.  Derivative is a horizontal line segment.

9. Sketching the Graph of f ’(x) using the Graph of f(x) Example – Sketch the graph of the derivative of the following function.

10. Left & Right Derivatives at a Point If in the definition of the derivative at a point, you use just the left or right hand limit, the derivative at a point can be considered from just one side or the other. Right-Hand Derivative at x 0 Left-Hand Derivative at x 0 If these are equal, then …

11. Left & Right Derivatives at a Point Example:

12. Where does a derivative NOT exist? Corner  left & right derivatives are different

13. Where does a derivative NOT exist? Corner Cusp  left & right derivatives are approaching  & – 

14. Where does a derivative NOT exist? Corner Cusp Vertical Tangent  The derivative limit is  or – 

15. Where does a derivative NOT exist? Corner Cusp Vertical Tangent Discontinuity  see the next theorem

16. Differentiability & Continuity If f ’(c) exists, then f(x) is continuous at x = c.  Proof …

17. Differentiability & Continuity If f ’(c) exists, then f(x) is continuous at x = c. Or … the contrapositive implies … If f(x) is NOT continuous at x = c, then f ’(c) does not exist. NOTES  If the derivative does not exist, that does not mean the function is not continuous.  If the function is continuous, that does not mean that the derivative exists.  Example … the Absolute Value function.