3.1 Derivatives

1. 3.1 Derivatives

2. Derivative • A derivative of a function is the instantaneous rate of change of the function at any point in its domain. • We say this is the derivative of f with respect to the variable x. • If this limit exists, then the function is differentiable.

3. Symbols Used to Denote Derivatives

4. Note on Notations • dx does not mean “d times x!!” • dy does not mean “d times y!!”

5. Example • Find the derivative of the function f(x) = x3.

6. Note • Your book talks about an “alternate definition.” Do not worry about using the “alternate definition.” You will never see it on an AP exam! • If the directions on your HW say to use the alternate definition, use the regular definition of the derivative.

7. Example • Find the derivative of (Multiply by the conjugate)

8. A Note from the Example • From the previous example: • What was the domain of f? • [0, ∞) • What was the domain of f’? • (0, ∞) • Significance??? • Sometimes the domain of the derivative of a function may be smaller than the domain of the function.

9. Functions and Derivatives Graphically The function f(x)has the following graph: What does the graph of y’ look like? Remember: y’ is the slope of y.

10. The derivative is defined at the end points of a function on a closed interval. Functions/Derivatives Graphically

11. One-Sided Derivatives • Since derivatives involve limits, in order for a derivative to exist at a certain point, its derivative from the left has to equal its derivative from the right.

12. Example • Show that the following function has a left- and right-hand derivatives at x = 0, but no derivative there.