1. Math 1304 Calculus I 2.8 – The Derivative

2. Definition of Derivative Definition: The derivative of a function f at a number a, denoted by f’(a) is given by the formula

3. Definition of Derivative Definition: The derivative of a function f the function whose formula is given by

4. Definition of Derivative An alternate formula for derivative at a point a is

5. Graphing Example Given a graph of a function, graph its derivative (do y=sin(x) in class)

6. Examples Find the derivative of

7. Example f(x) = |x|

8. Differentiable Definition (differentiable at a point): A function is said to be differentiable at a if f’(a) exists. Definition (differentiable on an interval): A function is said to be differentiable on an interval if f’(a) exists for all points a of the interval.

9. Differentiable implies Continuous Theorem: If f is differentiable at a then it is continuous at a. Note: the reverse is not true.

10. Various Notations There are several different ways of denoting the derivative of a function y=f(x) The symbols D and d/dx are called differentiation operators.

11. Second Derivatives We may take the derivative of the derivative. This is called the second derivative. Some notation: Alternate notations:

12. Third Derivatives We may continue taking derivatives to get the third, fourth, and more derivatives Some notation for the third: Alternate notations:

13. Nth Derivatives Notations for the nth derivative: