Math 1304 Calculus I

1. Math 1304 Calculus I 2.8 – Derivative as a Function

2. Notation for Derivative • Definition: The derivative of a function f at a number a, denoted by f’(a) is given by the formula • This is defines a new function f’ called the derivative of f.

3. Alternative Formula for Derivative • An alternate formula for derivative at a point a is

4. Examples • Find the derivative of

5. 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.

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

7. 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.

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

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

10. Nth Derivatives • Notations for the nth derivative: