1. 3.1 Derivatives of a Function
AP Calculus AB
3.1 Derivatives of a Function

2. In section 2.4, the slope of the curve of y = f(x) was defined
at the point where x = a to be
This limit is also called the derivative of f at a.
We write:
“The derivative of f with respect to x is …”

3. Example 1:

4. Definition (Alternate) Derivative at a Point
The derivative of a function at a point x = a is:
provided that the limit exists.

5. Example 2: Use the alternate definition to differentiate−1
For a = 2:
There are many ways to write the derivative of

6. “the derivative of f with respect to x”
“f prime x” or
“y prime”
“the derivative of y with respect to x”
“dee why dee ecks” or
“the derivative of f with respect to x”
“dee eff dee ecks” or
“the derivative of f of x”
“dee dee ecks uv eff uv ecks” or

7. Note:
dx does not mean d times x !
dy does not mean d times y !

8. Note: does not mean ! does not mean !
(except when it is convenient to think of it as division.)
does not mean !
(except when it is convenient to think of it as division.)

9. Note: does not mean times !
(except when it is convenient to treat it that way.)

10. In the future, all will become clear.

11. Example 3: Graph the derivative of the function f.
The derivative is the slope of the original function.
The derivative is defined at the end points of a function on a closed interval.

12. Graph y and y′ where

13. A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. p