1. Derivatives
Sec. 3.1

2. Mathematical Synonyms
Instantaneous Rate of Change of f(x) at x = a
Slope of the tangent line of f(x) at x = a
Derivative of f(x) at x = a
Reminder: The slope of the secant line at x=a is the AVERAGE rate of change at a.

3. There are many ways to write the derivative of
is called the derivative of at .
We write:
“The derivative of f with respect to x is …”
There are many ways to write the derivative of

4. “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

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

6. Note:
does not mean !
does not mean !

7. Note:
does not mean times !

8. The derivative is the slope of the original function.
The derivative is defined at the end points of a function on a closed interval.

9. y y' 41

12. Let y=sin(x). Graph y’.

15. 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

16. HOMEWORK
P #2, 4, 10, 11, 13-22, 25-28