1. Table of Contents
9. Section 3.1 Definition of Derivative

2. Section 3.1 Definition of Derivative
Essential Question – What is a derivative?

3. Calculus overview reminder
3 main concepts
Limits
Derivatives
Integrals

4. Slope of secant line Remember back to Section 2.1….
As h shrinks to zero, slope of secant line approaches slope of tangent line
Slope of tangent line =

5. Difference quotient
The formula for slope of a secant line (without the limit) is called the difference quotient
We will use it A LOT when we differentiate

6. Derivative
The formula for slope of a tangent line (with the limit) is called the derivative
Can also be written

7. All these have the same meaning
Slope of f(x) at x = a
Slope of tangent line to f(x) at x = a
Instantaneous rate of change of f(x) at x = a
Derivative at x = a

8. Example
Find the slope of y=x2 at the point x=2 and the equation of the tangent line at this point.
This is the slope, use it
to find the equation of the tangent line.

9. Example cont…..

10. Example
Find the derivative of f(x) = x2

11. Example
Find the derivative of f(x) = 4x2-7x at x=3

12. Example cont.
Find the derivative of f(x) = 4x2-7x at x=3 (use second equation)

13. Example
Find the derivative of f(x) = 1/x at x=2

14. Example cont.
Find the derivative of f(x) = 1/x at x=2 (use second equation)

15. Example cont….
Now find the equation of the tangent line at x=2

16. Example cont…. Now find the equation of the normal line at x=2
Normal lines are perpendicular to a point

17. Derivatives of constant functions
If f(x) = b is a constant function, f’(a)=0 for all a
f(x)=b

18. Derivatives of linear functions
If f(x) = mx + b is a linear function, f’(a)=m for all a
f(x)=mx + b

19. Assignment
Pg 124: #1-7 odd, 8, 9-29 odd, 41, 43, 52, odd, 61, 63