1. 3-2 The Derivative Wed Oct 5
Find the slope of the tangent line to y = f(x) at x = a
x^2 -4, a = 2
2x^3, a = 0

2. The derivative of a function
The derivative of f(x) is the function f’(x):
Lots of simplification will be involved!

3. Ex 1
Find the derivative of

4. Ex 2
Find the derivative of f(x) = 2x^2 - 5x + 7

5. Ex 3
Find the derivative of y = x^-2

6. Alternative derivative notations
There are several ways to denote the derivative. We already know f’(x)
The following notations are all equivalent:
These notations indicate “the derivative of y in terms of x”

7. Differentiability and Continuity
If f is differentiable at x = c (the derivative is defined at c) then f is also continuous at c 7

8. When a function is not differentiable at a point
When a function is not differentiable at a point x = a, the one sided limits will not be equal. There are several cases:
A jump discontinuity (piecewise function)
Vertical asymptote
Cusp (piecewise function)
Vertical tangent line 8

9. Derivative Info
The derivative can tell us when a function is increasing (+), decreasing (-), or horizontal (0)
This makes finding the vertex of a function easier 9

10. Benefits of the Derivative
The derivative also gives us a good view of the behavior of the original function f(x)
Slope
Velocity
Rates of change

11. Closure Hand in: Find the derivative f’(x) of
HW: p.139 #1-6, 43, 45, 66, 70

12. Practice Problems Find the derivative: 1) 2) 3) 4) 5)

13. 3.2 The Power Rule Thurs Oct 6
Do Now
Find the derivative of: 13

14. HW Review: p.139 #1-6 43 45 66 70 1) f’(x) = 3 2) f’(x) = 2x + 3
43) A - 3, B - 1, C - 2, D - 3
45) A = f(x), B = h(x), C = g(x)
66) A - 3, B - 1, C - 2
70) a: x = 1
b: x = 1, x = 2, x = 3 14

15. Computations of Derivatives
Thm- For any constant c,
Note, when y = c, the slope of that line is always horizontal. Therefore, its derivative must equal 0 15

16. Thm- Let f(x) = x, then Proof:
Note: This means that the derivative of any linear function is equal to the coefficient 16

17. Power Rule F(x) F’(x) 1 X^1 X^2 2x X^3 3x^2 X^4 4x^3
Let’s take a look at the different powers of x. Can you see the pattern in the table?
F(x)
F’(x) 1
X^1
X^2 2x
X^3
3x^2
X^4
4x^3 17

18. Power Rule cont’d Power Rule - For any real number n,
Note: The power rule works for negative exponents, as well as fraction exponents. 18

19. Ex 3.1
Find the derivatives of

20. Ex 3.2
Find the derivatives of

21. Derivative of e^x
The derivative of f(x) = e^x is 21

22. General Derivative Rules
Thm- If f(x) and g(x) are differentiable at x and c is any constant, then 1) 2) 3) 22

23. General Deriv. Rules
Remember, to rewrite any expressions so they have exponents! And split the expression into separate terms!
You try: Find the derivative of each: 1) 2) 3) 23

24. Closure Hand in: Find the derivative of: 1) 2)
HW: p.139 #15-41 odds, 49,
Small quiz tomorrow? (10 min) 24