1. DO NOW:
Write each expression as a sum of powers of x:

2. 3.3 - Rules for Differentiation
HW: Pg. 124 #2-22e

3. Derivative of a Constant Function
If f is the function with the constant value c, then
EX: f(x) = 4
f’(x) = 0

4. Power Rule for Positive Integer Powers of x
If n is a positive integer, then
EX: f(x) = x3 + x2

5. The Constant Multiple Rule
If f(x) is a differentiable function of x and c is a constant, then
EX: f(x) = 5x5

6. The Sum and Difference Rule
If f(x) and g(x) are differentiable functions of x, then their sum and difference are differentiable at every point where f(x) and g(x) are differentiable. At such points,

7. Example 1
Find dp/dt if p(t) = t4 + 5t3 - 2t2 + 6/7t + 4

8. Example 2
Does the curve y = x4 - 2x2 + 2 have any horizontal tangents?

9. Example 3
Differentiate:
(a) (b)

10. Example 4
Find the derivative of f(x) = (x + 4)(x - 5)

11. The Product Rule
The product of two differentiable functions f(x) and g(x) is differentiable, and
“The derivative of two functions equals the first function times the derivative of the second plus the second function times the derivative of the first”

12. Example 4 (cont) Find the derivative of f(x) using the product rule:
F(x) = (x + 4)(x - 5)

13. Example 5
Find f’(x) if f(x) = (x3 + 4)(x2 + 5)

14. The Quotient Rule
At a point where g(x) ≠ 0, the quotient y = f(x)/g(x) of two differentiable functions is differentiable, and

15. Example 6
Find f’(x) if

16. Example 7 Find the equation of the line tangent to the curve
y = (x2 + 3)/2x at the point (2,1).

17. Second and Higher Order Derivatives

18. Example 8 Find the first four derivatives of
y = x8 + 12x5 - 4x4 + 10x3 - 6x + 5

19. Example 9
If f(x) = (√x)g(x) , g(4) = 2 and g’(4) = 3

20. Example 10
Let y = uv be the product of the functions u and v. Find y’(2) if
u(2) = 3, u’(2) = -4, v(2) = 1, and v’(2) = 2.