Review Rules

1
Review Rules

• Power Rule If f(x) xn, then f ?(x) nxn-1
• If f(x) mx b, then f ?(x) m
• If f(x) c, then f ?(x) 0

2
Additional Rules

• Sum Rule
• If y f(x) g(x), then y? f ?(x) g ?(x)
• Difference Rule
• If y f(x) - g(x), then y? f ?(x) - g ?(x)

3
What about Products of functions?

• If y f(x) ? g(x), does y? f ?(x) ? g ?(x) ???
• Lets see! Let y (3x 2)(x2 4)
• What is f(x)? What is g(x)?
• What is f ?(x) ? What is g ?(x) ?

4
Make a guess

• f(x) 3x 2 ? f ?(x) 3
• g(x) x2 4 ? g ?(x) 2x
• So if y f(x) g(x), then our guess for the
  derivative is y ? 3(2x) 6x
• How could we check to see if this is correct?

5
Check the guess

• y (3x 2)(x2 4) ?
• y 3x3 2x2 12x 8
• So y? 9x2 4x 12
• How accurate was our guess???

6
We need a new rule!

• Let y f(x) g(x)
• Then y? lim f(x h)(g(x h) f(x)g(x)
• h?0 h

7
Product Rule

• If y f(x) g(x), then
• y? g(x) f ?(x) f(x) g ?(x)

8
Problem Revisited

• Find the derivative of y (3x 2)(x2 4)
• With f(x) 3x 2 and g(x) x2 4 and the
  Product Rule, we have
• y? (x2 4)(3) (3x 2)(2x)
• 3x2 12 6x2 4x
• 9x2 4x 12
• Which agrees with our previous result!

9
Find the derivative of

• f(x) 2x5(4x 3)
• f(x) (2x2 5)(x3 4)
• f(x) (x5 3x 7)(2x 1)
• f(x) (?x 3)(?x 6)

10
Homework for Tuesday Feb 8

• Section 2.4, page 140
• 1 17 odd, 57