Derivative Rules

1
Derivative Rules

• Power Rule If y xn, then y? nxn-1
• Product Rule If y f(x)g(x), then
• y? g(x)f ?(x) f(x)g ?(x)
• Quotient Rule If y f(x)/g(x), then
• y? g(x)f ?(x) - f(x)g ?(x)
• g(x)2
• General Power Rule If y f(x)n, then
• y? nf(x)n-1

2
Find the derivative

• y 4x3 2x2 7x 11
• y (3x2 5)4
• y 5/(3x 2)
• y (2x-3 5x)(x4 7)
• y ?(6x3 4x)

3
Combine the rules

• It is possible to combine the product and
  quotient rules with the Chain Rule!
• When the product and/or quotient rule is used in
  conjunction with the Chain rule, factoring and
  algebraic simplification is always necessary!

4
Find y? for

• y 8x5(2x3)3
• y (3x2 5)4(4x3 3)2
• y 4x6
• (5 2x)3
• y (2x-1)2(3x4)5

5

• The population of a city x years from 1998 is
  given by P(x) 4?(x2 1) million people. How
  fast is the population growing in 2000? In 2004?

6

• You should have noticed by now that each time we
  find the derivative of a function, we obtain
  another function.
• y 4x3 2x2 7x 11
• then y? 8x2 4x 7
• What is the next derivative?

7
Second derivative

• The second derivative is the derivative of the
  first derivative.
• Symbolically we denote the second derivative as
• y?? or f ?? or d2y/dx2 or D2

8
Find y?? for

• y 7x-4 3x-2 9x
• y (3x 5)4
• y (3x2 5)4
• y 5/(3x 2)
• y ?(6x3 4x)

9
What does a second derivative mean?

• Rate at which the instantaneous rate of change is
  changing
• Acceleration of a projectile

10

• The height in feet of a ball thrown upward is
  given by h(t) - 3.2t2 16t 5, where t is
  seconds after the ball is thrown.
• Evaluate h(2), h?(2), and h??(2) and interpret
  the answers.

11

• The population of Orlando since 1965 has grown
  according to P(x) .1712x.62 where x is years
  after 1965 and P is population in millions.
• Find the instantaneous rate of change for 2000.
• At what rate was the growth rate increasing in
  2004?

12

• Do all functions have a derivative?
• The derivative of a function does not exist if
• there is a point of discontinuity
• there is a vertical tangent line
• there is not a unique tangent line