Product and Quotient Rules HigherOrder Derivatives

1
Product and Quotient Rules Higher-Order
Derivatives
2
The Product Rule

• If f(x) and g(x) are differentiable at x,
  then so is their product P(x) f(x)g(x), and

• The derivative of the product fg is f times
  the derivative of g plus g times the
  derivative of f.

3
Proof of the Product Rule for a Specific Case
f(x) x2 and g(x) x3
4
EXAMPLE
5
ALTERNATE METHOD
First, multiply the functions, then take the
derivative.
6
EXAMPLE

• Let y uv be the product of the functions u
  and v. Find y'(3) if u(3) 1, u'(3) -2,
  v(3) 5, and v'(3) 4.

SOLUTION y' (uv)' uv' vu' y'(3) u(3)v'(3)
v(3)u'(3) (1)(4) (5)(-2)
4 - 10 -6
7
The Quotient Rule

• If f(x) and g(x) are differentiable at x,
  and g(x) does not equal 0, then the quotient
  Q(x) f(x)/g(x) is differentiable at x, and

8
The Quotient Rule

• The derivative of the quotient f /g is g
  times the derivative of the numerator f minus
  f times the derivative of the denominator g,
  all over g2.

9
EXAMPLE
10
EXAMPLE
11
NOTE

• The choice of which rules to use in solving a
  differentiation problem can make a difference in
  how much work you have to do.
• You may be able to avoid using the product and/or
  quotient rules by simplifying (multiply and/or
  divide) before taking the derivative.

12
EXAMPLE
13
EXAMPLE

• Find an equation for the tangent line to the
  curveat the point where x 1.

14
SOLUTION...
15
SOLUTION
The equation of the tangent line at the point
P(1, 0) with slope m -6 is y -6x 6
16
EXAMPLE

• The profit derived from the sale of x units of
  a certain commodity isthousand dollars. At
  what rate is profit changing with respect to
  sales when x 5?

17
SOLUTION...
18
SOLUTION
The profit is changing (increasing) at the rate
of15,111 per unit.
(This must be an expensive commodity, like a
freight truck, or a luxury car, or a house.)
19
The Second Derivative

• The second derivative of a function is the
  derivative of its derivative.
• If y f(x), the second derivative is denoted by

• The second derivative gives the rate of change of
  the rate of change of the original function.

20
Example
Find the second derivative of the function.
21
Example
Find the second derivative of the function.
22
Solution
23
Rates of Change mean Derivative !!!!

• The first derivative of a function is its rate of
  change.
• The second derivative of a function is the rate
  of change of the rate of change of the function.
• For example, consider the position s of a moving
  object with respect to time t. Then s f(t).
• The velocity v of an object is its rate of
  change with respect to time that is, v s'(t).
• The acceleration a of an object is the rate of
  change of the object's velocity that is, a
  v'(t) s"(t).