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. First, multiply the functions, then take the derivative.
ALTERNATE METHOD
First, multiply the functions, then take the derivative.

6. EXAMPLE SOLUTION y' = (uv)' = uv' + vu' y'(3) = u(3)v'(3) + v(3)u'(3)
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)
= = -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 curve at 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 is thousand dollars. At what rate is profit changing with respect to sales when x = 5?

17. SOLUTION...

18. The profit is changing (increasing) at the rate of $15,111 per unit.
…SOLUTION
The profit is changing (increasing) at the rate of $15,111 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).