1. Do Now Find the tangents to the curve at the points where
the slope is 4. What is the smallest slope of the curve? At what
value of x does the curve have this slope?

2. Product and Quotient Rules
Section 3.3b

3. Do Now Graphical support???
Find the tangents to the curve at the points where
the slope is 4. What is the smallest slope of the curve? At what
value of x does the curve have this slope?
The derivative:
Slope 4, points (1,2) and (–1,–2).
Find where the slope is 4:
Tangent lines:
For smallest slope, minimize
The smallest slope is 1,
and occurs at x = 0.
Graphical support???

4. We need to derive a new rule for products…
As we learned last class, the derivative of the sum of two
functions is the sum of their derivatives (and the same holds
true for differences of functions). Is there a similar rule for
the product of two functions?
Let
The derivative:
However,
We need to derive a new rule for products…

5. Let
The derivative:
Subtract and add u(x + h)v(x) in the numerator:

6. Let
The derivative:

7. Rule 5: The Product Rule To find the derivative of a product of two
The product of two differentiable functions u and v is
differentiable, and
To find the derivative of a product of two
functions: “The first times the derivative of the
second plus the second times the derivative of
the first.”

8. How about when we have a quotient?...
The derivative:
Subtract and add v(x)u(x) in the numerator:

9. How about when we have a quotient?...
The derivative:

10. Rule 6: The Quotient Rule
At a point where , the quotient of two
differentiable functions is differentiable, and
To find the derivative of a quotient of two
functions: “The bottom times the derivative of
the top minus the top times the derivative of
the bottom, all divided by the bottom squared.”

11. Practice Problems Any other method for finding this answer? Find if
Let’s use the product rule with
and
Any other method for
finding this answer?

12. Practice Problems Graphical support: Differentiate
Use the quotient rule with
and
Graphical support:

13. Practice Problems Let be the product of the functions u and v. Find if
From the Product Rule:
At our particular point:

14. Practice Problems
Suppose u and v are functions of x that are differentiable at x = 2.
Also suppose that
Find the values of the following derivatives at x = 2.
(a)

15. Practice Problems
Suppose u and v are functions of x that are differentiable at x = 2.
Also suppose that
Find the values of the following derivatives at x = 2.
(b)

16. Practice Problems
Suppose u and v are functions of x that are differentiable at x = 2.
Also suppose that
Find the values of the following derivatives at x = 2.
(c)