1. 2.3 The Product and Quotient Rules and Higher Order Derivatives

2. After this lesson, you should be able to:
Find the derivative of a function using the Product Rule
Find the derivative of a function using the Quotient Rule
Find the derivative of a trigonometric function
Find a higher-order derivative of a function

3. Theorem 2.7 The Product Rule
The product of two differentiable functions f and g is itself differentiable. The derivative is
where
Example Find dy/dx

4. Proof of Theorem 2.7
Since f (x) and g(x) are differentiable, so g(x) is continuous.
and
exist.
Therefore
exist.
and
exist.
Therefore
exist.

5. Proof of Theorem 2.7

6. The Product Rule Example
Example Find dy / dx :

7. The Product Rule Example
Example Find f ’(x)

8. The Product Rule Example
Example Find y’
Ans:

9. The Quotient Rule
The quotient of two differentiable functions f and g is itself differentiable at all values of x for which g(x) ≠ 0. The derivative is:
where
Example Find y ’

10. Proof of Theorem 2.8, The Quotient Rule
Since f (x) and g(x) are differentiable, so g(x) is continuous.
and
exist.
Therefore
exist.
and
exist.
Therefore
exist.

11. Proof of Theorem 2.8, The Quotient Rule

12. Proof of Theorem 2.8, The Quotient Rule

13. The Quotient Rule Example
Example Find f ’(x):

14. Equation of a Horizontal Tangent Line
Find the equations of the horizontal tangent lines for
Since we are asked to find horizontal tangent lines, we know the slopes of these lines are 0. So, set the 1st derivative equal to 0, then solve for x. This gives us the x-values of points on the graph where there are horizontal tangent lines. Then, we can find the points and write the equations.
Therefore, the points where there are horizontal tangent lines are
Since the tangent lines are horizontal, the equations are

15. The Quotient Rule Example

16. More Quotient Rule Examples
Sometimes, you need rewrite or simplify the function before you try to take the derivative(s).
Example
Rewrite
Then

17. More Quotient Rule Examples
In Section 2.2, the Power Rule was provided only for the case where the exponent n is a positive integer greater then 1. By using the Quotient Rule, we can generalize the Power Rule to n is an integer.
If n is a negative integer, there exists a positive integer k such that n = –k So, by the quotient rule

18. The Derivative of Tangent

19. The Derivative of Secant

20. The Derivatives of Cosecant and Cotangent

21. Equation of the Tangent Line
Example Given
Write the equation of the tangent line at
We can find the derivative of f (x) by using the rules.
The equation of the tangent line at is

22. Note
Because of the trigonometric identities, the derivative of a trigonometric function may have many forms. This presents a challenge when you are trying to match your answers to those given ones.

23. Higher Order Derivatives
First Derivative
Ex:
Second Derivative
Ex:
Third Derivative
Ex:
Fourth Derivative
Ex:
n-th Derivative
Ex:

24. Higher Order Derivatives of sin(x)

25. Homework
Section 2.3 page 124 #1-23odd, 27, 41, 43, 51, 89, 91, 101, 103