1. 2.3 The Product and Quotient Rules and Higher-Order Derivatives (Part 2)
London Bridge, Lake Havasu City, Arizona
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2001

2. Objectives Find the derivative of a trigonometric function.
Find a higher-order derivative of a function.

3. Consider the function
slope
We could make a graph of the slope:
Now we connect the dots!
The resulting curve is a cosine curve.

4. We can do the same thing for
slope
The resulting curve is a sine curve that has been reflected about the x-axis.

5. We can find the derivative of tangent x by using the quotient rule.

6. Derivatives of the remaining trig functions can be determined the same way.

7. Example: Find the derivative

8. Example: Find the derivative

9. Find the derivative:

10. Higher Order Derivatives:
is the first derivative of y with respect to x.
is the second derivative.
(y double prime)
is the third derivative.
is the fourth derivative.

11. Higher Order Derivatives:
See notation on page 123: y
f(x) y'
f '(x)
dy/dx
y''
f ''(x)
d2y/dx2
y'''
f '''(x)
d3y/dx3
y (4)
f (4)(x)
d4y/dx4

12. Homework
2.3 (page 124)
#43-71 odd
#83-91 odd
#95, 97, 101, 103