1. 1. Find the derivative 2. Draw the graph of y’ from the graph of f(x)
Warm-up
1. Find the derivative
2. Draw the graph of y’ from the graph of f(x)

2. 12. Section 3.5 and 3.6 Higher Derivatives and Trig Functions
Table of Contents
12. Section 3.5 and 3.6 Higher Derivatives and Trig Functions

3. Higher Derivatives and Trig Functions
Essential Question – How do you find derivatives for trig functions?

4. Second and Higher Order Derivatives
y’ is called 1st derivative
y’’ is called 2nd derivative
y’’’ is called 3rd derivative

5. Example
Find 1st 4 derivatives of and evaluate y’’’(-1)

6. Example
Calculate f’’’(x) for f(x) = xex

7. Acceleration Acceleration is a common second derivative
Find the acceleration of a ball tossed vertically in the air from ground level with an initial velocity of 40 ft/s. s(t) = so+vot-16t2

8. Graphing calculator
You can graph a derivative function without calculating it by using the nDeriv function (Math 8)
Put nDeriv(f(x), x, x) into y=
f(x) is the function, the first x means you are taking derivative with respect to x, the second x means that you want for all values of x (you could put a number here and it would just calculate that one value)

9. Sin and Cos Graph Nderiv(sin x, x, x)
Can you tell what the derivative of sin is?
y’(sin x) = cos x
Graph Nderiv(cos x, x, x)
y’(cos x) = - sin x

10. Example

11. Example

12. Simple Harmonic Motion
A weighted spring is an example of simple harmonic motion
Its motion is modeled by cos.

13. Example
A weight hanging from a spring is stretched 5 units past its rest position (s=0) and released at time 0 (t=0) to bob up and down. Its position at any later time is
Find the velocity and acceleration at time t.

14. Other trig function rules

15. Example
Verify the formula

16. Example
Find the tangent line to the function below at x=2.

17. Example
Find y” if y = sec x

18. Assignment
Pg. 165: # 1-17 odd
Pg. 170: # 1-33 odd, all