1. Derivatives of Trigonometric Functions
Section 3.5
Derivatives of Trigonometric Functions

2. Derivative of sinx
𝑑 𝑑𝑥 𝑠𝑖𝑛𝑥 =𝑐𝑜𝑠𝑥

3. Derivative of cosx
𝑑 𝑑𝑥 𝑐𝑜𝑠𝑥 =−𝑠𝑖𝑛𝑥

4. Example 1
Find the derivative of 𝑦= 𝑥 2 𝑠𝑖𝑛𝑥.

5. Example 2
Find the derivative of 𝑦= 𝑠𝑖𝑛𝑥 1+𝑐𝑜𝑠𝑥 .

6. Simple Harmonic Motion
Definition 1: The motion of a weight bobbing up and down on the end of a spring is an example of simple harmonic motion.

7. Example 3
A weight hanging from a spring is stretched 5 units beyond its rest position (s = 0) and release at time t = 0 to bob up and down. Its position at any later time t is
𝑠=5𝑐𝑜𝑠𝑡
What are its velocity and acceleration at time t? Describe its motion.

8. TOTD
Find dy/dx.
𝑦= 𝑥 1+𝑐𝑜𝑠𝑥

9. Jerk
Definition 2: Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is
𝑗 𝑡 = 𝑑𝑎 𝑑𝑡

10. Example 4
The jerk caused by the constant acceleration of gravity

11. Example 5
The jerk of the simple harmonic motion in Example 3 is

12. Derivatives of the Other Basic Trig Functions
𝑑 𝑑𝑥 𝑡𝑎𝑛𝑥 =𝑠𝑒 𝑐 2 𝑥
𝑑 𝑑𝑥 𝑠𝑒𝑐𝑥 =𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥
𝑑 𝑑𝑥 𝑐𝑜𝑡𝑥 =−𝑐𝑠 𝑐 2 𝑥
𝑑 𝑑𝑥 𝑐𝑠𝑐𝑥 =−𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥
Tangent & Secant
Cotangent & Cosecant

13. Example 6
Find the equations for the lines that are tangent and normal to the graph of 𝑓 𝑥 = 𝑠𝑒𝑐𝑥 1+𝑡𝑎𝑛𝑥 @ 𝑥= 𝜋 4 .

14. Example 7
Find y’’ if y = secx.

15. TOTD
A body is moving in simple harmonic motion with position function 𝑠=𝑓(𝑡) (s in meters, t in seconds).
Find the body’s velocity, speed, and acceleration at time t.
Find the body’s velocity, speed, and acceleration at time t = π/4.
𝑠=2𝑠𝑖𝑛𝑡+3𝑐𝑜𝑠𝑡