3.5 Derivatives of Trigonometric Functions

What you’ll learn about….

•Derivatives of the Sine and Cosine Functions•Simple Harmonic Motion•Jerk•Derivatives of the Tangent, Cotangent, Secant and Cosecant functions

Warm Up: Exploration 1Page 141

Graph sin x and nderiv(sin (x), x, x) in a trigonometric window

Make a conjecture, what function do you believe is the derivative of sin x?

If f(x) = sin x, then f’(x) =

Derivatives of Trigonometric Functions

xxdx

dcossin

xxdx

dsincos

xxdx

d 2sectan

xxxdx

dcotcsccsc

xxxdx

dtansecsec

xxdx

d 2csccot

Find

x

x

dx

dx

dx

d

cos

sintan

xdx

dtan

Example 1: Use the product & quotient rules to find the following derivatives

xxdx

dcos2

x

x

dx

d

cos1

sin

Simple Harmonic MotionThe motion of a weight bobbing up and down on the end of a spring is an

example of simple harmonic motion. If a weight hanging from a spring is stretched 5 units beyond its resting position (s=0) and released at time t = 0 to bob up and down, its position at any later time t is

s = 5 cos t.

What is its velocity and acceleration at time t?

v(t) = a(t) =

Describe its motion.

Starts at Amplitude Period

Greatest Velocity Greatest Acceleration

What about JERK?

A sudden change in acceleration is called a “jerk”. The derivative responsible for jerk is the third derivative of position.

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

3

3

)(dt

sd

dt

datj

Example 3: A Couple of Jerks

a) FYI The jerk caused by the constant acceleration of gravity (g = -32 ft/sec2) is zero.

This explains why we don’t experience motion sickness sitting around!

b) Find the jerk of the simple harmonic motion in Example 2.

Finding Tangent and Normal Lines

Find equations for the lines that are tangent and normal to

the graph of

Slope of tangent

Tangent line

Normal line

x

xxf

sin)(

Example 5 A Trigonometric Second Derivative

Find y” if y = csc x.

Find more examples & Practice

http://archives.math.utk.edu/visual.calculus/index.html

Select Derivatives

Derivatives of the Trigonometric Functions

Tutorial

Drill with Power Rule

Drill with Quotient Rule

Homework

Page 146

Quick Review 1-10

Exercises 1-11 odds

Agenda for Today

Opener Page 155 Exercises 70-73

No Calculator! (turn in when completed)

Present homework under document camera

Class / Homework Page 146

Exercises 13-31 odds, 37, 40, 42