Lesson 3-5: Derivatives of Trig Functions

AP Calculus

Mrs. Mongold

2

0

2

Consider the function siny

We could make a graph of the slope: slope

1

0

1

0

1Now we connect the dots!

The resulting curve is a cosine curve.

sin cosd

x xdx

2

0

2

We can do the same thing for cosy slope

0

1

0

1

0The resulting curve is a sine curve that has been reflected about the x-axis.

cos sind

x xdx

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

tand

xdx

sin

cos

d x

dx x

2

cos cos sin sin

cos

x x x x

x

2 2

2

cos sin

cos

x x

x

2

1

cos x

2sec x

2tan secd

x xdx

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

sin cosd

x xdx

cos sind

x xdx

2tan secd

x xdx

2cot cscd

x xdx

sec sec tand

x x xdx

csc csc cotd

x x xdx

Example

• A wight hanging from a spring is stretched 5 units beyond its rest position (s=0) and released at time t=0 to bob up and down. It’s position at any later time t is s=5cost

• What are the velocity and acceleration? Describe its motion.

Jerk

• The derivative of acceleration is jerk. If a body’s position at time t is s(t) the body’s jerk at time t is

• When a ride in a car or bus is jerky, it is not that the accelerations are large but the changes in acceleration are abrupt

3

3

)(dt

sd

dt

datj

Example: A couple of jerks

• A) the jerk caused by the constant acceleration of gravity (g = -32 ft/sec2) is zero.– This is why we don’t get motion sickness

when we are sitting around doing nothing

B) The jerk of a simple harmonic motion from example 1 is what?

Example

• Find equations for the lines that are tangent and normal to the graph of

at x = 2. x

xxf

tan)(

Example

• Find y’’ if y = secx

Homework

• Page 140/1-10, 12-14, 17, 20-22