3.8 Derivatives of Inverse Trigonometric Functions

What you’ll learn about…•Derivatives of Inverse Functions

Why?

The relationship between the graph of a function and its inverse allows us to see the relationship between their derivatives.

Derivatives of Inverse Trigonometric functions

dx

du

ux

dx

d

2

1

1

1cos

dx

du

ux

dx

d

2

1

1

1sin

dx

du

ux

dx

d

21

1

1tan

dx

du

uux

dx

d

1||

1csc

2

1

dx

du

uux

dx

d

1||

1sec

2

1

dx

du

ux

dx

d

2

1

1

1cot

Calculator Conversions

)/1(sincsc 11 xx

)/1(cossec 11 xx

xx 11 tan2

cot

Where do these come from?Derivative of the arcsine…

We can simplify using the identity

xy 1sin

1sincos 22 xx

yy 2sin1cos

Find

Use dx

du

ux

dx

d

2

1

1

1sin

)1(sin 1 xdx

d

What about the sec-1x?

xy 1sec 11sectan 22 xyy

Finding the derivative of the Arctangent

xy 1tan Using the trig identity yy 22 tan1sec

A particle moves along the x-axis so that its position at any time t ≥ 0 is given by x(t) = tan-1(t 2). Find the velocity at the t = 1.

Evaluate at t = 1. dx

du

ux

dx

d

21

1

1tan

What can we do with these derivatives?

We can find slopes and write the equations of tangent & normal lines to the curve at x=a!

• Find an equation for the line tangent to the graph of y = cot-1x at x = -1

How?• Use the given equation and x=a to find a point on the curve.• Find f ’ and evaluate it at a to get the slope of the tangent.• Recall that -1/a is the slope of the normal line.• Write an equation in point slope form, simplify if needed.

Homework

p170 Quick Review 1-10

Exercises 3-27 (3n, nЄI)

Today’s Agenda

Correct Homework: Q & A

Free Response Practice: No Calculator

Homework

Page 170

Exercises 15-27 (3n, nЄI), 35-40