3.8 Derivatives of Inverse Trigonometric Functions

S

3.8Derivatives of Inverse Trigonometric Functions

Quick Review

Slide 3- 2

In Exercises 1-5, give the and of the function,

and evaluate the function at 1.

domain range

x

1

1

1

1

1

1. sin

2. cos

3. tan

4. sec

5. tan tan

y x

y x

y x

y x

y x

Quick Review

Slide 3- 3

In Exercises 6-10, find the inverse of the given function.

3

6. 3 8

7. 5

88.

3 29.

10. arctan 3

y x

y x

yxx

yx

xy

Quick Review Solutions

Slide 3- 4

In Exercises 1-5, give the and of the function,

and evaluate the function at 1.

domain range

x

1

1

1

1

1. sin

2. cos

3. tan

Domain: 1,1 Range: - , At 1:2 2 2

Domain: 1,1 Range: 0, At 1:0

Domain:All R

4.

eals Range: - , At 1:2 2 4

Domain: , 1 1,

Range: 0, ,2 2

sec

y x

y x

y x

y x

∪

∪

1

At 1:0

Domain:All Reals Rang5. e:Al tan l Reals At n 11a :ty x

Quick Review Solutions

Slide 3- 5

1

1 3

1

1

1

3

6. 3 8

7. 5

88.

3 29.

8

3

5

8

2

3

3tan ,

10. arct23 2

an

xf x

f x x

f xx

f xx

f

y x

y x

yxx

y

xx

x

y x x

In Exercises 6-10, find the inverse of the given function.

What you’ll learn about

Derivatives of Inverse Functions

Derivatives of the Arcsine

Derivatives of the Arctangent

Derivatives of the Arcsecant

Derivatives of the Other Three

… and why

The relationship between the graph of a function and its inverse

allows us to see the relationship between their derivatives.

Slide 3- 6

Derivatives of Inverse Functions

If f is differentiable at every point of an interval I and dy

dxis never zero on I , then f has an inverse and f 1 is differentiable

at every point on the interval f I .

Slide 3- 7

Derivative of the Arcsine

If u is a differentiable function of x with u 1, we apply the

Chain Rule to get

d

dxsin 1 u

1

1 u2

du

dx, u 1.

Slide 3- 8

Let f(x) = sin x and g(x) = sin-1 x to verify the formula for the derivative of sin-1 x.

Example Derivative of the Arcsine

If ysin 1 8x2 , find

dy

dx.

Slide 3- 10

Example Derivative of the Arcsine

If ysin 1(1 t), find

dy

dx.

Slide 3- 11

Derivative of the Arctangent

The derivative is defined for all real numbers.

If u is a differentiable function of x, we apply the

Chain Rule to get

d

dxtan 1 u

1

1u2

du

dx.

Slide 3- 12

y = tan-1 (4x)

Determine

dy

dx.

y = x tan-1x

Determine

dy

dx.

Derivative of the Arcsecant

If u is a differentiable function of x with u 1, we have the

formula

d

dxsec 1 u

1

u u2 1

du

dx, u 1.

Slide 3- 15

Example Derivative of the Arcsecant

1Given sec 3 4 , find .dy

y xdx

Slide 3- 16

A particle moves along the x – axis so that its position at any time t ≥ 0 is given by x(t). Find the velocity at the indicated value of t.

x(t) sin 1 t

4

, t 4

Assignment 3.8.1

page 170,

# 3 – 11 odds

Inverse Function – Inverse Cofunction

Identities

1 1

1 1

1 1

cos sin2

cot tan2

csc sec2

x x

x x

x x

Slide 3- 19

Determine

dy

dx if y cos 1 x.

Derivatives of Inverse Trig Functions

Function

arcsin x

arccos x

arctan x

arcsec x

Derivative

2

1

1 x 2

1

1 x

2

1

1 x2

1

1x x

Example Derivative of the Arccotangent

Slide 3- 22

1 2Find the derivative of cot .y x

Calculator Conversion Identities

1 1

1 1

1 1

1sec cos

cot tan2

1csc sin

xx

x x

xx

Slide 3- 23

Determine the derivative of y with respect to the variable.

y cos 1 1

x

Determine the derivative of y with respect to the variable.

y sec 1 5s

Determine the derivative of y with respect to the variable.

y csc 1 x

2

Determine the derivative of y with respect to the variable.

y s2 1 sec 1 s

Find an equation for the tangent to the graph of y at the indicated point.

y tan 1 x, x 2

Find an equation for the tangent to the graph of y at the indicated point.

y cos 1 x

4

, x 5

Let f(x) = cos x + 3x

Show that f(x) has a differentiable inverse.

Let f(x) = cos x + 3x

Determine f(0) and f’(0).

Let f(x) = cos x + 3x

Determine f-1(1) and f-1(1).

y = cot-1 x

Determine the right end behavior model.

y = cot-1 x

Determine the left end behavior model.

y = cot-1 x

Does the function have any horizontal tangents?

Assignment 3.8.2

pages 170 – 171,

# 1, 13 – 29 odds, 32 and 41 – 45 odds