DERIVATIVES OF THE INVERSE TRIGONOMETRIC FUNCTIONS

Section 3.8

Derivatives of Inverse FunctionsTheorem: If is differentiable at every pointof an interval I and is never zero on I,then has an inverse and is differentiableat every point of the interval I.

fdf dx

f 1f

Derivatives of Inverse Functionsy

x

1 1

f a

a

dfdfdxdx

y f x

1y f x ,a f a

,f a a

The slopes of inverse functions are reciprocals,at the corresponding points… in math symbols

Derivatives of Inverse FunctionsLet . Given that the point is onthe graph of , find the slope of the inverse of at .

5 2 1f x x x f

1,2f 2x

Our new rule: The slope of at is the reciprocal ofthe slope of at .

f 1x 1f 2x

45 2df xdx

4

1

5 1 2 7x

dfdx

1

2

17x

dfdx

First, recall the graph:

x

y

–1 1So, should this function be

differentiable across itsentire domain???

Everywhere except at x = –1 or 1

2

2

: 1,1D

: 2, 2R

Derivative of the Arcsine1siny x

Derivative of the Arcsine1siny x

sin y x

1cos

dydx y

(sin )d dy xdx dx

2

1

1 (sin )

dydx y

cos 1dyydx

2

1

1

dydx x

Derivative of the Arcsine

1

2

1sin ,1

d duudx dxu

If is a differentiable function of with ,applying the Chain Rule:u x 1u

1u

Derivative of the Arctangent1tany x

tan y x2

1sec

dydx y

(tan )d dy xdx dx

2

11 (tan )

dydx y

2sec 1dyydx

2

11

dydx x

Derivative of the Arctangent

12

1tan1

d duudx u dx

If is a differentiable function of , again usingthe Chain Rule form:u x

Derivative of the Arcsecant1secy x

sec y x

1sec tan

dydx y y

(sec )d dy xdx dx

2

1

sec sec 1

dydx y y

sec tan 1dyy ydx

2

1

1

dydx x x

2

1

1

dydx x x

Derivative of the ArcsecantIf is a differentiable function of with , and“chaining” once again, we have:u x

1

2

1sec ,1

d duudx dxu u

u

1u

Derivative of the Others

The derivatives of the inverse cofunctionsare the opposites (negatives) of the derivativesof the corresponding inverse functions

1 1cot 2 tanx x

1 1cos 2 sinx x

1 1csc 2 secx x

Inverse Function – Inverse Cofunction Identities:

Guided Practice

Find if

2

22

1

1

dy d xdx dxx

4

2

1

x

x

dydx

1 2siny x

Guided Practice

Find ifdydx

1 4sec 5y x

4

24 4

1 55 5 1

dy d xdx dxx x

3

4 8

1 205 25 1

xx x

8

4

25 1x x

Guided PracticeA particle moves along the x-axis so that its positionat any time is . What is thevelocity of the particle when ?

1tandv t tdt

2

1

1

d tdtt

1 11 2t t

First, find the general equation for velocity:

0t 1tanx t t16t

Guided PracticeA particle moves along the x-axis so that its positionat any time is . What is thevelocity of the particle when ?

0t 1tanx t t16t

1 1161 16 2 16

v

1

136

Now, at the particular time: