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Lesson 9

Differentiation of

Inverse Trigonometric

Functions

OBJECTIVES

• to derive the formula for the derivatives of the

inverse trigonometric functions;

• to apply the derivative formulas to solve for the

derivatives of inverse trigonometric functions;

and

• to solve problems involving derivatives of

inverse trigonometric functions.

DIFFERENTIATION FORMULA

Derivative of Inverse Trigonometric

Function

Derivative of Inverse Trigonometric

Function

( )

( )

( )

( )

( )

( ) dx

du

u u 1

1 csc u dx

d

dx

du

u u 1

1 sec u dx

d

dx

du

1 u

1 cot u dx

d

dx

du

1 u

1 tan u dx

d

dx

du

1 u

1 cos u dx

d

dx

du

1 u

1 sin u dx

d

Differentiation formulas forinverse trigonometric functions:

2

1

2

1

2

1

2

1

2

1

2

1

−

=−

−

=

=−

=

−

=−

−

=

−

−

−

−

−

−

( ) ( )

1 x sin 2 e 2

1

5. h x

−

( )

x^2

x

1 2 e

2 e

2

1 h' x

−

= ⋅ 2 x

x

1 4 e

e

−

=

6. g ( )t sec 5 t csc 5 t

− 1 − 1

g' ( )t 0

5

5t 25t 1

( 1) 5

5t 25t 1

1 g' t 2 2

=

−

−

−

=

x

2

7. g x cot

= 1



  

 −

 

  



−

2 2 x

2

x

2 1

1 g' x 2 2

x x

4 1

2

^ ⋅ 

  



= (^) ( )

x 4

g' x

2

8. f( )x x tan ( 3 x)

2 − 1

) x

5

9. y Sec (csc

− 1

1 x

5 csc x

5 csc

x

5

x

5 cot x

5 csc

y' 2

2

^ − 

  



 



 

 − −

= x

5 cot x

5 1 cot x

5 but, csc

2 2

^ = 

  

  − = 

  



2

'

x

5 y =