Calculus Examples

Derivatives of Inverse Trigonometric Functions

 

1 - Derivative of arcsin(x) or sin-1(x)

The derivative of f(x) = arcsin x is given by

 
f '(x) = 1 / sqrt(1 - x 2)


 

2 - Derivative of arccos(x) or cos-1(x)

The derivative of f(x) = arccos x is given by

 
f '(x) = - 1 / sqrt(1 - x 2)


 

3 - Derivative of arctan(x) or tan-1(x)

The derivative of f(x) = arctan x is given by

 
f '(x) = 1 / (1 + x 2)


 

4 - Derivative of arccot(x) or cot-1(x)

The derivative of f(x) = arccot x is given by

 
f '(x) = - 1 / (1 + x 2)


 

5 - Derivative of arcsec(x) or sec-1(x)

The derivative of f(x) = arcsec x tan x is given by

 
f '(x) = 1 / x sqrt(x 2 - 1)


 

6 - Derivative of arccsc(x) or csc-1(x)

The derivative of f(x) = arccsc x is given by

 
f '(x) = - 1 / x sqrt(x 2 - 1)

Example of 4,d):

This is the original equation.

Make t2 = u   &   tan-1(u)   so your derivative of tan-1(u) is 1 / (1 + u2)    &    the derivative of u is 2t    so   multiply (1 / (1 + u2)) (2t).  So you have,

                                                                                                                                                                                        (1 / (1 + (t2)2)) (2t)

Make 3t = u   &   tan(u)   so your derivative of tan(u) is sec2(u)    &    the derivative of u is 3   so    multiply (sec2(u)) (3). So you have,

                                                                                                                                                                            (sec2(3t)) (3)

Do a little manipulation...