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 t^{2} = u & tan^{1}(u)
so your derivative of tan^{1}(u) is 1 / (1 + u^{2})
& the derivative of u is 2t so
multiply (1 / (1 + u^{2})) •
(2t). So you have,
(1 / (1 + (t^{2})^{2})) •
(2t)
Make 3t = u & tan(u) so your
derivative of tan(u) is sec^{2}(u) &
the derivative of u is 3 so multiply (sec^{2}(u))
• (3). So you have,
(sec^{2}(3t)) • (3)
Do a little manipulation...