We know that the function x = tan y is differentiable for all real numbers.
From the definition of derivative of an inverse function, the inverse function y = Tan– 1 (x) is differentiable for all the real numbers.
We find the derivative of y = Tan– 1 (x) as follows:
| y | = | Tan– 1 (x) | ||
| tan y | = | x [∵ Inverse fun. relationship] | ||
 (tan y) |
= | (x) [∵ Diff. both sides w.r.t. x] |
||
sec2 y  |
= | 1 [∵ Implicit diff.] | ||
|
= |  |
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 |
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If 'y' is a differentiable function of 'x', then we apply the chain rule to get (Tan– 1 y) = 
Derivatives of other basic inverse trigonometric functions
We use the same technique to find the derivatives of remaining inverse trigonometric functions (i.e, arccosine, arccotangent, arcsecant, arccosecant). They are summarised below:
| Derivatives of Inverse Trigonometric Functions | |
|---|---|
i.  (Sin– 1 x) =  |
ii. (Cos– 1 x) =  |
iii. (Tan– 1 x) =  |
iv. (Cot– 1 x) =  |
v. (Sec– 1 x) =  |
vi. (Cosec– 1 x) =  |