Derivative of the arctangent

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 function relationship]
(tan y) = (x) [∵ Differentiating both sides w.r.t. x]
sec2 y = 1 [∵ Implicit differentiation]
=

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) =