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) =|