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The derivatives of remaining trigonometric functions, ie., and can also be found easily using the quotient rule. Their derivatives are given by the following formulae:
i. (tan x) = sec2 x
ii. (cot x) = – cosec2 x
iii. (sec x) = sec x tan x
iv (cosec x) = – cosec x cot x
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:
If 'y' is a differentiable function of 'x', then we apply the chain rule to get (Tan– 1 y) =
We use the same technique to find the derivatives of remaining inverse trigonometric functions (i.e, arccosine, arccotangent, arcsecant, arccosecant). They are summarised below:
We know that: sinh–1 x = log(x + ), cosh–1 x = log(x + ) and tanh–1 x = It can be proved that:
Did you note that the derivatives of tanh–1 x and coth–1 x are the same? (Their domains are disjoint though).
Most of the times, the domain is not explicitly stated. We find the derivative in its appropriate domain.