# Derivative of Cosecant Inverse

In this tutorial we shall explore the derivative of inverse trigonometric functions and we shall prove derivative of cosecant inverse.

Let the function of the form

By definition of inverse trigonometric function, $y = {\csc ^{ - 1}}x$ can be written as

Differentiating both sides with respect to the variable $x$, we have

We can write from the fundamental trigonometric rules $\cot y = \sqrt {{{\csc }^2}y - 1}$. Putting this value in above relation (i) and simplifying, we have

But we have $\csc y = x$, putting this value in above relation

Example: Find the derivative of

We have the given function as

Differentiation with respect to variable $x$, we get

Using the cosine inverse rule, $\frac{d}{{dx}}\left( {{{\csc }^{ - 1}}x} \right) = - \frac{1}{{x\sqrt {{x^2} - 1} }}$, we get

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