# Derivative of Inverse Hyperbolic Cosecant

In this tutorial we shall be concerned with the derivative of inverse hyperbolic cosecant function with an example.

Let the function of the form

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

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

We can write from the fundamental rules of inverse hyperbolic identities $\coth y = \sqrt {1 + \csc {{\text{h}}^2}x}$. Putting this value in above relation (i) and simplifying, we have

From the above we have $\operatorname{csch} y = x$, by putting this

Example: Find the derivative of

We have the given function as

Differentiation with respect to variable $x$, we get

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