# Derivative of Hyperbolic Cosecant

In this tutorial we shall prove the derivative of the hyperbolic cosecant function.

Let the function be of the form

By the definition of the hyperbolic function, the hyperbolic cosecant function is defined as

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

Using the formula of exponential differentiation $\frac{d}{{dx}}{e^x} = {e^x}$, we have

By definition, $\operatorname{csch} x = \frac{2}{{{e^x} - {e^{ - x}}}}$ and $\coth x = \frac{{{e^x} + {e^{ - x}}}}{{{e^x} - {e^{ - x}}}}$, we get

Example: Find the derivative of

We have the given function as

Differentiating with respect to variable $x$, we get

Using the rule, $\frac{d}{{dx}}\operatorname{csch} x = - \csc hx\coth x$, we get