# Derivative of Hyperbolic Cosecant

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

Let the function be of the form $y = f\left( x \right) = \operatorname{csch} x$

By the definition of the hyperbolic function, the hyperbolic cosecant function is defined as
$\operatorname{csch} x = \frac{2}{{{e^x} – {e^{ – x}}}}$

Differentiating both sides with respect to the variable $x$, we have
$\begin{gathered} \frac{d}{{dx}}\operatorname{csch} x = \frac{d}{{dx}}\left( {\frac{2}{{{e^x} – {e^{ – x}}}}} \right) \\ \Rightarrow \frac{d}{{dx}}\operatorname{csch} x = 2\frac{d}{{dx}}{\left( {{e^x} – {e^{ – x}}} \right)^{ – 1}} \\ \Rightarrow \frac{d}{{dx}}\operatorname{csch} x = – 2{\left( {{e^x} – {e^{ – x}}} \right)^{ – 2}}\frac{d}{{dx}}\left( {{e^x} – {e^{ – x}}} \right) \\ \end{gathered}$

Using the formula of exponential differentiation $\frac{d}{{dx}}{e^x} = {e^x}$, we have
$\begin{gathered} \frac{d}{{dx}}\operatorname{csch} x = – 2{\left( {{e^x} – {e^{ – x}}} \right)^{ – 2}}\left( {{e^x} + {e^{ – x}}} \right) \\ \Rightarrow \frac{d}{{dx}}\operatorname{csch} x = – 2\frac{{\left( {{e^x} + {e^{ – x}}} \right)}}{{{{\left( {{e^x} – {e^{ – x}}} \right)}^2}}} \\ \Rightarrow \frac{d}{{dx}}\operatorname{csch} x = – \frac{2}{{{e^x} – {e^{ – x}}}}\frac{{{e^x} + {e^{ – x}}}}{{{e^x} – {e^{ – x}}}} \\ \end{gathered}$

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
$\frac{d}{{dx}}\operatorname{csch} x = – \csc hx\coth x$

Example: Find the derivative of $y = f\left( x \right) = \operatorname{csch} 4{x^2}$

We have the given function as
$y = \operatorname{csch} 4{x^2}$

Differentiating with respect to variable $x$, we get
$\frac{{dy}}{{dx}} = \frac{d}{{dx}}\operatorname{csch} 4{x^2}$

Using the rule, $\frac{d}{{dx}}\operatorname{csch} x = – \csc hx\coth x$, we get
$\begin{gathered} \frac{{dy}}{{dx}} = – \operatorname{csch} 4{x^2}\coth 4{x^2}\frac{d}{{dx}}\left( {4{x^2}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = – \operatorname{csch} 4{x^2}\coth 4{x^2}\left( {8x} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = – 8x\operatorname{csch} 4{x^2}\coth 4{x^2} \\ \end{gathered}$