Derivative of Hyperbolic Cosecant

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

Let the function of the form

y = f\left( x \right) = \operatorname{csch} x

By definition of 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}

Differentiation 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}