# Derivative of Hyperbolic Sine

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

Let the function of the form

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

Now take this function for differentiation, we have

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, $\cosh x = \frac{{{e^x} + {e^{ - x}}}}{2}$, we get

Example: Find the derivative of

We have the given function as

Differentiation with respect to variable $x$, we get

Using the rule, $\frac{d}{{dx}}\left( {\sinh x} \right) = \cos hx$, we get