You’re probably familiar with the standard trig functions $\sin(x)$ and $\cos(x)$ and how they relate to a circle. There are also special functions known as the hyperbolic functions that are similar to the trig functions, but relate to hyperbolas instead of circles.
Here are the formulas for the hyperbolic cosine ($\cosh$) and hyperbolic sine ($\sinh$) functions:
$$ \cosh(x) = \frac{e^x + e^{-x}}{2} $$
$$ \sinh(x) = \frac{e^x - e^{-x}}{2} $$
With the standard trig functions, $\cos^2(x) + \sin^2(x) = 1$. The equivalent identity for the hyperbolic functions is $\cosh^2(x) - \sinh^2(x) = 1$ (notice the minus sign instead of the plus sign!) This identity can be derived from the above formulas for $\cosh(x)$ and $\sinh(x)$.
The hyperbolic functions $\cosh(x)$ and $\sinh(x)$ are derivatives of each other:
$$ \frac{d}{dx}\cosh(x) = \sinh(x) $$
$$ \frac{d}{dx}\sinh(x) = \cosh(x) $$
Notice how the derivative of $\cosh(x)$ is just $\sinh(x)$, not $-\sinh(x)$! This is different from how the derivative of $\cos(x)$ is $-\sin(x)$.
You can find these derivatives using basic derivative rules and the formulas for $\cosh(x)$ and $\sinh(x)$.