Derivative of exp^x-exp^-(x)-2

1. Differentiate $\left(e^{x} - e^{- x}\right) - 2$ term by term:

   1. Differentiate $e^{x} - e^{- x}$ term by term:

      1. The derivative of $e^{x}$ is itself.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Let $u = - x$.

         2. The derivative of $e^{u}$ is itself.

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x\right)$:

            1. The derivative of a constant times a function is the constant times the derivative of the function.

               1. Apply the power rule: $x$ goes to $1$

               So, the result is: $-1$

            The result of the chain rule is:

            $$- e^{- x}$$

         So, the result is: $e^{- x}$

      The result is: $e^{x} + e^{- x}$

   2. The derivative of the constant $-2$ is zero.

   The result is: $e^{x} + e^{- x}$

2. Now simplify:

   $$2 \cosh{\left(x \right)}$$

The answer is:

$$2 \cosh{\left(x \right)}$$