Derivative of 2^x-2^-x

1. Differentiate $2^{x} - 2^{- x}$ term by term:

   1. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$

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

      1. Let $u = - x$.

      2. $\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$

      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:

         $$- 2^{- x} \log{\left(2 \right)}$$

      So, the result is: $2^{- x} \log{\left(2 \right)}$

   The result is: $2^{x} \log{\left(2 \right)} + 2^{- x} \log{\left(2 \right)}$

2. Now simplify:

   $$2^{- x} \left(4^{x} + 1\right) \log{\left(2 \right)}$$

The answer is:

$$2^{- x} \left(4^{x} + 1\right) \log{\left(2 \right)}$$