Derivative of 3^x-3^(x*(-2))

1. Differentiate $3^{x} - 3^{\left(-2\right) x}$ term by term:

   1. $\frac{d}{d x} 3^{x} = 3^{x} \log{\left(3 \right)}$

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

      1. Let $u = \left(-2\right) x$.

      2. $\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(-2\right) x$:

         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: $-2$

         The result of the chain rule is:

         $$- 2 \cdot 3^{\left(-2\right) x} \log{\left(3 \right)}$$

      So, the result is: $2 \cdot 3^{\left(-2\right) x} \log{\left(3 \right)}$

   The result is: $3^{x} \log{\left(3 \right)} + 2 \cdot 3^{\left(-2\right) x} \log{\left(3 \right)}$

2. Now simplify:

   $$9^{- x} \left(27^{x} + 2\right) \log{\left(3 \right)}$$

The answer is:

$$9^{- x} \left(27^{x} + 2\right) \log{\left(3 \right)}$$