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

1. Apply the quotient rule, which is:

   $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

   $f{\left(x \right)} = 2^{3 x}$ and $g{\left(x \right)} = 3^{2 x}$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = 3 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$:

      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: $3$

      The result of the chain rule is:

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

   To find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 2 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$:

      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:

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

   Now plug in to the quotient rule:

   $3^{- 4 x} \left(- 2 \cdot 2^{3 x} 3^{2 x} \log{\left(3 \right)} + 3 \cdot 2^{3 x} 3^{2 x} \log{\left(2 \right)}\right)$

2. Now simplify:

   $$\log{\left(\left(\frac{8}{9}\right)^{72^{x} 81^{- x}} \right)}$$

The answer is:

$$\log{\left(\left(\frac{8}{9}\right)^{72^{x} 81^{- x}} \right)}$$