Derivative of -4*e^(2*x)-2*log(6*x)

1. Differentiate $- 4 e^{2 x} - 2 \log{\left(6 x \right)}$ term by term:

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

      1. Let $u = 2 x$.

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

      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:

         $$2 e^{2 x}$$

      So, the result is: $- 8 e^{2 x}$

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

      1. Let $u = 6 x$.

      2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 6 x$:

         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: $6$

         The result of the chain rule is:

         $$\frac{1}{x}$$

      So, the result is: $- \frac{2}{x}$

   The result is: $- 8 e^{2 x} - \frac{2}{x}$

The answer is:

$$- 8 e^{2 x} - \frac{2}{x}$$