Derivative of e^4*t*(-2)-2*log(5*t)

1. Differentiate $\left(-2\right) e^{4} t - 2 \log{\left(5 t \right)}$ term by term:

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

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

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

         So, the result is: $e^{4}$

      So, the result is: $- 2 e^{4}$

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

      1. Let $u = 5 t$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d t} 5 t$:

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

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

            So, the result is: $5$

         The result of the chain rule is:

         $$\frac{1}{t}$$

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

   The result is: $- 2 e^{4} - \frac{2}{t}$

The answer is:

$$- 2 e^{4} - \frac{2}{t}$$