Derivative of 6*e^(5*x)-3*log(2*x)

1. Differentiate $6 e^{5 x} - 3 \log{\left(2 x \right)}$ term by term:

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

      1. Let $u = 5 x$.

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

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

         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: $5$

         The result of the chain rule is:

         $$5 e^{5 x}$$

      So, the result is: $30 e^{5 x}$

   2. 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. Let $u = 2 x$.

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

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

            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:

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

         So, the result is: $\frac{3}{x}$

      So, the result is: $- \frac{3}{x}$

   The result is: $30 e^{5 x} - \frac{3}{x}$

The answer is:

$$30 e^{5 x} - \frac{3}{x}$$