Derivative of 2*e^(4*x)+5*log(4*x)

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

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

      1. Let $u = 4 x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$:

         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: $4$

         The result of the chain rule is:

         $$4 e^{4 x}$$

      So, the result is: $8 e^{4 x}$

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

      1. Let $u = 4 x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$:

         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: $4$

         The result of the chain rule is:

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

      So, the result is: $\frac{5}{x}$

   The result is: $8 e^{4 x} + \frac{5}{x}$

The answer is:

$$8 e^{4 x} + \frac{5}{x}$$