Derivative of 4xln(2x)−4x

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

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

      1. Apply the product rule:

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

         $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

         $g{\left(x \right)} = \log{\left(2 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

         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:

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

         The result is: $\log{\left(2 x \right)} + 1$

      So, the result is: $4 \log{\left(2 x \right)} + 4$

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

         So, the result is: $4$

      So, the result is: $-4$

   The result is: $4 \log{\left(2 x \right)}$

The answer is:

$$4 \log{\left(2 x \right)}$$