Derivative of (3cos*4x-1/2x)

1. Differentiate $- \frac{x}{2} + 3 \cos{\left(4 x \right)}$ 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 cosine is negative sine:

         $$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$$

      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:

         $$- 4 \sin{\left(4 x \right)}$$

      So, the result is: $- 12 \sin{\left(4 x \right)}$

   2. 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: $- \frac{1}{2}$

   The result is: $- 12 \sin{\left(4 x \right)} - \frac{1}{2}$

The answer is:

$$- 12 \sin{\left(4 x \right)} - \frac{1}{2}$$