Derivative of 12*sqrt(2)*cos(x)+12*x-3*pi+6

1. Differentiate $\left(\left(12 x + 12 \sqrt{2} \cos{\left(x \right)}\right) - 3 \pi\right) + 6$ term by term:

   1. Differentiate $\left(12 x + 12 \sqrt{2} \cos{\left(x \right)}\right) - 3 \pi$ term by term:

      1. Differentiate $12 x + 12 \sqrt{2} \cos{\left(x \right)}$ 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 cosine is negative sine:

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

            So, the result is: $- 12 \sqrt{2} \sin{\left(x \right)}$

         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: $12$

         The result is: $- 12 \sqrt{2} \sin{\left(x \right)} + 12$

      2. The derivative of the constant $- 3 \pi$ is zero.

      The result is: $- 12 \sqrt{2} \sin{\left(x \right)} + 12$

   2. The derivative of the constant $6$ is zero.

   The result is: $- 12 \sqrt{2} \sin{\left(x \right)} + 12$

The answer is:

$$- 12 \sqrt{2} \sin{\left(x \right)} + 12$$