Derivative of 6cos(2000t+5x+1000)

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

   1. Let $u = \left(2000 t + 5 x\right) + 1000$.

   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 $\frac{\partial}{\partial x} \left(\left(2000 t + 5 x\right) + 1000\right)$:

      1. Differentiate $\left(2000 t + 5 x\right) + 1000$ term by term:

         1. Differentiate $2000 t + 5 x$ term by term:

            1. The derivative of the constant $2000 t$ is zero.

            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: $5$

            The result is: $5$

         2. The derivative of the constant $1000$ is zero.

         The result is: $5$

      The result of the chain rule is:

      $$- 5 \sin{\left(\left(2000 t + 5 x\right) + 1000 \right)}$$

   So, the result is: $- 30 \sin{\left(\left(2000 t + 5 x\right) + 1000 \right)}$

2. Now simplify:

   $$- 30 \sin{\left(2000 t + 5 x + 1000 \right)}$$

The answer is:

$$- 30 \sin{\left(2000 t + 5 x + 1000 \right)}$$