Derivative of cos(k*x)

1. Let $u = k x$.

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 $\frac{\partial}{\partial x} k x$:

   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: $k$

   The result of the chain rule is:

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

The answer is:

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