What do your primes denote differentiation with respect to?
Write your expression as exp(ikx).exp(-iwt). Depending upon what you are differentiating with respect to, one of these terms can be treated as a constant. You know how to differentiate exp(x), right?
I'm reviewing Meirovitch's "Methods of Analytical Dynamics," and I don't understand the commutation of the derivative from r to dr:
$$
\mathbf{F} \cdot d\mathbf{r} = m \ddot{\mathbf{r}} \cdot d\mathbf{r} = m\mathbf{\dot{r}} \cdot d\mathbf{\dot{r}}
$$