How to use complex exponential to find higher derivatives of e^x cos(x√3)?

[dcl]

How would one use the complex exponential to find something like this:
<br /> \frac{{d^{10} }}{{dx^{10} }}e^x \cos (x\sqrt 3 )
I'm guessing we'd have to convert the cos into terms of e^{i\theta } but the only thing I can think of doing then is going through each of the derivatives. I am guessing there is another way?

thanks in advance.

 

[cookiemonster]

e^{i \theta} = \cos{\theta} + i\sin{\theta}

so

\cos{\theta} = \frac{e^{i \theta} + e^{-i \theta}}{2}

Then each successive derivative just adds a power to the coefficient of e.

cookiemonster

 

[dcl]

Ahhh, fair enough :)
Thought it was going to be tedious, but isn't nearly that bad..