# Using the Complex Exponential

1. Jun 5, 2004

### dcl

How would one use the complex exponential to find something like this:
$$\frac{{d^{10} }}{{dx^{10} }}e^x \cos (x\sqrt 3 )$$
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. Im guessing there is another way?

2. Jun 5, 2004

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

so

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

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