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

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To find the tenth derivative of e^x cos(x√3) using complex exponentials, one can convert cos(x√3) into exponential form using e^{iθ} = cos(θ) + i sin(θ). This leads to the expression cos(x√3) = (e^{ix√3} + e^{-ix√3})/2. Each successive derivative of the resulting expression simplifies the process, as it involves adding powers to the coefficients of e. This method avoids tedious calculations and streamlines the differentiation process. The discussion highlights the efficiency of using complex exponentials for higher derivatives.
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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.
 
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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
 
Last edited:
Ahhh, fair enough :)
Thought it was going to be tedious, but isn't nearly that bad..
 

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