1. ## complex derivative

The following problem appears in my complex analysis text and I was wondering how you solve it in this context.
Find the 10th derivative of [e^(0.5x)]*sin((sqrt(3)/2)x) (obviously without taking the derivative ten times).

Thanks for any input.

2. Originally Posted by PvtBillPilgrim
The following problem appears in my complex analysis text and I was wondering how you solve it in this context.
Find the 10th derivative of [e^(0.5x)]*sin((sqrt(3)/2)x) (obviously without taking the derivative ten times).

Thanks for any input.
we still have to take quite a few derivatives from what i can see, but they should be easier as you can see a pattern.

note that the derivative you seek is the real part of the 10th derivative of $-i \text{exp}\left( \frac x2 + i\frac {\sqrt{3}}2x\right)$ which is the real part of $-i \left( \frac 12 + i \frac {\sqrt{3}}2\right)^{10} \text{exp} \left( \frac x2 + i\frac {\sqrt{3}}2x\right) = -i \text{exp} \left( i \frac {10 \pi}3\right) \cdot \text{exp} \left( \frac x2 + i\frac {\sqrt{3}}2x\right)$ $= -i \text{exp} \left[ \frac x2 + i \left( \frac {10 \pi}3 + \frac {\sqrt{3}}2 \right) x\right]$