# Thread: nth derivative

1. ## nth derivative

Show that

$\frac{d^n}{dx^n} (e^{ax}sinbx) = r^ne^{ax}sin(bx+n\theta)$

where $a$ and $b$ are positive numbers, $r^2 = a^2 + b^2$ and $\theta =tan^{-1}(b/a)$

2. ## Rewrite the sin

Try using
$
e ^ {ix } = cos (x) + i sin (x)
$

to write
$
sin (x) = Im ( e ^ {ix } )
$

where Im means "imaginary part of".

3. I really don't know much about imaginary numbers, so I don't know where to go from there. Is there possibly any other way to solve this problem?

4. ## Try to learn about complex numbers

Complex numbers are very useful. They also have some interesting properties. One is, if:
$
z = r ( cos (\theta) + i sin (\theta) )
$

then
$
z^n = r^n ( cos (n\theta) + i sin (n\theta) )
$

Check your text for a review on complex numbers.

5. ok i guess i'll try to learn about them and then i'll get back to you