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Thread: nth derivative

  1. #1
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    nth derivative

    Show that

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

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

    Try using
    $\displaystyle
    e ^ {ix } = cos (x) + i sin (x)
    $
    to write
    $\displaystyle
    sin (x) = Im ( e ^ {ix } )
    $
    where Im means "imaginary part of".
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  3. #3
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    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?
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  4. #4
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    Try to learn about complex numbers

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

    Check your text for a review on complex numbers.
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  5. #5
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    ok i guess i'll try to learn about them and then i'll get back to you
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