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Math Help - nth Derivative notation question

  1. #1
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    nth Derivative notation question

    Find an expression for the nth derivative of f(x)=e^(x) * sin(x)

    so what would the nth derivative of f be: f^(n) (x)

    You can make a different expression for odd or even numbers, but I still can't quiet get one that works.
    Please show me one that would work!!!
    THANKS
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  2. #2
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    Re: nth Derivative notation question

    \displaystyle \begin{align*} f(x) &= e^x\sin{x}\\ \\ f'(x) &= e^x\sin{x} + e^x\cos{x}\\ &= f(x) + e^x\cos{x}\\ \\ f''(x) &= f'(x) + e^x\cos{x} - e^x\sin{x} \\ &= f'(x) - f(x) + e^x\cos{x}\\ \\ f'''(x) &= f''(x) - f'(x) + e^x\cos{x} - e^x\sin{x} \\ &= f''(x) - f'(x) - f(x) + e^x\cos{x}\\ \\ f^{(\textrm{iv})}(x) &= f'''(x) - f''(x) - f'(x) + e^x\cos{x} - e^x\sin{x} \\ &= f'''(x) - f''(x) - f'(x) - f(x) + e^x\cos{x} \end{align*}

    I think you can see a pattern here...
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  3. #3
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    Re: nth Derivative notation question

    Yeah I know the pattern but how could you put it all into only a few expressions?
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  4. #4
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    Re: nth Derivative notation question

    You could use the sum notation. For the nth derivative, you start with (n-1)th derivative + e^x \cos x, and then do the sum of all the lower order derivatives and the original function.. don't forget the minus sign... It should look something like this:

    y = e^x \sin x

    \frac{\mathrm{d}^n y}{\mathrm{d} x^n} = e^x \cos x + \frac{\mathrm{d}^{n - 1} y}{\mathrm{d} x^{n - 1}} - \sum_{i=0}^{n-2} \left [\frac{\mathrm{d}^i y}{\mathrm{d} x^i} \right \]

    This gives the expression for all the nth derivatives of the function (so n > 0), assuming the sum term is ignored if n -2 < 0.

    That being said, I don't really see the point in doing this. It's not like that new expression is easier to use or something. Do you have any other information associated with this, perhaps the context would make it easier to know what you are really asking for (because I'm not sure this is what you want).
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  5. #5
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    Re: nth Derivative notation question

    Quote Originally Posted by NeedDirection View Post
    Find an expression for the nth derivative of f(x)=e^(x) * sin(x)

    so what would the nth derivative of f be: f^(n) (x)

    You can make a different expression for odd or even numbers, but I still can't quiet get one that works.
    Please show me one that would work!!!
    THANKS
    Try the Leibniz Rule for the nth derivative of a product.
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  6. #6
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    Re: nth Derivative notation question

    Quote Originally Posted by NeedDirection View Post
    Find an expression for the nth derivative of f(x)=e^(x) * sin(x)

    so what would the nth derivative of f be: f^(n) (x)

    You can make a different expression for odd or even numbers, but I still can't quiet get one that works.
    Please show me one that would work!!!
    THANKS
    \begin{aligned} f^{(n)}(x) & = \sum_{k=0}^{n}\binom{n}{k} e^{x}\sin\left(\frac{k\pi}{2}+x\right) \\& = \Im \bigg[ e^{x} \sum_{k=0}^{n} \binom{n}{k}e^{i(\frac{k\pi}{2}+x)}\bigg] \\ & = \Im\bigg[e^{x}e^{ix}\sum_{k=0}^{n}\binom{n}{k}(e^\frac{i\pi  }{2})^k}\bigg] \\ & =\Im\bigg[e^{x}e^{ix}(1+e^\frac{\i\pi}{2})^n\bigg]  \\ & = e^{x}~2^{\frac{n}{2}}~\sin\left(\frac{n\pi}{4}+x \right). \end{aligned}
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  7. #7
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    Re: nth Derivative notation question

    Wow, thanks a lot everyone! This makes perfect sense now!
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