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Math Help - What Is The nth Derivative of sin(x)

  1. #1
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    What Is The nth Derivative of sin(x)

    How can you write out the nth derivative of sin(x)?

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  2. #2
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    Let f(x)=\sin x

    f'(x)=\cos x

    f''(x)=-\sin x

    f'''(x)=-\cos x

    f''''(x)=\sin x and then the cycle repeats


    So you could just write it out using if's and keep dividing the n of  f^{(n)} by 4 to get the remainder

    But upon further analysis, we realize

    f'(x)=\cos x=\sin (\frac{\pi}{2}+x)

    f''(x)=-\sin x=\sin(\frac{2\pi}{2}+x)

    f'''(x)=-\cos x=\sin(\frac{3\pi}{2}+x)

    f''''(x)=\sin x=\sin(\frac{4\pi}{2}+x)

    So f^{(n)}(x)=\sin(\frac{n\pi}{2}+x)
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  3. #3
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    Smile

    Works for me, thanks!
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