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Calculus

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If f(x) = (e^(x))sin(x) what is the 1000th derivative?

Step are needed and the general rule/pattern

  • Calculus - ,

    y = e^x(sinx)
    yI = e^xcosx + e^xsinx
    = e^x(cosx + sinx)
    yII = e^x(-sinx + cosx) + e^x(cosx + sinx)
    = e^x(2cosx) = 2e^x(cosx)
    yIII = 2e^x(-sinx) + 2e^x(cosx)
    = 2e^x(cosx - sinx)

    yIV = 2e^x(-sinx-cosx) + 2e^x(cosx - sinx)
    = 2e^x(-2sinx_
    = -4e^x(sinx)

    Ahhh so it took 4 derivatives to reach -4(what we started with)
    so yVIII would be (-4)(-4)e^x(sinx)
    etc.
    so y1000= (-4)^250(e^x(sinx))
    = (4^250)(e^x(sinx))

  • Calculus - ,

    where did u get the 250 from?

  • Calculus - ,

    every fourth one is 4 times four before it.
    and
    1000/4 is 250

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