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12 June, 07:00

Find the 75th derivative of y = cos (2x).

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  1. 12 June, 07:14
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    Given: y = cos (2x).

    Let y () denote the n-th derivative of y wrt x.

    The first few derivatives of y wrt x are the following:

    n=1: y (¹) = - 2¹ sin (2x)

    n=2: y (²) = - 2² cos (2x)

    n=3: y (³) = + 2³ sin (2x)

    n=4: y () = + 2⁴ cos (2x)

    n=5: y () = - 2⁵ sin (2x)

    and so on

    A pattern emerges

    y () = - 2ⁿ sin (2x) for n = 1, 5, 9, ...,

    = - 2ⁿ cos (2x) for n = 2, 6, 10, ...,

    = + 2ⁿ sin (2x) for n = 3, 7, 11, ...,

    = + 2ⁿ cos (2x) for n = 4, 8, 12, ...

    For n=75, we seek a constant, a, which begins with 1,2,3, or 4 such that

    a + 4 (n-1) = 75

    That is,

    n = (75-a) / 4 is an integer.

    Try a = 1: n = 18.5 (reject)

    a = 2: n = 18.25 (reject)

    a = 3: n = 18 (accept)

    Therefore, y (⁷⁵) = + 2⁷⁵ sin (2x).

    Answer: 2⁷⁵ sin (2x)
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