Given: y = cos(2x).
Let y⁽ⁿ⁾ 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⁽⁴⁾ = + 2⁴ cos(2x)
n=5: y⁽⁵⁾ = - 2⁵ sin(2x)
and so on
A pattern emerges
y⁽ⁿ⁾ = - 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)
