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

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)