if the nth derivative of y is denoted as y^(n) and y= -sinx, then y^(7) is the same as what?

Answer 1

it will be a 4-cycle:
d1 = -cos
d2 = sin
d3 = cos
d4 = -sin
then repeat. 7 ≡ 3 mod 4, so d7 = cos x. d77 ≡ 1 mod 4, so d77 = -cos x.

Answer 2

The derivatives of y = -sin x form a cycle
y'(x) = -cos x
y''(x) = sin x
y^(3)(x) = cos x
y^(4)(x) = -sin x
The 7th derivative would equal the third derivative of the fourth derivative of the function. Since the fourth derivative of the function is the function itself, the third derivative of the function is also the 7th derivative of the function. The answer to this problem is cos x.