Question

If y=sin x and y{ⁿ}, means the nth derivative of y with respect to x then the smallest positive integer n for which y⁽ⁿ⁾ =y is

a.2
b.4
c.5
d.6
e.8

Answer 1

The smallest positive integer n for which the nth derivative of y = sin x equals y itself is 4, which corresponds to option b.

Step-by-step explanation:

The student is asking for the smallest positive integer n for which the nth derivative of y with respect to x, where y equals sin x, is equal to y itself. To solve this, we take the derivatives of y = sin x:

• y'(x) = cos x
• y''(x) = -sin x
• y'''(x) = -cos x
• y''''(x) = sin x

We find that at n = 4, y(n) equals y. Therefore, the smallest positive integer n for which y(n) = y is 4, which corresponds to option b.