If $$ y=\sin (x) $$, then $$ \frac{d^{96} y}{d x^{96}} $$ a. $$ -\cos (x) $$ b. $$ \sin (x) $$ c. $$ -\sin (x) $$ d. $$ \cos (x) $$

Question

UpStudy AI · Accepted Answer

To solve the problem, notice that the derivatives of sin(x) repeat every 4 steps:

y = sin(x)  
  y' = cos(x)  
  y'' = –sin(x)  
  y''' = –cos(x)  
  y⁽⁴⁾ = sin(x)

Since the cycle repeats every 4, the 96th derivative is given by:

96 mod 4 = 0

Thus, the 96th derivative of sin(x) is sin(x).

The correct answer is b. sin(x).
The 96th derivative of $$ y = \sin(x) $$ is $$ \sin(x) $$.

If \( y=\sin (x) \), then \( \frac{d^{96} y}{d x^{96}} \) a. \( -\cos (x) \) b. \( \sin (x) \) c. \( -\sin (x) \) d. \( \cos (x) \)