Question

question
find the 82nd derivative of f(x) = sin(x).
select the correct answer below:
sin(x)
cos(x)

• sin(x)
• cos(x)

Explanation:

Step1: List first - few derivatives

$f'(x)=\cos(x)$; $f''(x)=-\sin(x)$; $f'''(x)=-\cos(x)$; $f^{(4)}(x)=\sin(x)$.

Step2: Identify the cycle

The derivatives of $y = \sin(x)$ repeat every 4 derivatives.

Step3: Divide the order of derivative by 4

$82\div4 = 20\cdots\cdots2$. The quotient is 20 and the remainder is 2.

Step4: Determine the 82nd derivative

Since the remainder is 2, the 82nd derivative is the same as the 2nd - derivative in the cycle.

Answer:

$-\sin(x)$