Question

If \(f(x) = \cos(x)\) then the 50th derivative of \(f(x)\) is:

• A. \(\sin x\)
• B. \(-\sin x\)
• C. \(\cos x\)
• D. \(-\cos x\)

Hint:

Derivatives of cos x repeat every 4 steps, so divide 50 by 4 and match the remainder to that step in the cycle.

Answer 1

The Correct Option is D

Step 1: Write out the first few derivatives.
We start from \(f(x) = \cos x\) and differentiate again and again, watching for a pattern.
\[ f'(x) = -\sin x \]
\[ f''(x) = -\cos x \]
\[ f'''(x) = \sin x \]
\[ f''''(x) = \cos x \]

Step 2: Spot the repeating cycle.
After the 4th derivative we are back to \(\cos x\), the same as \(f(x)\) itself.
This means the derivatives repeat in a cycle of length 4: \(-\sin x, -\cos x, \sin x, \cos x\), then it starts over.
So to find the 50th derivative, we only need to know where 50 lands inside this 4 step cycle.

Step 3: Reduce 50 using the cycle length.
Divide 50 by 4:
\[ 50 = 4 \times 12 + 2 \]
So 50 leaves a remainder of 2 when divided by 4. This means the 50th derivative sits in the same position as the 2nd derivative in the cycle.

Step 4: Read off the answer from the cycle.
From Step 1, the 2nd derivative is \(f''(x) = -\cos x\).
Since 50 matches position 2 in the cycle, the 50th derivative must equal the 2nd derivative.

Final Answer:
The 50th derivative of \(\cos x\) is \(-\cos x\). \[ \boxed{f^{(50)}(x) = -\cos x} \]