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: Use the general derivative formula for cosine.
Instead of listing derivatives one at a time, there is a direct formula for the $n$th derivative of $\cos x$:
\[ \dfrac{d^n}{dx^n} \cos x = \cos\left(x + \dfrac{n\pi}{2}\right) \]
This formula comes from the fact that each derivative shifts the cosine graph by a quarter turn, a phase of $\pi/2$, which matches the 4 step repeating pattern that cosine derivatives follow.

Step 2: Plug in $n = 50$.
\[ f^{(50)}(x) = \cos\left(x + \dfrac{50\pi}{2}\right) = \cos(x + 25\pi) \]

Step 3: Simplify the angle $25\pi$.
Cosine repeats every $2\pi$, so we only care about $25\pi$ modulo $2\pi$.
\[ 25\pi = 12(2\pi) + \pi \]
So $25\pi$ and $\pi$ point to the same position on the circle, meaning $\cos(x + 25\pi) = \cos(x + \pi)$.

Step 4: Use the identity for $\cos(x + \pi)$.
Adding $\pi$ to an angle flips the cosine to its negative:
\[ \cos(x + \pi) = -\cos x \]

Step 5: State the result.
So $f^{(50)}(x) = -\cos x$, which matches the value found by tracking the 4 step cycle directly.
\[ \boxed{f^{(50)}(x) = -\cos x} \]