Questions: Find the 47th derivative of the function f(x) = cos(x).

Transcript text: Find the 47th derivative of the function $f(x)=\cos (x)$.

Solution

Solution Steps

To find the 47th derivative of the function $ f(x) = \cos(x) $, we need to recognize the cyclical pattern of derivatives for trigonometric functions. The derivatives of $ \cos(x) $ cycle every four derivatives: $ \cos(x) $, $-\sin(x)$, $-\cos(x)$, $\sin(x)$, and then back to $ \cos(x) $. Therefore, to find the 47th derivative, we determine the position within this cycle by calculating $ 47 \mod 4 $.

Step 1: Identify the Function and Its Derivatives

We start with the function $ f(x) = \cos(x) $. The derivatives of $ \cos(x) $ follow a cyclical pattern every four derivatives:

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

Step 2: Determine the Cycle Position

To find the 47th derivative, we calculate the position within the cycle by evaluating $ 47 \mod 4 $: \[ 47 \div 4 = 11 \quad \text{remainder} \quad 3 \] Thus, $ 47 \mod 4 = 3 $.

Step 3: Find the 47th Derivative

Since the remainder is 3, the 47th derivative corresponds to the third derivative in the cycle: \[ f^{(47)}(x) = f^{(3)}(x) = \sin(x) \]