Question

let ( f(x)=cos (x) ), find each of the following: find the second derivative, ( f^{prime prime}(x)= ) find the third derivative, ( f^{prime prime prime}(x)= ) find the fourth derivative, ( f^{prime prime prime prime}(x)= )

Explanation:

Step1: Find the first derivative

The derivative of \(f(x)=\cos(x)\) is \(f^{\prime}(x)=-\sin(x)\) (using the derivative formula \(\frac{d}{dx}\cos(x)=-\sin(x)\)).

Step2: Find the second derivative

Differentiate \(f^{\prime}(x)=-\sin(x)\). Using the formula \(\frac{d}{dx}\sin(x)=\cos(x)\), we get \(f^{\prime\prime}(x)=-\cos(x)\).

Step3: Find the third derivative

Differentiate \(f^{\prime\prime}(x)=-\cos(x)\). Using the formula \(\frac{d}{dx}\cos(x)=-\sin(x)\), we get \(f^{\prime\prime\prime}(x)=\sin(x)\).

Step4: Find the fourth derivative

Differentiate \(f^{\prime\prime\prime}(x)=\sin(x)\). Using the formula \(\frac{d}{dx}\sin(x)=\cos(x)\), we get \(f^{(4)}(x)=\cos(x)\).

Answer:

The second derivative \(f^{\prime\prime}(x)=-\cos(x)\).
The third derivative \(f^{\prime\prime\prime}(x)=\sin(x)\).
The fourth derivative \(f^{(4)}(x)=\cos(x)\).