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

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)).