Question

let ( f(x)=sin (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)= )
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Explanation:

Step1: Find the first derivative

The derivative of \( y = \sin(x) \) is \( y^{\prime}=\cos(x) \), so \( f^{\prime}(x)=\cos(x) \).

Step2: Find the second derivative

Differentiate \( f^{\prime}(x)=\cos(x) \). The derivative of \( \cos(x) \) is \( -\sin(x) \), so \( f^{\prime\prime}(x)=-\sin(x) \).

Step3: Find the third derivative

Differentiate \( f^{\prime\prime}(x)=-\sin(x) \). The derivative of \( -\sin(x) \) is \( -\cos(x) \), so \( f^{\prime\prime\prime}(x)=-\cos(x) \).

Step4: Find the fourth derivative

Differentiate \( f^{\prime\prime\prime}(x)=-\cos(x) \). The derivative of \( -\cos(x) \) is \( \sin(x) \), so \( f^{\prime\prime\prime\prime}(x)=\sin(x) \).

Answer:

• \( f^{\prime\prime}(x)=-\sin(x) \)
• \( f^{\prime\prime\prime}(x)=-\cos(x) \)
• \( f^{\prime\prime\prime\prime}(x)=\sin(x) \)