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

let ( f(x)=sin (x) ), find each of the following:\nfind the second derivative, ( f^{prime prime}(x)= )\nfind the third derivative, ( f^{prime prime prime}(x)= )\nfind the fourth derivative, ( f^{prime prime prime prime}(x)= )\nquestion help: video message instructor\nsubmit question jump to answer

let ( f(x)=sin (x) ), find each of the following:\nfind the second derivative, ( f^{prime prime}(x)= )\nfind the third derivative, ( f^{prime prime prime}(x)= )\nfind the fourth derivative, ( f^{prime prime prime prime}(x)= )\nquestion help: video message instructor\nsubmit question jump to answer

Answer

Answer:

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

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) ).