Question

exercises 3.5 derivatives of trigonometric functions
score: 32/37 answered: 16/18
question 17
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)=)
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Explanation:

Step1: Find the first derivative

The derivative of \( \cos(x) \) is \( -\sin(x) \), so \( f'(x)=-\sin(x) \).

Step2: Find the second derivative

The derivative of \( -\sin(x) \) is \( -\cos(x) \), so \( f''(x)=-\cos(x) \).

Step3: Find the third derivative

The derivative of \( -\cos(x) \) is \( \sin(x) \), so \( f'''(x)=\sin(x) \).

Step4: Find the fourth derivative

The derivative of \( \sin(x) \) is \( \cos(x) \), so \( f''''(x)=\cos(x) \).

Answer:

\( f''(x)=-\cos(x) \)
\( f'''(x)=\sin(x) \)
\( f''''(x)=\cos(x) \)