consider the following function.
\f(x) = \sin x\
step 5 of 5: find $f^{(41)}(x)$ by observing the pattern of the derivative from the previous steps.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Trigonometric Derivatives", "Higher-Order Derivatives" ], "new_concepts": [], "current_concepts": [ "Trigonometric Derivatives", "Higher-Order Derivatives", "Periodic Derivatives" ] } ## Compute the first four derivatives of $f(x) = \sin x$ $$ \begin{aligned} f'(x) &= \cos x \ f''(x) &= -\sin x \ f'''(x) &= -\cos x \ f^{(4)}(x) &= \sin x \end{aligned} $$ ## Determine the periodic pattern of the derivatives $$ f^{(n)}(x) = \begin{cases} \cos x & \text{if } n \equiv 1 \pmod 4 \ -\sin x & \text{if } n \equiv 2 \pmod 4 \ -\cos x & \text{if } n \equiv 3 \pmod 4 \ \sin x & \text{if } n \equiv 0 \pmod 4 \end{cases} $$ ## Evaluate the 41st derivative $$ \begin{aligned} 41 &= 4 \times 10 + 1 \implies 41 \equiv 1 \pmod 4 \ f^{(41)}(x) &= f'(x) = \cos x \end{aligned} $$ Find $f^{(41)}(x)$ by observing the pattern of the derivative from the previous steps. $f^{(41)}(x) =$ $\cos(x)$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Higher-Order Derivatives" ] }

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Explanation:

Compute the first four derivatives of \(f(x) = \sin x\)

Determine the periodic pattern of the derivatives

Evaluate the 41st derivative

Answer: