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MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

For the function f(x)=sin⁡xf(x)=\sin x, successive derivatives exhibit a repeating pattern. (1) Derive a general expression for f(n)(x)f^{(n)}(x), where nn is a positive integer. (2) Using your result from part (a), determine the exact value of f(37) ⁣(π6)f^{(37)}\!\left(\dfrac{\pi}{6}\right).

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