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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 y=arctan⁡(x3−2x)y=\arctan\bigl(x^{3}-2x\bigr), consider its behavior near x=1x=1. (a) Calculate dydx\dfrac{dy}{dx}. Express your answer in simplest rational form. (b) Determine d2ydx2\dfrac{d^{2}y}{dx^{2}} and simplify completely. (c) Evaluate d2ydx2\dfrac{d^{2}y}{dx^{2}} at x=1x=1. Show all algebraic steps, and state the exact value.

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