Apentix/AP Calculus AB/Unit 3: Differentiation: Composite, Implicit, and Inverse Functions/Question
AP Calculus ABhardmcq1 pt

If f(x)=arctan⁡xf(x)=\arctan x, then f′′(x)=f''(x)=

A.11+x2\dfrac{1}{1+x^2}
B.−1(1+x2)2-\dfrac{1}{(1+x^2)^2}
C.2x(1+x2)2\dfrac{2x}{(1+x^2)^2}
D.−2x(1+x2)2-\dfrac{2x}{(1+x^2)^2}

Explanation

Core Concept

First, f′(x)=11+x2=(1+x2)−1f'(x)=\frac{1}{1+x^2}=(1+x^2)^{-1}. Differentiating gives −1⋅(1+x2)−2⋅2x=−2x(1+x2)2-1\cdot(1+x^2)^{-2}\cdot 2x=-\frac{2x}{(1+x^2)^2}.

Correct Answer

D−2x(1+x2)2-\dfrac{2x}{(1+x^2)^2}

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