Apentix/AP Calculus AB/Unit 3: Differentiation: Composite, Implicit, and Inverse Functions/Question
AP Calculus ABmediummcq1 pt

If f(x)=arctan⁡(x)f(x)=\arctan(\sqrt{x}), then f′(x)=f'(x)=

A.12x(1+x)\dfrac{1}{2\sqrt{x}(1+x)}
B.11+x\dfrac{1}{1+x}
C.12x(1+x)\dfrac{1}{2\sqrt{x}(1+\sqrt{x})}
D.1x(1+x)\dfrac{1}{\sqrt{x}(1+x)}

Explanation

Core Concept

Apply the chain rule: 11+(x)2⋅12x\frac{1}{1+(\sqrt{x})^2}\cdot\frac{1}{2\sqrt{x}}. Since (x)2=x(\sqrt{x})^2=x, the result is 12x(1+x)\frac{1}{2\sqrt{x}(1+x)}.

Correct Answer

A12x(1+x)\dfrac{1}{2\sqrt{x}(1+x)}

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