Let . Which of the following gives for all in the domain of…

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Let g(x)=arcsin⁡(arctan⁡(3x))g(x)=\arcsin(\arctan(3x)). Which of the following gives g′(x)g'(x) for all xx in the domain of gg? (a) 31+9x21−(arctan⁡(3x))2\displaystyle \frac{3}{1+9x^{2}}\sqrt{1-(\arctan(3x))^{2}} (b) 31−(3x)2 (1+9x2)\displaystyle \frac{3}{\sqrt{1-(3x)^{2}}\,(1+9x^{2})} (c) 3(1+9x2)1−(arctan⁡(3x))2\displaystyle \frac{3}{(1+9x^{2})\sqrt{1-(\arctan(3x))^{2}}} (d) 31+9x2 [1−(arctan⁡(3x))2]\displaystyle \frac{3}{\sqrt{1+9x^{2}}\,[1-(\arctan(3x))^{2}]}

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