Problem

Let $f(x)=1+x^2$ for $x\in(-\infty,\infty)$ and let $g(x)=\tan(x)$ for $x\in(-\pi/2,\pi/2)$. Which of the following statement is true?

(a) $\dfrac{d}{dx}\dfrac{g(x)}{f(x)}=\dfrac{2x\tan(x)}{(1+x^2)^2}$ for $x\in(-\pi/2,\pi/2)$.

(b) $\dfrac{d}{dx}(f(x)+g(x))=2x+\sec^2(x)$ for $x\in(-\pi/2,\pi/2)$.

(c) $\dfrac{d}{dx}(f(x)g(x))=\sec^2(x)(1+x^2)+2x\tan(x)$ for $x\in(-\pi/2,\pi/2)$.

(d) $\dfrac{d}{dx}f(g(x))=2g(x)$ for $x\in(-\pi/2,\pi/2)$.

(e) None of the above statements is true.