(a) Find the local linearization of $$f(x) = \frac{1}{1 + 3 x}$$ near $x = 0$:
$\frac{1}{1 + 3 x}\approx$

(b) Using your answer to (a), what quadratic function would you expect to approximate $\displaystyle g(x) = \frac{1}{1 + 3 x^2}$?
$\frac{1}{1 + 3 x^2} \approx$

(c) Using your answer to (b), what would you expect the derivative of $\frac{1}{1 + 3 x^2}$ to be even without doing any differentiation?
$\frac{d}{dx}\left(\frac{1}{1 + 3 x^2}\right)\bigg|_{x=0} =$