# How do you use the first and second derivatives to sketch  g(x)= ( 1 + x^2 ) / ( 1 - x^2 )?

Oct 12, 2017

$g \left(x\right) = \frac{1 + {x}^{2}}{1 - {x}^{2}}$
=(1+x^2)(1+x^2+x^4+x^6…) Binomial Theorem. $| x | < 1$
=(1+2x^2+2x^4+2x^6…)
g'(x)=4x+8x^3+12x^5…
g''(x)=4+24x^2+60x^4…

#### Explanation:

Assuming that you want $g \left(x\right)$, $g ' \left(x\right)$ and $g ' ' \left(x\right)$ for small $x$, then you might as well use the Binomial Theorem to avoid differentiating a quotient. So $g \left(0\right) = 1$, $g ' \left(0\right) = 0$, $g ' ' \left(0\right) = 4$.

For small $x$ the graph is shown below.

Notice that the curvature of $g$ at $x = 0$, being $\frac{1}{g ' ' \left(0\right)}$, is the same as the curvature of a circle of radius $\frac{1}{4}$.

For larger $x$ you need the asymptotes.