Question

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

Answer 1

`g(x)=(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(x)`, `g'(x)` and `g''(x)` for small `x`, then you might as well use the Binomial Theorem to avoid differentiating a quotient. So `g(0)=1`, `g'(0)=0`, `g''(0)=4`.

For small `x` the graph is shown below.

Notice that the curvature of `g` at `x=0`, being `1/(g''(0))`, is the same as the curvature of a circle of radius `1/4`.

For larger `x` you need the asymptotes.