Question

What is the derivative of a hyperbola?

Answer 1

I assume you're referring to the equilateral hyperbola, as it's the only hyperbola that can be expressed as real function of one real variable.

The function is defined by `f(x)=1/x`.

By definition, `forall x in (-infty,0) cup (0,+infty)` the derivative is:

`f'(x)=lim_{h to 0}{f(x+h)-f(x)}/{h}=lim_{h to 0}{1/{x+h}-1/x}/{h}=lim_{h to 0}{{x-(x+h)}/{(x+h)x}}/{h} = lim_{h to 0}{-h}/{xh(x+h)}= lim_{h to 0}{-1}/{x^2+hx}=-1/x^2`

This can also be obtained by the following derivation rule `forall alpha ne 1`:
`(x^alpha)'=alpha x^{alpha-1}`.
In this case, for `alpha=-1`, you get
`(1/x)'=(x^{-1})'=(-1)x^{-2}=-1/x^2`