What type of function is this (derivative of a hyperbola)?

Question

The derivative of the hyperbola $$f(x)=\frac{b}{a}\sqrt {a^2+x^2}$$

is

$$f'(x)=\frac{bx}{a\sqrt {a^2+x^2}}$$

The graph (for $a=b=1$) looks somewhat like a Sigmoid function, but I honestly cannot see the connection.

Can anybody help me out by telling me what type of function this is? Since I am not that good at maths, can you please thorougly explain exactly why it is that type of function?

Futhermore, the double-derivative is

$$f''(x)=\frac{ab}{(x^2+a^2)^\frac{3}{2}}$$

What type of function is this? The graph (for $a=b=1$) looks somewhat like a Bell curve.

I am looking forwards to your answers. Thank you in advance.

Answer 1

Your function $f'$ is a bounded, odd, strictly increasing algebraic function: The graph $$ y = f'(x) = \frac{bx}{a\sqrt{a^{2} + x^{2}}} \tag{1} $$ is part of the algebraic plane curve $$ b^{2}x^{2} - a^{2}y^{2}(a^{2} + x^{2}) = 0. \tag{2} $$ (The locus (2) consists of the graph (1) and the graph $y = -f'(x)$ obtained by reflection across the $x$-axis. Because $f'$ is odd, reflecting its graph across the $y$-axis gives the same graph as reflecting across the $x$-axis.)

If $b \neq 0$, the function $f'$ is not rational: There are distinct horizontal asymptotes at $\infty$ and $-\infty$, while a rational function with a horizontal asymptote has the same limit at $\pm\infty$.

The function $f''$ is, similarly, an even, irrational algebraic function. The graph $$ y = f''(x) = \frac{ab}{(a^{2} + x^{2})^{3/2}} $$ is defined by the two-variable polynomial $$ y^{2}(a^{2} + x^{2})^{3} - a^{2} b^{2} = 0. $$

Though the functions $f'$ and $f''$ bear qualitative resemblance to a logistic curve, or to a Gaussian bell curve and its cumulative distribution, they are substantially different, approaching their respective asymptotes polynomially fast rather than exponentially fast.