I'm trying to figure out what families of functions would have a constant ratio in their derivatives when horizontally stretching or compressing. In other words:
$$ \frac{f'(x)}{f'(yx)} = z $$
where $z$ and $y$ are constants, and we are deriving with respect to $x$.
I noticed that $ln(x)$ and $\sqrt{x}$ follow that rule, having a ratio of derivative of $z = 1/y$ and $z=1/\sqrt{y}$, respectively. However, this rule does not seem to apply to every family of functions. For example, hyperbolic functions like the logistic map $ f(x) = 1/(1+e^x)$ does not follow that rule. In the hyperbolic case, $z$ is dependent on $e^x$.
Is there any way to find which functions would follow the above rule? Is anybody already given a name to functions that follow this rule?
I am not a professional mathematician so I'm not familiar with the professional jargon, and I'm having trouble even finding a way to google for this rule.