Sderivative

 Scale-derivative

$$f`(x):=\lim_{h\to 1}\sqrt[h]{\frac{f(hx)}{f(x)}}$$

s-linear approximation:

Let

$$h=\frac{x}{a},$$

so

$$f`(a):=\lim_{x\to a}\sqrt[x/a]{\frac{f(x)}{f(a)}},$$

therefore

$$f(x)\approx f(a) \left(f`(a)\right)^{\frac{x}{a}}.$$