VorosDerivative
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Derivative
\( \displaystyle \mathrm{D} (f)(x) = \lim_{t\rightarrow 0} \frac{f(t+g(x))}{t} \)
Strange derivative
\( \displaystyle \lim_{h\rightarrow \infty} \left( \frac{y(x+1/h)}{y(x)} \right)^h \) \( \displaystyle = \lim_{h\rightarrow \infty} \exp\left( h \ln \left(\frac{y(x+1/h)}{y(x)} \right) \right) \) \( \displaystyle = \lim_{h\rightarrow \infty} \exp\left( h \ln(y(x)+y'(x)/h)+.. - h \ln (y(x)) \right) \) \( \displaystyle = \exp\left( \frac{y'(x)}{y(x)} \right) \) \( \displaystyle = \exp \left( \frac{\mathrm d}{\mathrm d x} \ln\left( y(x) \right) \right) \)
Voros Derivative
Define operation \(oD\) with
\( \displaystyle oD ~ y= \lim_{h\rightarrow \infty} \left( y\left(x+\frac{1}{h}\right) \circ y^{\circ -1}(x) \right)^{\circ h} \)