# Tetration Forum

Full Version: Differential equation for exp^[0.5]
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We know the logarithmic derivative is f ' / f dz.
We know the solve derivative is dz / ( f ' (inv f) ).

Let f = exp^[0.5].
Then solve f = inv f = ln(f).

THUS f ' / f = 1 / ( f ' (ln(f)) )

I wonder how good approximations of f satisfy the differential equation. (2sinh , base change , fake etc )

regards

tommy1729
(07/14/2014, 12:55 AM)tommy1729 Wrote: [ -> ]We know the logarithmic derivative is f ' / f dz.
We know the solve derivative is dz / ( f ' (inv f) ).

Let f = exp^[0.5].
Then solve f = inv f = ln(f).

THUS f ' / f = 1 / ( f ' (ln(f)) )

I wonder how good approximations of f satisfy the differential equation. (2sinh , base change , fake etc )

regards

tommy1729

I wonder if g ' / g = 1 / (g ' (h(g)))

where h is a fake ln would make a good fake g(z) such that g^[2](z) - exp(z) is small ?

regards

tommy1729