Regression and derivation

[Jul Herb]

Hello,
Maybe it's a stupid question but a friend asked me for help on Stata. We have for each individual a time and a percentage of completion.
He first asked me to find a regression to estimate the percentage versus time curve. I did this and found Y = B0 + B1 * ln (t) to be very good.
Now he asks me the speed with respect to time, and if we follow the logic it would suffice to differentiate Y with t. This would give the function B1 * (1 / t) as the estimated speed

Is it good? Or, is it better to estimate the speed as being (V1-V0)/(T1-T0) for each time (For example if percentage is 0% in second 0 and 20% in second 1, the speed would be 20%/s at the beggining)? And then do a regression with this new variable?

Here is what I would have done.

* Example generated by -dataex-. To install: ssc install dataex
clear
input float id int time float(gender perc)
1  0 1  0
1  1 1 15
1  2 1 23
1  3 1 28
1  4 1 32
1  5 1 36
1  6 1 39
1  7 1 41
1  8 1 43
1  9 1 44
1 10 1 46
1 11 1 46
1 12 1 47
1 13 1 48
1 14 1 49
1 15 1 50
1 16 1 50
1 17 1 50
1 18 1 51
1 19 1 51
1 20 1 51
2  0 1  0
2  1 1 10
2  2 1 15
2  3 1 20
2  4 1 23
2  5 1 26
2  6 1 29
2  7 1 31
2  8 1 34
2  9 1 36
2 10 1 38
2 11 1 39
2 12 1 41
2 13 1 42
2 14 1 43
2 15 1 45
2 16 1 47
2 17 1 48
2 18 1 49
2 19 1 50
2 20 1 51
3  0 0  0
3  1 0 17
3  2 0 27
3  3 0 35
3  4 0 41
3  5 0 46
3  6 0 50
3  7 0 53
3  8 0 55
3  9 0 58
3 10 0 61
3 11 0 63
3 12 0 65
3 13 0 68
3 14 0 70
3 15 0 71
3 16 0 72
3 17 0 74
3 18 0 75
3 19 0 76
3 20 0 77
4  0 0  0
4  1 0 14
4  2 0 20
4  3 0 24
4  4 0 29
4  5 0 33
4  6 0 36
4  7 0 39
4  8 0 40
4  9 0 43
4 10 0 45
4 11 0 47
4 12 0 47
4 13 0 47
4 14 0 48
4 15 0 49
4 16 0 48
4 17 0 48
4 18 0 47
4 19 0 48
4 20 0 49
end

gen ln_time=ln(time)
reg perc ln_time if gender==1
gen reg= _b[_cons]+_b[ln_time]*ln_time if gender==1
gen reg_speed=_b[ln_time]/time if gender==1
reg perc ln_time if gender==0
replace reg= _b[_cons]+_b[ln_time]*ln_time if gender==0
replace reg_speed=_b[ln_time]/time if gender==0

gen real_speed=.
replace real_speed=(perc[_n]-perc[_n-1])/1 if time>0