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I would like to find the derivate from some combined logistic and exponential functions that all describe the same data numerically. $$f'(t)=\frac{f(t+h)-f(t)}{h}$$ seems not the best choise for these functions.

I don't have a mathematical background but how do I choose an approximation method like taylor series or richardson? And based on what do I make that choise?


the formules:

$$f(t)=a_1\left[\frac{1-p}{1+e^{-b_1(t-c_1)}}+\frac{p}{1+e^{-b_2(t-c_2)}}\right]+\frac{a_2}{1+e^{-b_3(t-c_3)}}$$

$$f(t)= a\left(1-\frac{1}{1+(b_1(t+e))^{c_1}+(b_2(t+e))^{c_2}+(b_3(t+e))^{c_3}}\right)$$

$$f(t)= h_1-\frac{2(h_1-h_\theta)}{e^{s_\theta(t-\theta)}+e^{s_1(t-\theta)}}$$

$$f(t)=\frac{a_1}{1+e^{-b_1t-c_1)^{d_1}}}+\frac{a_2}{1+e^{-b_2t-c_2)^{d_2}}}+\frac{a_3}{1+e^{-b_3t-c_3)^{d_3}}}$$

$$(t)= h_1\left(1-\frac{1}{1+\frac{(t+0.75)}{d_1}^{c_1}+\frac{(t+0.75)}{d_2}^{c_2}+\frac{(t+0.75)}{d_3}^{c_3}}\right)$$


enter image description here

The figure shown above gives the general form of the curve (red line) and its derivate (blue line).

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2  
Are you supposed to know only the data points? If you know the function, you can explicitly compute the derivative, no point to use linear approximation. –  Shuhao Cao Jun 24 '13 at 16:53
    
it is based on yearly measurments and curve fitted by using the different functions. –  Niels De Blende Jun 24 '13 at 17:17
    
I know that the derivative can be computed symbolically, but I would still like to know how to do this numerically. This because a lot of biologist will just read points from the graphs ('numerically') so I was looking for a way to mimick that mathematically to see what the effect would be. –  Niels De Blende Jun 24 '13 at 17:40
1  
@NielsDeBlende: Perhaps this will help with the numerical approaches if the exact approaches are not available. Regards –  Amzoti Jun 24 '13 at 17:48

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