calculus - Calculate the first derivative without the function?

I am a complete newbie when it comes to advanced mathematics, and am trying to learn calculus on my own. I wanted to know - is it possible to calculate the first derivative if you don't know the function that created a curve, but you DO have all of the points along the curve?

Edit: I created the curve using a cubic Spline interpolation

If so, can you point me to a place where I can learn how this would be accomplished?

Thanks!!

Answer

If you have the curve, then geometrically, that is all you need to find a derivative value at a given point. You could estimate the direction of the tangent line at a given $x=a$. The slope of that tangent line is the value of $f'(a)$.

If you have a table of values, let's say you know $f(2.9), f(3), f(3.1)$, etc., but perhaps you have no info about $f(3.05)$. Then you can still estimate $f'(3)$ (in this what-if), by calculating the average rate of change over the smallest interval available in the data. For example, $f'(3) \approx \frac{f(3.1) - f(3)}{0.1} \approx \frac{f(3) - f(2.9)}{0.1}$. Perhaps a better estimate can be had by averaging those two to get: $f'(3) \approx \frac{f(3.1) - f(2.9)}{0.2}$.

Hope this helps!