What would be the roots of the derivative of this polynomial I got this $$ f=a(x-1)(x-2)...(x-101) $$ knowing a and that $1 ,2 \ldots ,101$ are it's roots ... I need the roots of $f'$ , how should I do it ? More specifc how can I derivate a polynomial like that, is there a formula or something ?
You can get the derivative thus.

Taking logs, we get $$\log f=\log a+\sum_{i=1}^{101}{\log(x-i)},$$ so that differentiating gives $$f'=f\sum_{i=1}^{101}{\frac{1}{x-i}}.$$
Original Answer

To answer the question in the title, I've asked Mathematica:


  NSolve[D[f[x, 101], x] == 0, x]


where 

$$f_n(x)=\prod \limits_{k=1}^n (x-k)$$

Turns out the result $x_1=x_2=...=x_{100}=51$. I'm still thinking about how you can do this with some brain power or pen and pencil, but I'm sure there are brighter minds than me around here. 

Update

After fixing the Mathematica code, these are the (numerically found) roots:

 &gt;1.18083  
 &gt;2.21008  
 &gt;3.22935  
 &gt;4.2444  
 &gt;5.25705  
 &gt;6.26813  
 &gt;7.27808  
 &gt;8.28718  
 &gt;9.29562  
 &gt;10.3035  
 &gt;11.311  
 &gt;12.3181  
 &gt;13.3248  
 &gt;14.3313  
 &gt;15.3375  
 &gt;16.3436  
 &gt;17.3494  
 &gt;18.3551  
 &gt;19.3606  
 &gt;20.366  
 &gt;21.3712  
 &gt;22.3764  
 &gt;23.3814  
 &gt;24.3863  
 &gt;25.3912  
 &gt;26.396  
 &gt;27.4007  
 &gt;28.4053  
 &gt;29.4099  
 &gt;30.4144  
 &gt;31.4189  
 &gt;32.4233  
 &gt;33.4277  
 &gt;34.432  
 &gt;35.4363  
 &gt;36.4406  
 &gt;37.4448  
 &gt;38.449  
 &gt;39.4532  
 &gt;40.4573  
 &gt;41.4615  
 &gt;42.4656  
 &gt;43.4696  
 &gt;44.4737  
 &gt;45.4778  
 &gt;46.4818  
 &gt;47.4859  
 &gt;48.4899  
 &gt;49.494  
 &gt;50.498  
 &gt;51.502  
 &gt;52.506  
 &gt;53.5101  
 &gt;54.5141  
 &gt;55.5182  
 &gt;56.5222  
 &gt;57.5263  
 &gt;58.5304  
 &gt;59.5344  
 &gt;60.5385  
 &gt;61.5427  
 &gt;62.5468  
 &gt;63.551  
 &gt;64.5552  
 &gt;65.5594  
 &gt;66.5637  
 &gt;67.568  
 &gt;68.5723  
 &gt;69.5767  
 &gt;70.5811  
 &gt;71.5856  
 &gt;72.5901  
 &gt;73.5947  
 &gt;74.5993  
 &gt;75.604  
 &gt;76.6088  
 &gt;77.6137  
 &gt;78.6186  
 &gt;79.6236  
 &gt;80.6288  
 &gt;81.634  
 &gt;82.6394  
 &gt;83.6449  
 &gt;84.6506  
 &gt;85.6564  
 &gt;86.6625  
 &gt;87.6687  
 &gt;88.6752  
 &gt;89.6819  
 &gt;90.689  
 &gt;91.6965  
 &gt;92.7044  
 &gt;93.7128  
 &gt;94.7219  
 &gt;95.7319  
 &gt;96.743  
 &gt;97.7556  
 &gt;98.7707  
 &gt;99.7899  
 &gt;100.819