Is d/dx x^2 really equal to d/dx (x+x+x+...+x)? Inserting some value for x seems to break that part. Say x=5:
d/dx 5^2=10 and
d/dx (x+x+x+x+x)=d/dx 5x=5
10 ≠ 5
I understand that x^2, (x+x+x+...+x), and x*x are the same. But its equivalent to doing d/dx x*x =x as d/dx n*x=n. Again this results in 2x=x or 2=1.
The mistake is to treat a variable as a constant and deriving in that way. Doing a sum x times means you have an unaccounted variable when you do d/dx (x+x+x+...+x), this is not 1+1+1+...+1 x times. i.e. The rate of change of one of the x is not incorporated into the derivative.
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