Spot the mistake

Calculus Level 1

Here's my attempt at proving that 2=12=1. In which step did I first commit a flaw in my logic?

12=122=2+2(2 times)32=3+3+3(3 times)\begin{aligned} 1^2&=1\\ 2^2&=2+2&\text{(2 times)}\\ 3^2&=3+3+3&\text{(3 times)} \end{aligned}

Step 1: For any positive integer:

x2=x+x++x(x times)x^2=x+x+\cdots+x\quad\quad\text{(x times)}

Step 2: Now, differentiating with respect to xx:

2x=1+1++1(x times). 2x=1+1+\cdots+1\text{(x times)}.

Step 3: Summing this up, we get

2x=x. 2x = x.

Step 4: Dividing by xx, we get

2=1. 2 = 1.


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