Spot the mistake
Calculus
Level
1
Here's my attempt at proving that \(2=1\). In which step did I first commit a flaw in my logic?
\[\begin{align} 1^2&=1\\ 2^2&=2+2&\text{(2 times)}\\ 3^2&=3+3+3&\text{(3 times)} \end{align}\]
Step 1: For any positive integer:
\[x^2=x+x+\cdots+x\quad\quad\text{(x times)}\]
Step 2: Now, differentiating with respect to \(x\):
\[ 2x=1+1+\cdots+1\text{(x times)}. \]
Step 3: Summing this up, we get
\[ 2x = x. \]
Step 4: Dividing by \(x\), we get
\[ 2 = 1. \]
by
Pankaj Shegokar