We can express a square number as the repeated addition of that number in this manner:

$1^2 = 1$

$2^2 = 2 + 2$

$3^2 = 3 + 3 + 3$

Generalising this, we get:

$x^2 = x + x + x...$ $x$ $times$

If we differentiate with respect to $x$ on both sides, we get:

$2x = 1 + 1 + 1...$ $x$ $times$

$2x = x$

$2 = 1$

This is obviously wrong. What's the mistake in my proof?