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False Proof

A student (A.K.) passed this “gem” on to me. How would you respond?

We know that x2 = (x + x + x + .... + x + x) i.e. x added to itself x-times.

Now, take the derivative on both sides, to get: 2x = (1 + 1 + 1 + …. + 1 + 1) i.e. 1 added to itself x-times.

Thus, 2x = x.

And, canceling the common term, we get 2 = 1.

This is a fact we always knew was true but could not prove until now!