I want to say that, assuming there was a way to make the notation make sense for non-integral x, the derivative of (x + x + … + x) (x times) is being taken wrong. The contribution from each x is 1, but there is an additional contribution from the fact that the stack of x’s is growing with the size of x that is being ignored. The notation makes it awkward, but that is kind of the point.
The derivative actually being taken is that of nx, with n set to be equal to x afterward, which is not the same thing as the derivative of xx.
Here’s one possible “proof” of 2=1 using calculus.
Two students were trying to evaluate the indefinite integral of 1/(2x). Using S to stand for the integral sign, that’s S(1/(2x))dx.
One of them decided to use substitution.
Let u = 2x.
Then du = 2dx, so dx = (1/2)du.
So S(1/(2x))dx = S(1/u)(1/2)du = (1/2)ln|u| + C = (1/2)ln|2x| + C
The other student factored 1/(2x) into (1/2)(1/x).
So S(1/(2x))dx = S(1/2)(1/x)dx = (1/2)ln|x| + C
Neither student made any errors, and both answers, when differentiated, yield the original integrand of 1/(2x).
So (1/2)ln|2x| + C = (1/2)ln|x| + C, from which we can derive that 2 = 1.
I leave it as an exercise for the interested reader to find the flaw in the proof.
I’m surprised so many people miss that one, it’s actually a surprisingly useful property of integrals, I’ve used it quite a few times, though I don’t recall the situation.
The simplification is an example of the wrongness.
The wrongness is in trying to use square roots in the proof of LHS=RHS.
Actually after the square root, the mathematician knows to check for if the LHS is absurd compares to the RHS… Where the result is absurd, he blames the square root process for introducing an error. The idea is to check for the +ve and -ve of each square root to see if specific choices of +ve and -ve leads to a non-absurd result.
Every such “proof” either involves division by 0 or taking the wrong square root.
Every incorrect geometric proof, on the other hand, involves ignoring “betweenness”. For example, I know a simple proof that every angle is a right angle.
The flaw in the second is more obvious because we know about square inches and square yards, but we don’t have a concept of square cents or square dollars.
I’ll admit I don’t see what the “obvious” flaw is supposed to be, but maybe I missed something. It doesn’t make enough sense to me to have an obvious flaw.
If you say you are going to take the positive square root, and then you take the negative square root, your reasoning is clear but you made a mistake.
But if you say:
x2 = x+…+x with x number x’s
g(x) = x+…+x (with x number of x’s)
g’(x) = 1+…+1 (with x number of 1’s) = x
Then I am at a loss to understand your reasoning, so there are any number of possible flaws.
Writing a nonsensical statement (“x number of x’s for x a real number”)
Taking the derivative of something nonsensical (flaw: use of magic)
Taking the derivative of a function whose domain is the natural numbers
Assuming a property of natural numbers holds for all real numbers
Treating the first x in “x number of x’s” as a constant when differentiating
I take it you mean flaw #5 to be the obvious one? But it doesn’t really work because the rest doesn’t make sense either. (again, I could be missing something)
I swear to god if I ever have to grade a math-heavy class again and students just pull answers out of their asses, from now on I’m going to write “-<points> Use of magic”.