Weird disproof of d/dx(x^2)=2x

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<p>d/dx(x^2)=2x</p>

<p>Disproof
d/dx(x^2)=d/dx(x+x+x+x+x+x+x+……) ……<Add x, x times
d/dx(x+x+x+x+x+……)=d/dx(x)+d/dx(x)+d/dx(x)……add d/dx(x), x times
d/dx(x)+d/dx(x)+d/dx(x)+….=1+1+1+1+1+1+……add 1, x times
1+1+1+1+1+1+……= x</p>

<p>Hm…I just saw this on somebody’s facebook wall :slight_smile:
Can anyone explain what’s wrong with it?
Is it because adding x, x times assumes that x is only an integer?</p>

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<p>I think the main reason is that the sum rule only works when the number of things being added is fixed. The theorem for the derivative of a sum says that, when the derivatives exist,
d/dx(f(x) + g(x)) = d/dx(f(x)) + d/dx(g(x))</p>

<p>For a sum that contains 10 terms, you can apply this rule 9 times to break the sum into 10 derivatives. However, when the number of things being added isn’t fixed, you can’t apply the rule some number of times to get the derivative of each term.</p>

<p>Also, like you said, I don’t think it really makes sense to take the derivative of half of a term, if x isn’t an integer.</p>