can anybody please tell me what's happening here ?
$$1^2=1$$
$$2^2=2+2$$
$$3^2=3+3+3$$
$$x^2 = x+x+\cdots+x \mbox{ ($x$ times)}$$
differentiating both the sides $$2x = 1 + 1 + \cdots+1 \mbox{ ($x$ times)}$$
thank u
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Well, by that reasoning, you could also, $$x=\sum_{k=1}^x 1$$ differentiate both sides and get $$1=\sum_{k=1}^x 0 = 0. $$ This issue here is that differentiating involves the $\sum$ itself and not just the variables being added. It is just like when we are asked to differentiate $$f(x):=\int_0^{x^2} y^7 dy$$ with respect to $x$. Shouldn't the summation limits play just as much a role as the integrand? – Bobby Ocean Sep 22 '14 at 22:20
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$$\bigg(\sum_1^xx\bigg)'=\sum_1^xx'+\sum_1^{x'}x=\sum_1^x1+\sum_1^1x=x+x+2x.$$ – Lucian Sep 23 '14 at 01:09
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When you are getting $$x^2 = x+x+\cdots+x \mbox{ ($x$ times)}$$ then $x$ is a constant no not a variable. So, differentiating both side you will get $0=0$ which is true.
Ri-Li
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$$g(x)=x^2 ,domain=\mathbb{R}$$but in$g(x)=x+x+x+...x $ can you mean g(-1.3) ? or even g(2.76) =? – Khosrotash Aug 04 '15 at 09:15