What is wrong with my integration of a function?

Question

Doing some derivative problems I have encountered function $y$ = $\mathrm{e}^{-x^2}$. I tried to differentiate it, but could not get the right result, I got $\dfrac{\mathrm{e}^{-x^2}}{2x} + C$, but using online calculator for derivatives it should be $\dfrac{\sqrt{{\pi}}\operatorname{erf}\left(x\right)}{2}$. Why is that so? Can anyone explain what is wrong with my derivation, what am I missing. I used if $dy/dx = \mathrm{e}^{u}({du}/dx)$, then y = $\mathrm{e}^{u}+C$ if u is a function of x. Thanks in advance.

Answer 1

I think there is some confusion, to differentiate: $$\frac{d}{dx}e^{-x^2}=\frac{d}{dx}[-x^2]\times e^{-x^2}=-2xe^{-x^2}$$

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If you want to integrate this function, there is no elementary antiderivative and so we use the function defined as: $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}dt$$ so when your calculator gives you $\operatorname{erf}$ as an answer it is just trying to define it in terms of known functions. The reason why the constant is in front is to do with looking at $|x|\to\infty$ as this function is used in statistical modelling.