Intuition behind the derivative of dirac delta function

Suppose the spikes in the smooth approximation to $\delta'(x)$ are located at $x=-h$ and $x=h$.

When $\bar{x} \approx x+h$, the smooth approximation to $\delta'(x-\bar{x})$ will be large and positive, so the integral will roughly pick up "something large" times $f(x+h)$. Similary, for $\bar{x} \approx x-h$, the integral will pick up the same large factor times $f(x-h)$, but with the opposite sign. So if that large factor turns out to be of the magnitude $\frac{1}{2h}$, the integral will be roughly $$ \frac{f(x+h)-f(x-h)}{2h} = \frac{\bigl(f(x) + h \, f'(x) + O(h^2)\bigr) - \bigl(f(x) - h \, f'(x) + O(h^2)\bigr)}{2h} , $$ which tends to $f'(x)$ as $h \to 0$.