Past exam of the differential equations and applications course of the University of Oxford / 2010 / A / 1 / ii / c / Solution

The distributional derivative is defined by moving the derivative onto a test function and changing the sign:Since $⟨{\delta }_a,\varphi ⟩=\varphi (a)$, one has $⟨{\delta }_0^{\prime },\varphi ⟩=-{\varphi }^{\prime }(0)$. HenceThe minus sign in front of ${\delta }_0^{\prime }$ is essential: it makes its action the positive derivative evaluation required here.