\langle\nabla\delta(\vec r-\vec\lambda),\varphi\rangle=\langle\nabla\delta,\varphi(\vec r+\vec\lambda)\rangle=
(\langle\frac{\partial\delta}{\partial x},\varphi(\vec r+\vec\lambda)\rangle,
\langle\frac{\partial\delta}{\partial y},\varphi(\vec r+\vec\lambda)\rangle,
\langle\frac{\partial\delta}{\partial z},\varphi(\vec r+\vec\lambda)\rangle)=
(-\langle\delta,\frac{\partial\varphi}{\partial x}(\vec r+\vec\lambda)\rangle,
-\langle\delta,\frac{\partial\varphi}{\partial y}(\vec r+\vec\lambda)\rangle,
-\langle\delta,\frac{\partial\varphi}{\partial z}(\vec r+\vec\lambda)\rangle)=
(-\frac{\partial\varphi}{\partial x}(\vec\lambda),
-\frac{\partial\varphi}{\partial y}(\vec\lambda),
-\frac{\partial\varphi}{\partial z}(\vec\lambda))
=-\nabla\varphi(\vec\lambda)