Differentiating a Convolution through the Kernel

Let $n$, $\Omega$, $f$, $\delta$, $\rho$, the set $\Omega^{\delta}$ and the convolution $f*\rho$ be as in Convolution of a Continuous Function with a Compactly Supported Continuous Kernel, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on $\mathbb{R}^n$ and let $d$ denote the Euclidean distance. Assume in addition that $\rho$ is of class $C^1$ on $\mathbb{R}^n$, and let $i\in\{1,\dots,n\}$.

1. (The differentiated kernel) The partial derivative of $\rho$ with respect to the $i$th variable exists at every point of $\mathbb{R}^n$; the resulting function $\partial_i\rho:\mathbb{R}^n\to\mathbb{R}$ is continuous on $\mathbb{R}^n$ as a map from $(\mathbb{R}^n,d)$ to $(\mathbb{R},d)$, and $\partial_i\rho(y)=0$ for every $y\in\mathbb{R}^n$ with $\lVert y\rVert>\delta$. Consequently Convolution of a Continuous Function with a Compactly Supported Continuous Kernel applies verbatim with the kernel $\partial_i\rho$ in place of $\rho$ and the same $n$, $\Omega$, $f$ and $\delta$, and the convolution $f*(\partial_i\rho)$ is a real-valued function on the same set $\Omega^{\delta}$.

2. (Differentiation through the kernel) The set $\Omega^{\delta}$ is open in $\mathbb{R}^n$, the partial derivative of $f*\rho$ with respect to the $i$th variable exists at every point of $\Omega^{\delta}$, and

$$\partial_i(f*\rho)(x)=\bigl(f*(\partial_i\rho)\bigr)(x)\qquad\text{for every }x\in\Omega^{\delta}.$$