What is the inverse of the partial derivative of a function?

Question

If $\partial_i:C^1(\mathbb{R}^m,\mathbb{R}^n)\to C^0(\mathbb{R}^m,\mathbb{R}^n)$ is the partial derivative w.r.t. the $i^{th}$ component, and $$ (\partial_if)^{-1}=\partial_i(H_f)\ :\ \mathbb{R}^n\to\mathbb{R}^m $$ can an expression for $H_f$ be determined simply in terms of $f$? Since $f:\mathbb{R}^m\to\mathbb{R}^n$, I know that $f^{-1}$ must be a factor in $H_f$ but I don't know how you'd manipulate it to make this identify hold.