Assume first that I have a one-dimensional function monotone and invertible function $y=f(x):\mathbb{R}\rightarrow\mathbb{R}$. In that case, we have:
$$\frac{\partial}{\partial y}f^{-1}(y)=\left(\frac{\partial}{\partial x}f(x)\right)^{-1}=\frac{1}{\frac{\partial}{\partial x}f(x)} \tag{1}$$
Now assume I have a monotone and invertible multivariate function $f(\mathbf{x}):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, that is $\frac{\partial}{\partial x_{i}}f(\mathbf{x}) > 0$ for all dimensions $i=1,...,n$. I am interested in learning whether, anologous to the one-dimensional case, the following is true:
$$\nabla_{\mathbf{y}} f^{-1}\left(\mathbf{y}\right) = \left(\nabla_{\mathbf{x}} f\left(\mathbf{x}\right)\right)^{-1} \tag{2}$$
where $\nabla_{\mathbf{x}} f$ is the (square) matrix of partial derivatives of $f$ with respect to the dimensions of $\mathbf{x}$. In other words: are the partial derivatives of the function's inverse $f^{-1}$ the same as the inverse of the partial derivatives of the function $f$?