Can somebody tell me how Tipping (2001) in his classical paper about the Relevance Vector Machine arrives at the following expressions \begin{align} \mathbf{D} &= (\mathbf{C}^{-1}\mathbf{t}\mathbf{t}^{T}\mathbf{C}^{-1}-\mathbf{C}^{-1}) \mathbf{\Phi} \mathbf{A}^{-1} \\ &= \beta[(\mathbf{t}-\mathbf{y})\boldsymbol{\mu}^{T} - \mathbf{\Phi}\Sigma] \end{align}

where $$\mathbf{C} = \beta^{-1}\mathbf{I} + \mathbf{\Phi} \mathbf{A}^{-1} \mathbf{\Phi}^{T}, \\ \boldsymbol{\mu} = \beta^{-1}\mathbf{\Sigma}\mathbf{\Phi}^{T}\mathbf{t}, \\ \boldsymbol{\Sigma} = \left( \mathbf{A} + \beta \mathbf{\Phi}^{T} \mathbf{\Phi} \right)^{-1}$$

which contain the derivatives of likelihood $$\mathcal{L} = -\frac{1}{2}\big[N\log 2\pi + \log\left|\mathbf{C}\right| + \mathbf{t}^{T}\mathbf{C}^{-1}\mathbf{t} \big]$$

with respect to the elements $\phi_{mn}$ of the design matrix $\mathbf{\Phi}$ (i.e. $\mathbf{D}_{mn} = \partial \mathcal{L} / \partial \phi_{mn}$)?

Note that bold small letters e.g $\mathbf{t}$ denote $N\times 1$ vectors, bold capital letters $N\times N$ matrices, and $\mathbf{A}$ is diagonal. Also, the author makes frequent use of the following identity (by the Woodbury inversion lemma) which might be helpful:

\begin{align} \mathbf{C}^{-1} = (\beta^{-1}\mathbf{I} + \mathbf{\Phi} \mathbf{A}^{-1} \mathbf{\Phi}^{T} )^{-1} = \beta \mathbf{I} - \beta \mathbf{\Phi}\left( \mathbf{A} + \beta \mathbf{\Phi}^{T} \mathbf{\Phi} \right)^{-1} \mathbf{\Phi}^{T} \beta = \beta \mathbf{I} - \beta \mathbf{\Phi} \mathbf{\Sigma} \mathbf{\Phi}^{T} \beta \end{align}