= Solution
The <Jacobian matrix> of $F_\mu$ in $(x,y)$ is
$$
J(v)=\begin{pmatrix}H(x,y)&-A(x)^T\\Y A(x)&C(x)\end{pmatrix}.
$$
The top-left block is the <Hessian matrix> of $f-y^Tc$ with $y$ held fixed. In the second block, differentiating $C(x)y$ in $x$ gives $YA$ and in $y$ gives $C$. The explicit derivative with respect to $\mu$ is $(0,-e)^T$. Differentiate $F_\mu(v(\mu))=0$ by the <chain rule>:
$$
\boxed{\begin{pmatrix}H&-A^T\\YA&C\end{pmatrix}\begin{pmatrix}x'(\mu)\\y'(\mu)\end{pmatrix}=\begin{pmatrix}0\\e\end{pmatrix}.}
$$
All blocks are evaluated on the current <central path> point.
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