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It suffices to consider the case $n=2$, $m=1$. Namely, write $Y=[Y_1|Y_2]$ etc, then $L=Y_1$ is upper triangular with positives on the diagonal, and \begin{align*} X &= Y^\top Y = \begin{pmatrix} Y_1^\top Y_1 & Y_1^\top Y_2 \\ Y_2^\top Y_1 & Y_2^\top Y_2\end{pmatrix}\,, \\ dX &= \begin{pmatrix} (dY_1)^\top. Y_1 + Y_1^\top.dY_1 & (dY_1)^\top. Y_2 + Y_1^\top.dY_2 \\ (dY_2)^\top. Y_1 + Y_2^\top. dY_1 & (dY_2)^\top. Y_2 + Y_2^\top.dY_2 \end{pmatrix} \\ YB &= Y_1B_1 + Y_2B_2 \end{align*} Moreover, \begin{align*} df(X) =& -(YB)^{-1}.dY.B.(YB)^{-\top} - (YB)^{-1}.A.(YB)^{-\top}.(dY.B)^\top.(YB)^{-\top} \\=& -(YB)^{-1}.dY.B.f(X) - f(X)((YB)^{-1}.dY.B)^\top \\ dY.B =& dY_1.B_1 + dY_2.B_2 \end{align*} Now you can compute $dY_1$ from $dX_{1,1}$ as in the invertible situation, etc.