\begin{align*}

\frac{dL}{dz^{[1]}} = & \frac{dL}{da^{[1]}} \times \frac{da^{[1]}}{dz^{[1]}} \\

& = \frac{dz^{[2]}}{da^{[1]}} \frac{dL}{dz^{[2]}} \times \frac{da^{[1]}}{dz^{[1]}} \\
& = \frac{d(W^{[2]}{^T}a^{[1]})}{d a^{[1]}} {\bf dz^{[2]}} \times \frac{da^{[1]}}{dz^{[1]}} \\
& = W^{[2]}{^T} {\bf dz^{[2]}} \times \frac{da^{[1]}}{dz^{[1]}} \\

& = \frac{W^{[2]}{^T}}{dz^{[2]}} \times \frac{d}{dz^{[1]}}g^{[1]}(z^{[1]}) \\

& = \frac{W^{[2]}{^T}}{dz^{[2]}} \times g^{[1]}{'}(z^{[1]})

\end{align*}