\begin{align*} \frac{\partial \xi_j }{\partial str_j} = \underbrace{\frac{\partial \xi_j}{\partial f}}_{+}*\underbrace{f'(str_j)}_{+} + \underbrace{\frac{\partial \xi_j }{\partial g }}_{-}*\underbrace{g'(str_j)}_{+} . \end{align*}