This page describes the row-orientation or convention, which may be a useful reference if you come across a paper in which the author adopts this approach. We will never use this convention, and always use the column-orientation convention instead.
If derivatives are taken to be row vectors (and to represent the rows of the Jacobian for a vector-valued function), then
\[\begin{alignat*}{2} &\text{Inner product:} \hspace{8em} & \pf{\A \x}{\x} &= \A \\ &\text{Quadratic form:} & \pf{\x \Tr \A \Tr \x}{\x} &= \x \Tr (\A + \A \Tr) \\ &\text{Chain rule:} & \pf{\f}{\x} &= \pf{\f}{\y} \pf{\y}{\x} \\ &\text{Product rule:} & \pf{\f \Tr \g}{\x} &= \g \Tr \pf{\f}{\x} + \f \Tr \pf{\g}{\x} \\ &\text{Inverse function theorem:} & \pf{\y}{\x} &= \left(\pf{\x}{\y}\right)^{-1} \end{alignat*}\]For the inverse function theorem to apply, the derivative must be invertible.