Direct transformation:
$$
dx_1' = \frac{\partial x_1'}{\partial x_1}dx_1 + \frac{\partial x_1'}{\partial x_2}dx_2 \\
dx_2' = \frac{\partial x_2'}{\partial x_1}dx_1 + \frac{\partial x_2'}{\partial x_2}dx_2
$$
$$
\begin{bmatrix} dx_1' \\ dx_2' \end{bmatrix} = \large
\begin{bmatrix} \frac{\partial x_1'}{\partial x_1} & \frac{\partial x_1'}{\partial x_2} \\
\frac{\partial x_2'}{\partial x_1} & \frac{\partial x_2'}{\partial x_2}
\end{bmatrix} \normalsize
\begin{bmatrix} dx_1 \\ dx_2 \end{bmatrix}
$$
Inverse transformation:
$$
dx_1 = \frac{\partial x_1}{\partial x_1'}dx_1' + \frac{\partial x_1}{\partial x_2'}dx_2' \\
dx_2 = \frac{\partial x_2}{\partial x_1'}dx_1' + \frac{\partial x_2}{\partial x_2'}dx_2'
$$
$$
\begin{bmatrix} dx_1 \\ dx_2 \end{bmatrix} = \large
\begin{bmatrix} \frac{\partial x_1}{\partial x_1'} & \frac{\partial x_1}{\partial x_2'} \\
\frac{\partial x_2}{\partial x_1'} & \frac{\partial x_2}{\partial x_2'}
\end{bmatrix} \normalsize
\begin{bmatrix} dx_1' \\ dx_2' \end{bmatrix}
$$
With the direct transformation, calculating the inverse results in:
$$
\begin{bmatrix} dx_1 \\ dx_2 \end{bmatrix} = \large
\begin{bmatrix} \frac{\partial x_2'}{\partial x_2} & -\frac{\partial x_1'}{\partial x_2} \\
-\frac{\partial x_2'}{\partial x_1} & \frac{\partial x_1'}{\partial x_1}
\end{bmatrix} / \normalsize \det
\begin{bmatrix} dx_1' \\ dx_2' \end{bmatrix} \\ \mbox{with} \qquad
\det = \frac{\partial x_1'}{\partial x_1}\frac{\partial x_2'}{\partial x_2}
- \frac{\partial x_2'}{\partial x_1}\frac{\partial x_1'}{\partial x_2}
$$
With the inverse transformation, calculating the inverse results in:
$$
\begin{bmatrix} dx_1' \\ dx_2' \end{bmatrix} = \large
\begin{bmatrix} \frac{\partial x_2}{\partial x_2'} & -\frac{\partial x_1}{\partial x_2'} \\
-\frac{\partial x_2}{\partial x_1'} & \frac{\partial x_1}{\partial x_1'}
\end{bmatrix} / \normalsize \det'
\begin{bmatrix} dx_1 \\ dx_2 \end{bmatrix} \\ \mbox{with} \qquad
\det' = \frac{\partial x_1}{\partial x_1'}\frac{\partial x_2}{\partial x_2'}
- \frac{\partial x_2}{\partial x_1'}\frac{\partial x_1}{\partial x_2'}
$$
Hope you can take it from here.