I found the following proof in a paper:
$\frac{D\mathbf{F}}{Dt} = \frac{D\frac{\delta\mathbf{x}}{\delta\mathbf{X}}}{Dt} = \frac{\delta\frac{D\mathbf{x}}{Dt}}{\delta\mathbf{X}}=\frac{\delta \mathbf{u}}{\delta \mathbf{X}} = \frac{\delta \mathbf{u}}{\delta \mathbf{x}}\frac{\delta\mathbf{x}}{\delta\mathbf{X}}$
From which follows:
$\frac{D\mathbf{F}}{Dt}=\mathbf{F}\cdot\frac{\delta\mathbf{u}}{\delta\mathbf{x}}$
($\mathbf{X}$ = reference configuration, $\mathbf x$ = current configuration, $\mathbf{u}$ = velocity vector)
To be honest, I can't follow the proof exactly and am thus lost on how exactly the dot product on the right hand side of the second equation is carried out. Later on in the paper the statement is repeated as
$\frac{D\mathbf{F}}{Dt}=\frac{\delta\mathbf{u}}{\delta\mathbf{x}}\cdot\mathbf{F}$ (Note the reordering)
This would suggest that $\delta\mathbf{u}/\delta\mathbf{x}$ is a scalar. However, if I remember my continuum mechanics classes correctly $\delta\mathbf{u}/\delta\mathbf{x}$ is a 2nd order tensor of the form:
[$\delta\mathbf{u}/\delta\mathbf{x}]_{i,j} = \frac{\delta u_i}{\delta x_j}$
If so, which ordering of the matrix product is correct?