While I am able to see the differentiation of a matrix expression in the matrix cookbook of this form,
$$ \frac{\partial \mathbf{b}^T \mathbf{X}^T\mathbf{X}\mathbf{c}}{\partial \mathbf{X}} = \mathbf{X} (\mathbf{b} \mathbf{c}^T + \mathbf{c} \mathbf{b}^T)$$
I am unable to figure out the derivative of the numerator's transpose from the cookbook i.e.
$$ \frac{\partial \mathbf{c}^T \mathbf{X}\mathbf{X}^T \mathbf{b}}{\partial \mathbf{X}} = \ ?$$
For $f(X) = b^T X X^T c$ we have $$Df(X)[H] = b^T H X^T c + b^T X H^T c = tr(X^T cb^T H) + tr(X^Tbc^TH) .$$ So we have $$ \frac{\partial f(X)}{\partial X} = (bc^T + cb^T) X .$$