Identity for $\nabla[( \mathbf{a} \cdot \mathbf{b} )\mathbf{c}] \cdot \mathbf{d}$ I am looking for an identity for the following derivative:
$$\nabla[( \mathbf{a} \cdot \mathbf{b} )\mathbf{c}] \cdot \mathbf{d},$$
where $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are column vectors.
My approach to expand the expression is as follows:
\begin{align*}
\nabla[( \mathbf{a} \cdot \mathbf{b} )\mathbf{c}] \cdot \mathbf{d}
&= [ ( \mathbf{a} \cdot \mathbf{b} ) D \mathbf{c} + \mathbf{c} \otimes\nabla(\mathbf{a} \cdot \mathbf{b}) ] \cdot \mathbf{d} \\
&= ( \mathbf{a} \cdot \mathbf{b} ) D \mathbf{c} \cdot \mathbf{d} + \{ \mathbf{c} \otimes [ (D\mathbf{a})^\top \mathbf{b} + (D\mathbf{b})^\top \mathbf{a}] \} \cdot \mathbf{d}\\
&= ( \mathbf{a} \cdot \mathbf{b} ) D \mathbf{c} \cdot \mathbf{d} + \{ \mathbf{c} \otimes [ (D\mathbf{a})^\top \mathbf{b} + (D\mathbf{b})^\top \mathbf{a}] \}^\top \mathbf{d}\\
&= ( \mathbf{a} \cdot \mathbf{b} ) D \mathbf{c} \cdot \mathbf{d} + \{ [ (D\mathbf{a})^\top \mathbf{b} + (D\mathbf{b})^\top \mathbf{a}] \otimes \mathbf{c} \} \mathbf{d}\\
&= ( \mathbf{a} \cdot \mathbf{b} ) D \mathbf{c} \cdot \mathbf{d} + (\mathbf{c} \cdot \mathbf{d}) [ (D\mathbf{a})^\top \mathbf{b} + (D\mathbf{b})^\top \mathbf{a}].
\end{align*}
Here, $D \equiv \nabla^\top$, i.e., $\nabla$ stands for the gradient operator and $D$ denotes the Jacobian.
My question is, is this expansion correct? Moreover, in my understanding, the resulting expression is again a column vector. Is it right?

I think I arrive at the same conclusion but in slightly different way.
Using differentials
$$
d\mathbf{f} = (d\mathbf{a})^T\mathbf{b} \mathbf{c}+
\mathbf{a}^T d\mathbf{b} \mathbf{c}+
\mathbf{a}^T\mathbf{b} d\mathbf{c}
$$
Now using
$d\mathbf{a}=\mathbf{J}_a d\mathbf{x}$... and rearranging terms,
you will find that the Jacobian of $\mathbf{f}$ which is a matrix, is
$$
\mathbf{J}_f = 
\mathbf{c} 
\left[
\mathbf{b}^T \mathbf{J}_a + \mathbf{a}^T \mathbf{J}_b 
\right] +
\mathbf{a}^T\mathbf{b} \mathbf{J}_c 
$$
The directional derivative is $\mathbf{J}_f \mathbf{d}$.
