I am struggeling to find the derivative of $f(x) = a^T(x) b(x) c(x)$, where $a,b,c,x$ are all $\in R^{3x1}$.
Using single variable chain rule I get: $f'(x) = \frac{d}{dx}\left(a^T(x)\right) * b(x) * c(x) + ...$
However, already here I am hold back by a dimension mismatch: A vector derived by a vector should be a matrix, so $\frac{d}{dx}(a^T(x))$ should be a $3x3$ matrix. Multiplying this matrix with $b(x)$ would yield a $3x1$ column vector. And multiplying this with $c(x)$ would result in a dimension mismatch.
What am I missing? Thx in advance