Let $A \in R^{n \times d}$ and $B \in R^{d \times m}$, and let $ C = AB $. How would $ \frac{dC}{dB} $ be defined?
Intuitively, I feel it's $ A^T $, but I'm a little unclear how to arrive at it mathematically.
Any help or pointers would be much appreciated! Thanks!
Let's define $f(B)=AB$. Then the directional derivative along a matrix $D$ is $$ \lim_{h\to0}\frac{f(B+hD)-f(B)}{h}=\lim_{h\to0}\frac{A(B+hD)-AB}{h}=AD. $$ This means that the Gateaux derivative is given by $d_BfD=AD$ or simply $d_Bf=A$.