# Differential and derivative of matrix function

Compute the differential and the derivative of the following function: $f: M(m \times n) \rightarrow\Bbb R$ which is given by $f(X) = \cos(b(\text{transpose})X(\text{transpose})AXb)$, where $A$ is a constant matrix and $b$ is a constant column vector.

So far I did the following: I computed the differential of the composition function $b(\text{transpose})X(\text{transpose})AXb$, which I think is equal to $b'X'Ahb + b'h'AXb$. I also computed the derivative of this function, namely $b'(\text{Kronecker})(b'X'A) + b'(\text{Kronecker})(b'hA)$.

To find the differential and the derivative of $f(X)$, I think I need to use the chain rule. So I would get $-\sin(b(\text{transpose})X(\text{transpose})AXb)$ multiplied by the derivative or the differential of the composition function I computed. However, I am not sure with which of the two I should multiply, namely the derivative, or the differential. Could anyone help me out? Could you also please comment on whether my approach to this problem is correct?

the primes denote a transpose and the h=dX

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Yes with respect to X. X is a variable matrix. Oh sorry, with X' I mean X transpose, thats a notation commonly used in econometrics for it. –  901301 Feb 22 at 20:02