Let $A(x\_1,...,x\_n)$ be an $n\times n$ matrix field over $R^n$. 


I am interested in the partial derivative determinant of $A$ in respect to $x\_i$. In can be shown that:


$\frac{\partial{\det(A)}}{\partial{x\_i}} = \det(A)\cdot\sum\_{a=1}^{n}{\sum\_{b=1}^{n}{ A^{-1}\_{a,b} \cdot \frac{ \partial{A\_{b,a}} }{ \partial{x\_i} }}}$


I was able to prove this using induction and careful, boring calculations, but I was wondering if there was any intuition behind this formula? 



Derivative of a determinant of a matrix field
your identity follows simply by using $\log({\rm det}\; A)= {\rm tr}\; (\log A)$, so 


$$\frac{\partial}{\partial x\_i}{\rm det}\;A=
\frac{\partial}{\partial x\_i} \exp({\rm tr}\;\log A)=
({\rm det}\;A) \frac{\partial}{\partial x\_i}{\rm tr}\;\log A=
({\rm det}\;A)\;{\rm tr}\;\left(A^{-1}\frac{\partial}{\partial x\_i}A\right)$$


this identity is known as [Jacobi's formula](http://en.wikipedia.org/wiki/Jacobi_formula).


