Proof for the derivative of the determinant of a matrix

Another way to obtain the formula is to first consider the derivative of the determinant at the identity: $$ \frac{d}{dt} \det (I + t M) = \operatorname{tr} M. $$

Next, one has $$ \begin{split} \frac{d}{dt} \det A (t) &=\lim_{h \to 0} \frac{\det \bigl(A (t + h)\bigr) - \det A (t)}{h}\\ &=\det A (t) \lim_{h \to 0} \frac{\det \bigl(A (t)^{-1} A (t + h)\bigr) - 1}{h}\\ &=\det A (t) \operatorname{tr} \Bigl(A (t)^{-1}\frac{d A}{dt} (t) \Bigr). \end{split} $$

This is just Jacobi's formula in the case of $A$ invertible.

Most books with any matrix theory in it should have a proof. Even wikipedia has one.