Derivative of $\log^2(\det(A))$ w.r.t. matrix A

Question

Given that $A\in\mathbb{R}^{n\times n}$, is there an explicit matrix expression for

$$\frac{d}{dA}\log^2(\det(A))$$

(in numerator layout format)?

Attempt

$$=2\log(\det(A))\frac{d}{dA}\log(\det(A))$$ $$=2\log(\det(A))\frac{1}{\det(A)}\frac{d}{dA}\det(A)$$