I've never differentiated utility functions. I'm struggling to understand how the differentiation of this problem results in the answer that it does. Could someone please show me (step by step) how this is being solved?
The problem is: G*U(A) + (1-G)*U(A+L) = U(A+L-p)
The solution is: dP/dA = -[G*U'(A) + (1-G)*U'(A+L) - U'(A+L-p)]/[U'(A+L-p]
Yes. This appears to be a simple application of the implicit function theorem.
Think of $p$ as a function of $A$. Then your first equality must hold for all values of $A$. Hence the derivative of the difference must equal zero. Thus,
$$GU'(A) + (1-G) U'(A+L) - U'(A+L-p) \Big(1-\frac{dp}{dA}\Big)=0.$$
Solve for $dp/dA$.