I need to find the derivative of $L$ with respect to $A$ given the implicit function $$A F (L) = \lambda V(L),$$
where $F:{\Bbb R} \to {\Bbb R}$, $V:{\Bbb R} \to {\Bbb R}$, $\lambda \in {\Bbb R} $ and $A\in {\Bbb R} $.
What I did is the following: taking derivatives in both sides I obtain
\begin{equation} 1 \cdot F(L) + A \frac{\partial L}{\partial A} F'(L) = \lambda \frac{\partial L}{\partial A} V'(L) \end{equation}
rearranging and solving for $\partial L/\partial A$ I get the following solution
\begin{equation} \frac{\partial L}{\partial A} = \frac{F(L)}{\lambda V'(L) - A F'(L)}. \end{equation}
It is the first time I try to solve this kind of problems. Can I solve this problem this way?
Yes, this is correct. More briefly, $$L’=\frac{F}{\lambda V’-AF’},$$ assuming L is a function of the single variable $A$.