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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?

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  • $\begingroup$ It would help if you tell us in which space you're working and what are $A, F, V$ and $L$. $\endgroup$ May 23, 2021 at 15:39
  • $\begingroup$ @mathcounterexamples.net I have updated the equation by adding these informations (at least I hope). $\endgroup$
    – fennel
    May 23, 2021 at 15:46
  • $\begingroup$ Is $L$ a function of the single variable $A$? If so, then you can replace $\partial L/\partial A$ with $dL/dA$, or just $L’$. $\endgroup$
    – MPW
    May 23, 2021 at 15:49

1 Answer 1

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Yes, this is correct. More briefly, $$L’=\frac{F}{\lambda V’-AF’},$$ assuming L is a function of the single variable $A$.

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