# Help in taking the derivative of an implicit function

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

$$$$1 \cdot F(L) + A \frac{\partial L}{\partial A} F'(L) = \lambda \frac{\partial L}{\partial A} V'(L)$$$$

rearranging and solving for $$\partial L/\partial A$$ I get the following solution

$$$$\frac{\partial L}{\partial A} = \frac{F(L)}{\lambda V'(L) - A F'(L)}.$$$$

It is the first time I try to solve this kind of problems. Can I solve this problem this way?

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

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