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I have a question of partial derivatives for implicit functions. I have three equations:

$rho_k=rho_lfrac{lambda_l}{H_l}+rho_gfrac{(1-lambda_l)}{1-H_l}$
$Re=frac{rho_kv_mphi}{mu}$
$H_l=f(Re,lambda_l)$

$phi$ is a constant and $f()$ is a complex function. $rho_l,rho_g,lambda_l,v_m,mu$ are all functions of $P$ (and $Q$). I solve these iteratively for $H_l$ given $P$ and $Q$. Now I need $frac{partial H_l}{partial P}$. I do this by differentiating all three equations:

$frac{partial rho_k}{partial P}=frac{partial rho_k}{partial rho_l}frac{partial rho_l}{partial P}+frac{partial rho_k}{partial rho_g}frac{partial rho_g}{partial P}+frac{partial rho_k}{partial lambda_l}frac{partial lambda_l}{partial P}+frac{partial rho_k}{partial H_l}frac{partial H_l}{partial P}$
$frac{partial Re}{partial P}=frac{partial Re}{partial rho_k}frac{partial rho_k}{partial P}+frac{partial Re}{partial v_m}frac{partial v_m}{partial P}+frac{partial Re}{partial mu}frac{partial mu}{partial P}$
$frac{partial H_L}{partial P}=frac{partial H_L}{partial Re}frac{partial Re}{partial P}+frac{partial H_L}{partial lambda_l}frac{partial lambda_l}{partial P}$

I can solve these for $frac{partial H_L}{partial P}$ if I have all the other derivatives.

However I have a question on how I calculate some of the other derivatives needed, in particular $frac{partial rho_k}{partial lambda_l}$ ? Do I keep $H_l$ constant or do I need to take into consideration $frac{partial H_L}{partial lambda_l}$ ?

Mathematically, is $frac{partial rho_k}{partial lambda_l}$ equal to:

$rho_lfrac{2.0lambda_l}{H_l}-rho_gfrac{2.0(1.0-lambda_l)}{1.0-H_l}$, or

$rho_lfrac{2.0H_llambda_l-lambda_l^2frac{partial H_L}{partial lambda_l}}{H_l^2}-rho_gfrac{2.0(1.0-H_l)(1.0-lambda_l)+(1-lambda_l)^2frac{partial H_L}{partial lambda_l}}{(1.0-H_l)^2}$ ?

Neither of these formulations give the same analytical value of $frac{partial H_L}{partial P}$ as a numerical perturbation though, so I'm wondering if I am doing this all correctly.

It turns out the first formulation for $frac{partial rho_k}{partial lambda_l}$ is correct, that is:

$rho_lfrac{2.0lambda_l}{H_l}-rho_gfrac{2.0(1.0-lambda_l)}{1.0-H_l}$

The errors with $frac{partial H_L}{partial P}$ were in the derivatives I was calculating for $H_l=f(Re,lambda_l)$. This is a graph which I was using Bezier functions to interpolate. This works well in 1d, but 2d Bezier interpolation causes noise, resulting in erroneous derivatives.