# Implicit derivative of set of equations

I have an equation and I want to take its derivative with respect to two variables. However, it contains some parameters which need to solve them together to find them (Pn, omega, gamma). These parameters are implicit in each other. I defined first some parameters and I want now to take the partial derivative with respect to rho, x1, x2.

    alpha = sqrt (4*pi*beta*e^2/eps0)
eta3 = x1*rho*sigma1^3 + x2*rho*sigma2^3
delta = 1 - pi*eta3/6
rho1 = x1*rho
rho2 = x2*rho

gamma = alpha/
2*((rho1*(z1 -
pi/(2*delta)*sigma1^2*Pn/(1 + gamma*sigma1))) + (rho2*(z2 -
pi/(2*delta)*sigma2^2*Pn/(1 + gamma*sigma2))))^0.5

omega = 1 +
pi/(2*delta)*(rho1*sigma1^3*z1/(1 + gamma*sigma1) +
rho2*sigma2^3*z2/(1 + gamma*sigma2))

Pn = 1/omega*(rh1*sigma1*z1/(1 + gamma*sigma1) +
rh2*sigma2*z2/(1 + gamma*sigma2))

f[_rho, _x1, _x2] = -e^2/
eps0*(gamma*((rho1*
z1^2/(1 + gamma*sigma1) + (rho2*z2^2/(1 + gamma*sigma2)) +
pi/(2*delta)*omega*Pn^2)) + gamma^3/(3*pi)


I attached two pictures to illustrate the equations: - I want to take the derivative of the equation 28 with respect to xion, where rho_ion = xion*rho. Keeping in mid the following: The system of equations 29 need to be solved first to get the derivative since they depend on xion and on each other. Also, they are implicit.

• You also have a problem with your definition of gamma. You start by saying gamma= and so Mathematica makes note that gamma is being defined and it starts evaluating the right hand side. During that it looks up the values for alpha, rho1, rho2 and then it finds gamma on the right hand side. So it goes back to the definition of gamma and starts over. And over. And over. Until it has started over 1024 times and bails out. Is that the exactly correct definition of gamma? Can you explain precisely what you mean gamma to be? You also seem to have a missing ) inside your definition of f.
• You need to learn the difference between and the appropriate use Set ( = ) and SetDelayed ( := ). There is a discussion of it here Dec 17 '18 at 1:25