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.
Could you please help me? I'm new to Mathematica so I will appreciate any help. Thanks in advance.