I have an optimisation problem $\min f(x)$ s.t. $g(x) = E$, with Lagrangian $L = f(x) + \lambda(g - E)$, and I know that $g$ is binding. Can I argue that the shadow price can be locally approximated by the total derivative of the objective function $\lambda =\frac{\text{d} f}{\text{d} E} $ ?
To be clear, if $f(x) = f_1(x) + f_2(x)$, I want to take
$\lambda = \frac{\text{d} f}{\text{d} E} = \frac{\partial f}{\partial f_1} \frac{\text{d} f_1}{\text{d} E} + \frac{\partial f}{\partial f_2} \frac{\text{d} f_2}{\text{d} E}$