Quote (feanur @ Oct 3 2017 10:53am)

du_x/dx1 = Ax1^2 x3 + Bx2x3

u_x(x₁ , x₂ , x₃) = A.(x₁³ / 3).x₃ + B.x₁.x₂.x₃ + C(x₂ , x₃)
where C is a function that does not depend on x₁.

When derivating C with respect to x₁, only 0 remains (since, precisely, it doesn't depend on x₁).

Without any other information, it's impossible to know anything about C.
It could be as simple as : C(x₂ , x₃) = x₂ + x₃
or as complicated as : C(x₂ , x₃) = log(x₂² + x₃⁶ + 1) / (sin x₂ + cos x₃ + exp(x₂ - x₃) )



What you must add is a constant - with respect to the variable you're working on.
In other word, you add a function that depends on any other variables (see examples above).