total differential. please help.?

Question

what is the total differential of

z = (x - y)/(x + 1)

Answer 1

It depends on if you mean with respect to x or with respect to y. In general, dz/dx = ∂z/∂x + (∂z/∂y)(∂y/∂x), where ∂ is the partial derivative operator, and the second term is an application of the chain rule. Similarly, dz/dy = ∂z/∂y + (∂z/∂x)(∂x/∂y). Even though this expression doesn't involve a time variable t, you can still say dz/dt = (∂z/∂x)(∂x/∂t) + (∂z/∂y)(∂y/∂t). You haven't been given any information about the dependence of x and y on each other or on t, so we can assume there isn't one. If we're looking for the total differential with respect to x, then dz/dx = ∂z/∂x because (∂y/∂x) = 0. dz/dx = ∂z/∂x = ((x + 1)(1) - (x - y)(1)) / (x + 1)^2 = (x + 1 - x + y) / (x^2 + 2x + 1) = (y + 1) / (x^2 + 2x + 1).

Answer 2

complete differential demands partial derivatives dz= (partial of z wrt x)dx + (partial of z wrt y)dy partial of z wrt x holds y as a continuing so = (3y^2)x^(3y^2-a million) partial of z wrt y holds x as a continuing so = x^3y^2lnx dz = (3y^2)x^(3y^2-a million) dx + x^3y^2lnx dy