Finding ∂z/∂y when z = f(xz + y)

[aubergine]

I have a function z=f(xz+y) and I want to find the partial differential of z with respect to y (it's a general sort of question, I only need it in terms of the variables already given).
My answer would be just partial df/dy but this isn't the right answer. I'm not too hot on partial differentiation so can anyone give me a hint?

 

[LCKurtz]

I have a function z=f(xz+y) and I want to find the partial differential of z with respect to y (it's a general sort of question, I only need it in terms of the variables already given).
My answer would be just partial df/dy but this isn't the right answer. I'm not too hot on partial differentiation so can anyone give me a hint?

f is apparently a function of a single variable and z is a function of x and y. Since z is not solved for, you must differentiate implicitly. So begin by taking the partial derivative of both sides with respect to y. So you will start like this:

z_y = f_y(xz+y)

Now you must use the chain rule on the right side, remembering that in the argument of f, both y and z depend on y:

f_y(xz+y) = f'(xz+y)(xz+y)_y

Now finish executing the y partials and solve for z_y.