The discussion revolves around finding the partial derivatives of the function \( z = \frac{1}{x}[f(x-y) + g(x+y)] \) and proving a specific relationship involving these derivatives. The problem is situated within the context of multivariable calculus, specifically focusing on partial derivatives and the application of the chain rule.
The discussion is ongoing, with participants exploring different interpretations of the problem. There is an exchange of ideas regarding the use of the chain rule and how to handle the derivatives of composite functions. Some guidance has been offered on applying the chain rule, but no consensus has been reached on the overall approach.
Participants note the challenge of working with undefined functions \( f \) and \( g \), which impacts their ability to compute the required derivatives. The discussion reflects a common constraint in homework settings where specific forms of functions are not provided.
chmate said:Homework Statement
If [itex]z=\frac{1}{x}[f(x-y)+g(x+y)][/itex], prove that [itex]\frac{\partial }{\partial x}(x^2\frac{\partial z}{\partial x})=x^2\frac{\partial^2 z}{\partial y^2}[/itex]
Homework Equations
The Attempt at a Solution
I don't know how I'm supposed to find the partial derivative of z with respect to any of variables if the function f and g are not expressed.
Help me!
Thank you
LCKurtz said:Just leave the f and g in there, maybe they will cancel out. For example, if you want the partial of f(x-y) with respect to y it would be:$$
f_y(x-y) = -1\cdot f'(x-y)$$and so on.
chmate said:Can you explain me why [itex]f_y(x-y) = -1\cdot f'(x-y)[/itex]? I don't get it.
What is then [itex]f_x(x+y) = ?[/itex]