Computing the partial derivatives of a function defined as another compound function

Question

The problem is as follows:

My working is:

I don't know what I should be doing. Can you please help?

Answer 1

According to your first line,

$g(x,y)=f(u(x,y),v(x,y))$.

This does not imply $\frac{\partial g}{\partial x}=\frac{\partial f}{\partial x}$ (or "similarly for $y$").

In line 2, I believe what you are trying to say is $$ \frac{\partial g}{\partial x}=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x} $$.

Subject to these changes in interpreting the chain rule and the role of variables for partial derivatives, you are not far from executing the right logical steps to obtain the required answer.

---

You can avoid the problem of notation by working directly with respect to the correct variable:

$$g_x(x,y)=f_x(x+y,-x+y) \cdot \frac{\partial}{\partial x}(x+y)+f_y(x+y,-x+y) \cdot \frac{\ \partial}{\partial x} (-x+y)$$

$$g_y(x,y)=f_x(x+y,-x+y) \cdot \frac{\partial}{\partial y}(x+y)+f_y(x+y,-x+y) \cdot \frac{\ \partial}{\partial y} (-x+y)$$

which leads to Option (c).