Partial derivative on chain rule

Question

Could anyone please guide me whether the solution of this partial derivative is correct?

Solution from reference material:

I have tried to calculate my own solution but it is different.

My calculation:

Take the case of two functions with two variables each

The solution of reference material, which is using summation to add

1. both results of dy/dx_1
2. both results of dy/dx_2

May i know the theory behind it? Can we just simply add both results together?

Thank you.

Answer 1

Your solution for the two-variable case is on the right track, but there is no "or" in the derivative $\partial y/\partial x_1$. Since both $u_1$ and $u_2$ are functions of $x_1$, you have to add their contributions together:

$\qquad \begin{equation} \dfrac{\partial y}{\partial x_1}=\dfrac{\partial y}{\partial u_1}\,\dfrac{\partial u_1}{\partial x_1}+\dfrac{\partial y}{\partial u_2}\,\dfrac{\partial u_2}{\partial x_1} \end{equation} $

Of course, similar considerations apply to $\partial y/\partial x_2$, and to the case of $n$ independent variables $x_1,\ldots,x_n$.