Function

[kidia]

I need a help on this question,Show that if u=x-y,v=xy and w=f(u,v) then x\(\displaystyle \frac{dw}{dx}\)-y\(\displaystyle \frac{dw}{dy}\)=(x-y)\(\displaystyle \frac{dw}{du}\).

The point which I failed to understand is w=f(u,v).

 

[pka]

Are you sure that you have written this question correctly?
Are these partial derivatives?

 

[kidia]

Are you sure that you have written this question correctly?
Are these partial derivatives?

Yah it is absolute correct written,it seem like partial derivatives.

 

[pka]

Using the chain rule for partial derivatives:
\(\displaystyle \L
\begin{array}{l}
\frac{{\partial w}}{{\partial x}} = \frac{{\partial w}}{{\partial u}}\frac{{\partial u}}{{\partial x}} + \frac{{\partial w}}{{\partial v}}\frac{{\partial v}}{{\partial x}}\quad \Rightarrow \quad \frac{{\partial w}}{{\partial x}} = \frac{{\partial w}}{{\partial u}}(1) + \frac{{\partial w}}{{\partial v}}(y) \\
\frac{{\partial w}}{{\partial y}} = \frac{{\partial w}}{{\partial u}}\frac{{\partial u}}{{\partial y}} + \frac{{\partial w}}{{\partial v}}\frac{{\partial v}}{{\partial y}}\quad \Rightarrow \quad \frac{{\partial w}}{{\partial y}} = \frac{{\partial w}}{{\partial u}}( - 1) + \frac{{\partial w}}{{\partial v}}(x) \\
\end{array}\)

Multiply by x & -y:
\(\displaystyle \L
\begin{array}{l}
x\frac{{\partial w}}{{\partial x}} = \frac{{\partial w}}{{\partial u}}(x) + \frac{{\partial w}}{{\partial v}}(xy) \\
- y\frac{{\partial w}}{{\partial y}} = \frac{{\partial w}}{{\partial u}}(y) - \frac{{\partial w}}{{\partial v}}(xy) \\
\end{array}\)

Now add together:
\(\displaystyle \L
x\frac{{\partial w}}{{\partial x}} - y\frac{{\partial w}}{{\partial y}} = \left( {x + y} \right)\frac{{\partial w}}{{\partial u}}\)

 

[kidia]

Excellent!!!!!,Thanx very much pka I understood you.