# Thread: Chain Rule in Partial Derivative

1. ## Chain Rule in Partial Derivative

$\displaystyle f\left(u,v,w\right)$ is differentiable and $\displaystyle u=\;x-y,\; v=\;y-z,\;w=\;z-x$, show that $\displaystyle \frac{\partial{f}}{\partial{x}}\;+\;\frac{\partial {f}}{\partial{y}}\;+\;\frac{\partial{f}}{\partial{ z}}\;=\;0$.

I think i'll use chain rule and that equation will be $\displaystyle \frac{\partial{f}}{\partial{u}}\frac{\partial{u}}{ \partial{x}}\;+\;\frac{\partial{f}}{\partial{v}}\f rac{\partial{v}}{\partial{y}}\;+\;\frac{\partial{f }}{\partial{w}}\frac{\partial{w}}{\partial{z}}\;=\ ;0$
From now on i can not find any value for $\displaystyle \frac{\partial{f}}{\partial{u}}\;$ ,, $\displaystyle \frac{\partial{f}}{\partial{v}}\;$ and $\displaystyle \;\frac{\partial{f}}{\partial{x}}$. So how can i find these values?

If that's not the way of solution, i need some hints.

2. This is only intuition

$\displaystyle f\left(u,v,w\right) = f\left(x-y,y-z,z-x\right)$

$\displaystyle \frac{\partial f}{\partial u} = \frac{\partial f}{\partial (x - y)} = \frac{\partial f}{\partial x} - \frac{\partial f}{\partial y}$.

3. wow.! I didn't know, i can split derivative like that.
Thanks. =)

4. Neither do I lol, it`s just the only thing I can think of, may be wrong. Wait for a serious mathematician or something..

5. ## Hint

Hint: $\displaystyle \frac{{\partial f}}{{\partial x}} = \frac{{\partial f}}{{\partial u}} \times \frac{{\partial u}}{{\partial x}} + \frac{{\partial f}}{{\partial w}} \times \frac{{\partial w}}{{\partial x}}$

6. Originally Posted by Lafexlos
I think i'll use chain rule and that equation will be $\displaystyle \frac{\partial{f}}{\partial{u}}\frac{\partial{u}}{ \partial{x}}\;+\;\frac{\partial{f}}{\partial{v}}\f rac{\partial{v}}{\partial{y}}\;+\;\frac{\partial{f }}{\partial{w}}\frac{\partial{w}}{\partial{z}}\;=\ ;0$
From now on i can not find any value for $\displaystyle \frac{\partial{f}}{\partial{u}}\;$ ,, $\displaystyle \frac{\partial{f}}{\partial{v}}\;$ and $\displaystyle \;\frac{\partial{f}}{\partial{x}}$. So how can i find these values?

If that's not the way of solution, i need some hints.
.

7. $\displaystyle \frac{{\partial f}}{{\partial x}} = \frac{{\partial f}}{{\partial u}} \times \frac{{\partial u}}{{\partial x}} + \frac{{\partial f}}{{\partial w}} \times \frac{{\partial w}}{{\partial x}} = \frac{{\partial f}}{{\partial u}} - \frac{{\partial f}}{{\partial w}}$

$\displaystyle \frac{{\partial f}}{{\partial y}} = \frac{{\partial f}}{{\partial v}} \times \frac{{\partial v}}{{\partial y}} + \frac{{\partial f}}{{\partial u}} \times \frac{{\partial u}}{{\partial y}} = \frac{{\partial f}}{{\partial v}} - \frac{{\partial f}}{{\partial u}}$

$\displaystyle \frac{{\partial f}}{{\partial z}} = \frac{{\partial f}}{{\partial v}} \times \frac{{\partial v}}{{\partial x}} + \frac{{\partial f}}{{\partial w}} \times \frac{{\partial w}}{{\partial z}} = - \frac{{\partial f}}{{\partial v}} + \frac{{\partial f}}{{\partial w}}$