Partial derivatives, simple question about notations

Question

let $w=f(x,u,v)$ $u=(x,v,y)$ $v=h(x,y)$

Now textbook suggests that :

$$\frac{\partial w}{\partial x} = \frac{\partial w}{\partial x} + \frac{\partial w}{\partial u}\frac{\partial u}{\partial x}+ \frac{\partial w}{\partial v}\frac{\partial v}{\partial x} + \frac{\partial w}{\partial u}\frac{\partial u}{\partial v}\frac{\partial v}{\partial x}$$

Shouldn't this be equal to $\frac{dw}{dx}$ instead of $\frac{\partial w}{\partial x}$

Answer 1

Hint It is not $\frac{dw}{dx}$ since $w$ depends on two variables $x$ and $y$.

to obtain $\frac{\partial w}{\partial x}=w'_x$,

use the following

• $$dw=f'_xdx+f'_udu+f'_vdv$$

• $$du=g'_xdx+g'_vdv+g'_ydy$$

• $$dv=h'_xdx+h'_ydy$$

plug $du$ and $dv$ in $dw$ and

factor to get as you said $\frac{\partial w}{\partial x}.$

Answer 2

The book is right, since ultimately $w$ depends on $x$ and $y$, so a partial derivative is needed on the left-hand side.