Chain rule in partial derivative

[bigwuying]

There is a theorem in partial derivative
If x= x(t) , y= y(t), z= z(t) are differentiable at $$t_{0}$$, and if w= f(x,y,z) is differentiable at the point (x,y,z)=(x(t),y(t),z(t)),then w=f(x(t),y(t),z(t)) is differentiable at t and
$$\frac{dw}{dt}$$=$$\frac{\partial w}{\partial x}$$$$\frac{dx}{dt}$$ + $$\frac{\partial w}{\partial y}$$ $$\frac{dy}{dt}$$ + $$\frac{\partial w}{\partial z}$$ $$\frac{dz}{dt}$$
where the ordinary derivatives are evaluated at t and the partial derivatives are evaluated at (x,y,z)

My question are
1. What is the difference between dx and$$\partial x$$?
2. i don't know why i am not allowed to eliminate the dx and $$\partial x$$,also the same for y and z to get 3$$\frac{\partial w}{dt}$$

 

[HallsofIvy]

The problem appears to be that you do not understand the derivative itself. Neither "$\partial x$" nor "dx" have any meaning alone. And, when using the "ordinary" chain rule- (df/dx)(dx/dy)= df/dy- you are NOT "canceling" the "dx"s.

 

[bigwuying]

Yes, you are right. i only know how to use chain rule, but i don't know what it is actually.
When i start to learn Partial Derivative, i find it difficult to understand why dw/dt appear 3 term, and why do dx and dx not be eliminate.

To understand this problem ,i have read my text and start confusing

The definition of partial d of z with repect to x at $$x_{0},y_{0}$$ is
$$f_{x}(x_{0},y_{0})=lim_{x->x0}\frac{f(x,y0)-f(x0-y0)}{x-x0}= \frac {\partialf}{\partialx}$$

Consider a part of the chain rule, let say : dw/dt= $$\frac{\partial w}{\partial x}$$ $$\frac{dx}{dt}$$

it is equivalent to $$\left[lim_{x->x0}\frac{f(x,y0,z0)-f(x0,y0,z0)}{x-x0}\right]$$ $$\left[lim_{t->t0}\frac{x(t)-x(t0)}{t-t0}\right]$$

$$lim_{x->x0} x-x_{0}$$ isn't the same as $$lim_{t->t_{0}} x(t)-x(t_{0})$$ ??
So i get confusing why i can't eliminate them. or i cannot treat them as a fraction??