Partial differentials and total Differentials Why is it that the partial differential cannot cancel out, just like the total differential?
With $\frac{∂f}{∂x}\frac{∂x}{∂t}$, why is it that the $∂x$ in the middle cannot be canceled out, whereas with $\frac{\text{d}f}{\text{d}x}\frac{\text{d}x}{\text{d}t}$, the $\text{d}x$ in the middle can be canceled?
 A: For simplicity assume you have $f(x_1,\dots,x_n)$ and $x_1,\dots,x_n$ are functions only of $t$. Then the chain rule tells you that $\frac{df}{dt}=\sum_{i=1}^n \frac{\partial f}{\partial x_i} \frac{dx_i}{dt}$. The interpretation of each term of this sum is the contribution to the rate of change in $f$ as a result of the change in $x_i$. You need all of these contributions added together in order to get the overall rate of change of $f$.
That gets you the right intuition even when $x_i$ depend on more than one variable.
That said, a quick-and-dirty, low-rigor way to think about this is that a partial differential is not uniquely defined until you specify what is held fixed for it. Say for instance that $x_1,\dots,x_n$ depend on $t_1,\dots,t_k$. Then in $\frac{\partial f}{\partial x_1} \frac{\partial x_1}{\partial t_1}$, what is held fixed in the two $\partial x_1$ "expressions" is different. In the first factor, it is $x_2,\dots,x_n$; in the second, it is $t_2,\dots,t_k$. As a result you should not expect them to cancel out.
