Earlier in our tutorials, we discussed the treatment of differentials like $dx$ and $dy$, and whether you could simply manipulate the differentials as you would ordinary numbers.

From a *rigorous* sense, separating differentials should not be done until you can develop the theory of how to do it. This leads to a whole new theory called Non-standard analysis.

But let's not do that. Let's focus on where things go wrong in a *practical* sense. At least for the physicists, you may not care that it's not *rigorous* mathematics as long as it gives the right answer.

So let's see where it gives the wrong answer.

It's obvious that in a rigorous sense, you can't! The definition of the limit is \[ y'(x) = \lim_{\delta x\to 0} \frac{y(x + \delta x) - y(x)}{\delta x} \equiv \frac{dy}{dx}. \]

However, because both numerator and denominator are tending to zero, then you would have to identify $dx = 0$ and $dy = 0$. So a statement like \begin{equation} \label{dy} dy = y'(x) dx \end{equation}

is vacuous because it is expressing $0 = 0$. The nature of $y'(x)$ is that it is a very finely tuned quantity which corresponds to both numerators and denominators tending to zero *in a very particular way*. So $dx$ and $dy$ are intimately linked: one cannot exist without the other.

But then again, *“it works!”* many students will claim. For example, if $y = y(x)$ was our original curve, and we now parameterised $x = x(t)$, then by the chain rule,
\[
\frac{dy}{dt} = \frac{dx}{dt}\frac{dy}{dx}
\]

and we point to the above formula and say “see? it works!”.

We might also make an example of the rule for inverting functions. If $y = f(x)$ and we invert this using $x = f^{-1}(y)$, then the derivative of the inverse function is given by \[ \frac{df^{-1}(y)}{dy} = \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}. \]

See? They're basically fractions!

Another example is separation of variables when solving differential equations. If we have \[ \frac{dy}{dx} = \frac{1}{y}, \]

then we simply manipulate the differentials \[ y \ dy = dx, \]

integrate both sides \[ \int y \, dy = \int dx \]

and *voila*:
\[
\tfrac{1}{2} y^2 = x + C.
\]

It works, see?

Blindly treating differentials like fractions works well enough when you're in first year and working with functions of a single variable. But treating $dy/dx$ like a fraction is a gateway drug to treating $\partial y/\partial x$ like a fraction.

Here's an obvious example. Let $F(x,y) = 0$ be a function that defines $y$ implicitly. For example, suppose that \[ F(x,y) = x + y. \]

Then blindly treating the differentials as fractions, we have \[ \frac{dy}{dx} = \frac{{\partial F}}{\partial x}\frac{\partial y}{\partial F} = \frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}} = \frac{1}{1} = 1. \]

But obviously, $y = -x$ and so $dy/dx = -1$.

In fact, you can show that the *correct* formula is:
\[
\frac{dy}{dx} = -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}
\]

so treating differentials like fractions will be problematic even for the simplest of problems.

The reason why it works in 1D is because there is only one object being varied (the $dx$) and one object who's variation you are concerned about ($dy$). In 2D, for example, such as the case of $F(x,y)$, then the variation of $F$ depends on how we choose to vary *both* $x$ and $y$.

So while it makes sense to think of $dy/dx$ as dividing by a *number*, $dx$, it doesn't make sense to think of derivatives of $F$ as obtained by dividing by a vector $[dx, dy]$.

You will learn that there is a difference between $\partial F/\partial x$, which is the variation of $F$ as $x$ changes but $y$ is fixed, and the *total* variation of $F$ with respect to $x$, $dF/dx$. The latter is
\begin{equation} \label{dFdx}
\frac{dF}{dx} = \frac{dx}{dx} \frac{\partial F}{\partial x} + \frac{dy}{dx} \frac{\partial F}{\partial y}.
\end{equation}

The difference is that by considering $y = y(x)$, then when we vary $x$, we must also take in account the change in $y$. Thus $F$ is changed in response to variations of *both* $x$ and $y$, but there is only a single quantity being varied ($x$). If we are interested in the curve $F(x,y) = 0$, then we can set the \eqref{dFdx} to zero, and this gives the proper value of $dy/dx$.

If you fall into the trap of constantly thinking of differentials as equivalent to ordinary numbers, then what is the difference between $dx$, $\Delta x$, $\delta x$, and $\partial x$? Similarly, what is the difference between $dF$, $\Delta F$, $\delta F$, and $\partial F$? Is $\delta x = dx$? Is $\partial x = dx$? If it's not true then do we say that $dx > \partial x$ or $dx < \partial x$? These are all very silly questions because the infinitesimals are ill-defined by themselves and should not be thought-of as real numbers.

Treating differentials like fractions is a gateway drug to further misunderstanding.

And in more than one dimension, it's just plain *wrong*.