What justifies using the derivative $\frac{\mathrm{d}y}{\mathrm{d}x}$ as a fraction?

Question

For instance, one method of solving first-order equations uses separation of variables. $$\frac{\mathrm{d}y}{\mathrm{d}x}=yx$$ $$\frac{\mathrm{d}y}y=x\;\mathrm{d}x$$ $$\log |y|=x^2/2+C$$ $$y=Ce^{\frac{x^2}2}$$

Answer 1

It's definitely true that $\frac{dy}{dx}$ is approximately a ratio of very tiny numbers: \begin{equation} \frac{dy}{dx} \approx \frac{\Delta y}{\Delta x}, \end{equation} where $\Delta x$ is a tiny change in the input to $y$, and $\Delta y$ is the corresponding change in the value of $y$. We can do manipulations involving $\Delta x$ and $\Delta y$ separately, obtaining approximations rather than exact equations. We can then hope that "in the limit" we will obtain exact equality rather than mere approximations. Of course, if we're being careful, we will later have to find a rigorous proof that the exact equality holds.

Undergrad physics textbooks often contain arguments that involve manipulating infinitesimal quantities, but I think they are actually making an argument very similar to what I just described.

Another thing to be aware of is that, in differential equations textbooks, usually arguments that involve manipulating infinitesimals (such as the example given in your question) can easily be slightly rephrased to avoid using infinitesimals.

Answer 2

There is no direct justification for derivatives being represented as fractions since it is a historical convention by Leibniz.

But representing them as fractions does have its uses; for example, in the chain-rule for functions of one variable

$$\cfrac{\mathrm{d}y}{\mathrm{d}x}=\cfrac{\mathrm{d}y}{\mathrm{d}u}\cdot\cfrac{\mathrm{d}u}{\mathrm{d}x}$$

they 'behave' as though they were fractions so this is one justification for them being represented as fractions.

Another justification is that the reciprocal relation

$$ \cfrac{\mathrm{d}y}{\mathrm{d}x}=\cfrac{1}{\left(\cfrac{\mathrm{d}x}{\mathrm{d}y}\right )} $$

would make no sense whatsoever unless the derivatives were being represented as fractions.

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This part is not completely related to your question but I will place it here anyway:

Suppose $u$ is a function of two variables $x$ and $y$ or $u(x,y)$.

They don't need to be fractions as $u_x \equiv \cfrac{\partial u}{\partial x}$ where the subscript tells you which variable to differentiate with respect to. So the $x$ subscript has taken the role of the denominator of $\cfrac{\partial u}{\partial x}$.