When not to treat dy/dx as a fraction in single-variable calculus? While I do know that $\frac{dy}{dx}$ isn't a fraction and shouldn't be treated as such, in many situations, doing things like multiplying both sides by $dx$ and integrating, cancelling terms, doing things like $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$ works out just fine.
So I wanted to know: Are there any particular cases (in single-variable calculus) we have to look out for, where treating $\frac{dy}{dx}$ as a fraction gives incorrect answers, in particular, at an introductory level?
Note: Please provide specific instances and examples where treating $\frac{dy}{dx}$ as a fraction fails

I don't know if this qualifies as an answer or not, but the problems come in the second derivative.  While $\frac{dy}{dx}$ can be used as a fraction, the standard notation for the second derivative, $\frac{d^2y}{dx^2}$, cannot, at least in that form.  An alternative notation for the second derivative, which can be used as a fraction, is $\frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2}$, which can be derived simply from applying the quotient rule to the first derivative (which shows another place where $\frac{dy}{dx}$ can be treated as a quotient!).
Someone also mentioned partial derivatives.  You can treat those as a fraction, too, provided you use sufficient notation.  The reason why you can't split $\frac{\partial y}{\partial x}$ is because there is information about the numerator contained in the denominator.  In other words, the $\partial y$ in $\frac{\partial y}{\partial x}$ is a different term than the $\partial y$ in $\frac{\partial y}{\partial t}$.  If you used sufficient notation to distinguish these as different values (say, $\partial_ty$ and $\partial_xy$), then you can, in fact, split these fractions successfully.  If you do this, the problem specified by B. Goddard goes away.
