A lot of the time, in maths, especially when I'm trying to remember a formula, I'm taught to remember it in a way that is not notationally correct, but produces the right result.
e.g. when learning the chain rule, I learnt that, if $y=f(u), u=g(x),$ then $$\frac{dy}{dx}=\require{cancel} \frac{dy}{\cancel{du}}\cdot \frac{\cancel{du}}{dx}.$$ Now, I know that this 'cancelling' of $du$s is wrong as $\frac{dy}{dx}$ is not a quotient.
Another example is the cross product of two vectors in $\mathbb{R^3}$: $$\vec u \times \vec v \equiv\begin{vmatrix} \hat \imath & \hat \jmath &\hat k \\ \vec u_1 & \vec u_2 & \vec u_3\\ \vec v_1 & \vec v_2 & \vec v_3 \\ \end{vmatrix}.$$
Finally, a trap that I certainly fell into when first learning integration by substitution:
Suppose we want to evaluate $I=\int(x+3)^{20}dx$.
Let $u=x+3\implies \overbrace{\frac{du}{dx}=1 \iff du=dx}^{\text{we're treating} \ du \ \text{and } \ dx \ \text{as fractions here, which they are} \ \bf{not}} $
As you can see, a lot of this notation is misleading, and gives the wrong idea about its topic, but it returns the correct 'result'.
I want to try and find as many examples as possible, of 'misleading' notation, so that I don't inadvertently get the wrong idea in my head about what the statement/formula actually means.
Thanks a lot!
Another prominent example is the way we solve ODEs with separable variables:
$$y'=f(x)g(y)\\\frac{dy}{dx} = f(x)g(y)\\ dy=f(x)g(y)dx\\\frac{dy}{g(y)} = f(x)dx\\ \int\frac{dy}{g(y)} = \int f(x) dx.$$