I have often read that it is less precise to state the chain rule using Leibniz's notation as opposed to Lagrange's notation. I don't understand this claim because it seems to me that the two statements are identical, and, if anything, Leibniz's formulation is 'cleaner'. If $y=f(u)$, and $u=g(x)$ (and $f$ is differentiable at $u$, and $g$ differentiable at $x$), then
$$ \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} $$
Equally, we could write
$$ (f \circ g)'(x) = (f' \circ g)(x)g'(x) $$
It seems simple to show that these two statements mean the same thing:
$$ \frac{dy}{dx}\equiv(f \circ g)'(x) $$
because they both refer to the change in $y$ (where $y=(f \circ g)(x)$) divided by the change in $x$. More formally, they both refer to the same limit expression: $$ \lim_{\Delta x \to 0}\frac{f(g(x+\Delta x))-f(g(x))}{\Delta x} $$ We can also verify that $\frac{dy}{du}\equiv(f' \circ g)(x)$ and that $\frac{du}{dx} \equiv g'(x)$. However, the fact that the two notations refer to the same statement does not mean that we can't prefer one over the other. So what dangers are there in using Leibniz's notation for the chain rule? I have always found it to be simpler and more intuitive, but perhaps there are things which make Leibniz's notation misleading.
It is possibly because beginners may think there is "cancellation" in $$ \frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}\; $$ while there is really no.
As long as interpreted correctly, both sets of notations are "precise".
See also excellent discussions of notations in this post:
https://mathoverflow.net/a/366118/164038