So Lagrange's $y'$ and Leibniz' $\frac{d}{dx}y$ seems to be the two most common notations for differentiation, but it seems puzzling to me that there are two notations for this. I've been taught Lagrange's notation, and haven't really used Lebniz' notation. In most of the cases it seems to me like Lagrange's really is the best. But I'm pretty sure that since Leibniz' is so widespread and common, there must be some use for it. For instance I find it easier to write:
$$
\begin{align}
\sin'x&= \cos x &&\text{sine rule}\\[0.5em]
(uv)' &= uv'+u'v &&\text{multiplication rule}\\[0.5em]
(u(v))' &= u'(v)\times v' &&\text{chain rule}
\end{align}
$$
than:
$$
\begin{align}
\frac{d}{dx}\sin x&= \cos x &&\text{sine rule}\\[1em]
\frac{d(uv)}{dx} &= \frac{du}{dx}v+\frac{dv}{dx}u &&\text{multiplication rule}\\[1em]
\frac{d(u(v))}{dx} &= \frac{du}{dv}\times\frac{dv}{dx} &&\text{chain rule}
\end{align}
$$
This are just some cases where I find it a lot easier to use Lagrange's notation, so when is Leibniz' notation the best?
Answer
In many concrete situations, you may have some specific expression on hand. Consider the density of the normal distribution, $f(x,\mu,\sigma)=(2\pi\sigma^2)^{-1/2}\exp(-|x-\mu|^2/2\sigma^2)$. If you simply write
$$
\left((2\pi\sigma^2)^{-1/2}\exp(-|x-\mu|^2/2\sigma^2)\right)',
$$
it will of course be completely unclear what you mean. If you want, you can define $f_{\mu,\sigma}(x)=f(x,\mu,\sigma)$, and write $f_{\mu,\sigma}'$, but it quickly becomes cumbersome to define new functions every time you wish to take a derivative. It is much simpler to simply write, say
$$
\left.\frac{d}{dx}\right|_{x=1}(2\pi\sigma^2)^{-1/2}\exp(-|x-\mu|^2/2\sigma^2)
$$
in place of first defining $f_{\mu,\sigma}$ and then writing $f_{\mu,\sigma}'(1)$.
This is much the same reason that it is useful to have the $dx$ appearing somewhere when you integrate, instead of a more general $d\mu$, where $\mu$ is a measure. It is notationally less pretty, but much more flexible.
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