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?