When is Leibniz' notation for derivatives useful?

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?

• I think it suits best on the application of integration techniques such as substitution of integration by parts, for example: $y=x^2, dy=2xdx$. – Supermanco Oct 16 '14 at 12:22
• I often find it better to just convert $\int f(u) dx$ to $\int \frac{u'}{u'}\times F'(u) dx = \frac{1}{u'}\int (F(u))'dx$ which is easy to find. – Frank Vel Oct 16 '14 at 12:24
• There are some explanations which I don't actually remember, but as a matter of notations, I think people should use what they feel comfortable with. – Supermanco Oct 16 '14 at 12:27
• $\frac{d}{dx}$ is more useful when the expressions/functions become complicated, the prime $'$ can easily be missed. – gammatester Oct 16 '14 at 12:29

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.
• It's the only one that occurs to me. Also, Newton's notation $\dot y$ deserves a mention. It is quite common in physics, and usually denotes a derivative with respect to time (or any particular parameter which may have such an interpretation). In fact, it is not wholly uncommon to see two or three of these different notations used at the same time! For instance, a function $f_\alpha(x,t)$ depending on 'time' t, 'space' x and a parameter $\alpha$ might elicit the use of all three notations at the same time. – Jonas Dahlbæk Oct 16 '14 at 15:11