Question

What is Leibniz Notation ?

Answer 1

In the context of Calculus 1-3, it's just a more explicit and convenient way or notating derivatives.

`f(x) = (dy)/(dx) = (df)/(dx)` if `y = f(x)`

It makes it clearer why the chain rule works how it does.

`color(green)((df(u))/(dx) = (df)/(du)*(du)/(dx))`

which clearly cancels. So if `f(x) = e^(lnx)`, then `f(u)` could be `e^u`, with `u = lnx`. Then:

`(df)/(dx) = e^(lnx) * 1/x`

where

`(df)/(du) = e^u = e^(lnx)`

`(du)/(dx) = d/(dx) [lnx] = 1/x`

In regular function notation, you just see it as:
`color(green)(f'(u(x)) = f'(u)u'(x))`

which doesn't say much as to why it's written that way.

It's also easy to manipulate. If you write out:

`(du)/(dx) = x`

then you can just treat it like a fraction and say it's also:

`du = xdx`

and then integrate it to get:

`intdu = intxdx -> u = x^2/2 + C`

It's also somewhat easier to write:

`(d^6f)/(dx^6)`

rather than

`f''''''(x)`

if you're, let's say, doing the Taylor series.

`sum_(n=0)^oo f^n(a)/(n!)(x-a)^n`

Answer 2

Warning: history follows.

Explanation:

The notation `dy/dx` was proposed as a substitute for `(Delta y)/(Delta x)` used in certain situations.

Mathematicians used the idea of an infinitesimal quantity -- an infinitely small quantity -- for many years (centuries).
In fact even into the 1970s, we sometimes referred to "the Infinitesimal Calculus" or "the Calculus of Infinitesimals". (At Clemson University in 1975, I had professors who used these phrases.)

The idea of something being infinitely small seems to be an intuitively clear idea, but it turns out to be terribly difficult to make it precise.
The attempt to use a notion of " a number smaller than every positive number, but not equal to zero" caused difficulty and led to Bishop George Berkeley's attack on the careless thinking in analysis (calculus) called The Analyst

In any event, to return to the question about notation:
Leibniz used
`(Delta y)/(Delta x)` for "finite changes" in the values of the variables

and he used:
`dy/dx` for "infinitesimal changes" (Used in ways similar to what we now call the limits as `Deltax rarr 0`).

This notation has been extended to include the operator `d/dx()` to mean "the derivative with respect to `x` of ()" (The phrase "with respect to `x`" indicates that we are to consider `x` to be the independent variable.)

Another notation you may see (or may have already seen) for the derivative w.r.t. `x` is `D_x()`. The Leibnizian `d/dx()` means the same thing.