Leibniz's notation for differentiation

A notation for describing derivatives.

The derivative of the function f(x) is written:

If we have a variable representing a function, for example if we set:

then we can write the derivative as:

Using Lagrange's notation for differentiation, we can write:

Using the Newton's notation for differentiation, we can write:

For higher derivatives, we express them as follows:

or

denotes the nth derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is:

which we can loosely write as:

Now drop the brackets and we have:

or

The chain rule and integration by substitution rules are especially easy to express here, because the "d" terms appear to cancel:

etc.

and: