**Leibniz's Notation For Differentiation**

In Leibniz's notation for differentiation, 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, we can write:

Using Newton's notation, we can write:

For higher derivatives, we express them as follows:

denotes the *n*th derivative of ƒ(*x*) or *y* respectively. Historically, this came from the fact that, for example, the third derivative is:

which we can loosely write as:

Now drop the parentheses and we have:

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

etc., and:

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