Leibniz's Notation - Leibniz's Notation For Differentiation

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 nth 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:

\left(\frac{d}{dx}\right)^3 \bigl(f(x)\bigr) =
\frac{d^3}{\left(dx\right)^3} \bigl(f(x)\bigr)\,.

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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