Leibniz's notation for differentiation

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In Leibniz's notation for differentiation, the derivative of the function f(x) is written:

[\frac]

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

[y = f\left(x\right)]

then we can write the derivative as:

[\frac]

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

[\frac = f'\left(x\right)]

[\frac = \dot]

For higher derivatives, we express them as follows:

[\frac] or [\frac]

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

[\frac \right)} \right)} ]

which we can loosely write as:

[\left(\frac\right)^3 \left(f\left(x\right)\right) =\frac \left(f\left(x\right)\right)]

Now drop the brackets and we have:

[\frac\left(f\left(x\right)\right)\ \mbox\ \frac]

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

[\frac = \frac \cdot \frac \cdot \frac \cdot \frac] etc.

and:

[\int y \, dx = \int y \frac \, du.]

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