Leibniz's notation for differentiation

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See also Leibniz notation and separation of variables

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

\frac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x}

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{\mathrm{d}y}{\mathrm{d}x}

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

\frac{\mathrm{d}\left(f\left(x\right)\right)}{\mathrm{d}x} = f'\left(x\right)

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

\frac{\mathrm{d}x}{\mathrm{d}t} = \dot{x}

For higher derivatives, we express them as follows:

\frac{\mathrm{d}^n\left(f\left(x\right)\right)}{\mathrm{d}x^n} or \frac{\mathrm{d}^ny}{\mathrm{d}x^n}

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

\frac{\mathrm{d} \left(\frac{\mathrm{d} \left( \frac{\mathrm{d} \left(f\left(x\right)\right)} {\mathrm{d}x}\right)} {\mathrm{d}x}\right)} {\mathrm{d}x}

which we can loosely write as:

\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^3 \left(f\left(x\right)\right) = \frac{\mathrm{d}^3}{\left(\mathrm{d}x\right)^3} \left(f\left(x\right)\right)

Now drop the brackets and we have:

\frac{\mathrm{d}^3}{\mathrm{d}x^3}\left(f\left(x\right)\right)\ \mbox{or}\ \frac{\mathrm{d}^3y}{\mathrm{d}x^3}

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

\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}u} \cdot \frac{\mathrm{d}u}{\mathrm{d}v} \cdot \frac{\mathrm{d}v}{\mathrm{d}w} \cdot \frac{\mathrm{d}w}{\mathrm{d}x} etc.

and:

\int y \, \mathrm{d}x = \int y \frac{\mathrm{d}x}{\mathrm{d}u} \, \mathrm{d}u.
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