Leibniz notation

Alongside the notation $f'(x)$ due to Lagrange, one very often uses the notation introduced by Leibniz, which makes transparent the link between the derivative and the passage to the limit of the difference quotient.

Remark — From $\Delta$ to $d$

In Leibniz notation, the passage to the limit is indicated by replacing the $\Delta$ (finite increments) with the $d$ (differentials): $\frac{\Delta f}{\Delta x} \xrightarrow{\;\Delta x\to 0\;} \frac{df}{dx}.$ The symbol $\frac{df}{dx}$ is read ”$df$ over $dx$”, and is not a fraction in the usual algebraic sense: it is a single symbol denoting the derivative. Nevertheless, in many contexts it behaves like a fraction: in the chain rule ($\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$, the $du$ “cancel out”) and in the separation of variables of differential equations ($\frac{dy}{dx}=f(x)g(y) \implies \frac{dy}{g(y)}=f(x)\,dx$).

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Topics: Derivatives
Concepts: Derivative · Leibniz notation
People: Gottfried Leibniz