The differential

The differential turns the idea of an “infinitesimal variation along the tangent” into a self-standing object: it is the tool with which one approximates $f$ near a point and with which the notation $\frac{df}{dx}$ acquires a precise meaning.

Definition — The differential

The differential of $f$ at the point $x$ with increment $dx$ is: $df = f'(x)\cdot dx.$ Here $dx$ is an infinitesimal increment of the independent variable (it is the same $\Delta x$ in the limit $\Delta x\to 0$, but written with a “d” to indicate that we have already passed to the limit). The differential $df$ is the approximate variation of $f$ along the tangent.

Remark — Why the notation $\frac{df}{dx}$ makes sense

If one divides the differential $df = f'(x)\cdot dx$ by $dx$, one obtains $\frac{df}{dx} = f'(x)$. This is why the derivative can be “read” as a ratio between the differential of the function and the differential of the variable: it is not an abuse of notation but a genuine property of differentials.

Remark — The differential as an approximation

Geometrically, $df$ is the vertical displacement along the tangent line when one moves by $dx$ horizontally. The true function moves by $\Delta f = f(x+dx)-f(x)$, slightly different from $df$ (the difference is an “error” of order $dx^2$). But for small $dx$, $df\approx\Delta f$ is an excellent approximation.

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Topics: Derivatives
Concepts: Derivative · Differential · Leibniz notation
Skills: Estimating