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Basics of Differentials

Differentials capture the linear approximation of a function near a point, turning derivative information into a small-increment language. This section complements the derivative basics and focuses on definitions, properties, and quick applications.

Definition

If $f(x)$ is differentiable at $x$ , its differential at that point is

df = f'(x)\,dx,

where $df$ is the change in $f$ and $dx$ is a small change in the independent variable.

$dx$ (“dee x”): a tiny increment of $x$ , treated as a symbolic small change.
$df$ (“dee f”): the corresponding small change in $f(x)$ , linked by $df = f'(x)\,dx$ .

Form of the Differential

df = f'(x)\,dx

To compute a differential, find the derivative first, then multiply by $dx$ .

Geometric meaning

On the graph of $y = f(x)$ , $df$ is the vertical change along the tangent line produced by a horizontal step $dx$ . The pair $(dx, df)$ is the linearized version of the curve near the point.

Properties

Linearity

d(af + bg) = a \cdot df + b \cdot dg

Product rule

d(uv) = u \cdot dv + v \cdot du

Quotient rule

d\!\left(\frac{u}{v}\right) = \frac{v \cdot du - u \cdot dv}{v^2}

Quick uses

Linear approximation: $f(x + \Delta x) \approx f(x) + df$ for small $\Delta x$ .
Error estimation: bound output error by controlling $|dx|$ .
Change of variables: rewrite expressions via $dx$ and $df$ when substituting.

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$dx$	Mathematical symbol	“dee x”	A small increment of the independent variable
$df$	Mathematical symbol	“dee f”	A small increment of the function value

Physical Meaning of the Derivative

History of the Derivative

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