Differential of a Function

Summary

The differential is the linear approximation to the change in a function. In one variable, $df=f^{\prime }(x)dx$ . In two variables, $df=f_{x}dx+f_{y}dy$ .

Prerequisites

Derivatives, Partial Derivatives, Differentiability of a Function

Formula

One variable

If $f$ is differentiable,

The actual increment is $\mathrm{\Delta }f=f(x+\mathrm{\Delta }x)-f(x)=df+\epsilon$ with $\epsilon /\mathrm{\Delta }x\to 0$ as $\mathrm{\Delta }x\to 0$ .

Two variables

Conditions / Assumptions

• Differentiability at the expansion point.
• The increments $dx,dy$ are independent variables in the linear map; in applications they are small changes.

Worked Example

For $f(x)=x^2+3x-5$ , $df=(2x+3)dx$ . At $x=1$ , $dx=0.1$ , $df=0.5$ .

For $f(x,y)=x^2+3xy-4y^2$ ,

At $(1,2)$ with $dx=0.1$ , $dy=0.2$ :

Common Mistakes

• Treating $df$ as exact $\mathrm{\Delta }f$ for large increments.
• Omitting one of the partial terms in several variables.

Connections

• Tangent Plane, Chain Rules, error estimation in measurements

References

Differentials and linear approximation are covered in OpenStax Calculus.^[1]