Differentiability of a Function

Summary

Differentiability means the function admits a good local linear approximation. In one variable this is equivalent to existence of $f^{\prime }(x_0)$ . In several variables, existence of partial derivatives is weaker than differentiability.

Prerequisites

Derivatives, Partial Derivatives, Limits

Definition

One variable

$f$ is differentiable at $x_0$ if there exists $L$ such that

Then $L=f^{\prime }(x_0)$ .

Two variables

$f$ is differentiable at $(x_0,y_0)$ if there exist $A,B$ such that

with

Necessarily $A=f_x(x_0,y_0)$ and $B=f_y(x_0,y_0)$ when the partials exist.

Conditions / Assumptions

• Differentiability $⟹$ continuity.
• Continuity of partial derivatives on a neighborhood is a sufficient condition for differentiability at the point (not necessary).
• Existence of partials alone is not sufficient in several variables.

Worked Example

$f(x)=|x|$ is continuous everywhere but not differentiable at $0$ .

$f(x,y)=x^2+y^2$ has continuous partials $f_x=2x$ , $f_y=2y$ , hence is differentiable on ${\mathbb{R}}^2$ . The linear approximation at $(x_0,y_0)$ is

Common Mistakes

• Equating “partials exist” with “differentiable” in ${\mathbb{R}}^2$ .
• Forgetting that differentiability yields the tangent-plane approximation with little- $o$ error in the distance.

Connections

• Tangent Plane, Differential of a Function, Directional Derivative

References

Differentiability criteria appear in OpenStax Calculus Volume 3.^[1]