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Second-Order Partial Derivative

Definition

Let f ⁣:E⊆Rn→Rk\boldsymbol{f}\colon E\subseteq\mathbb{R}^n\to\mathbb{R}^k be a function and let xi,xjx_i,x_j be coordinate variables. A second-order partial derivative is obtained by differentiating a first-order partial derivative again:

∂2f∂xj∂xi=∂∂xj(∂f∂xi).\frac{\partial^2 \boldsymbol{f}}{\partial x_j\partial x_i} = \frac{\partial}{\partial x_j} \left( \frac{\partial \boldsymbol{f}}{\partial x_i} \right).

When i≠ji\neq j, this is called a mixed partial derivative or cross partial derivative.

Interpretation

A second-order partial derivative measures how one first-order rate of change changes as another input is varied. Mixed partial derivatives compare changes across two different coordinate directions.

Tree of partial derivatives of f(x, y): two first partials, then the four second partials obtained by differentiating each of them in x or in y.

The expression ∂2f/(∂xj∂xi)\partial^2\boldsymbol{f}/(\partial x_j\partial x_i) means differentiating first with respect to xix_i and then with respect to xjx_j. For a scalar-valued function, the n2n^2 ordered second partials form its Hessian. For a vector-valued function, each scalar component has its own Hessian. Pure partials measure curvature along an axis; mixed partials measure how a slope changes in another coordinate direction and agree in symmetric pairs when f\boldsymbol{f} is C2C^2.

The entries of Hf(x0)\boldsymbol{H}_f(\boldsymbol{x}_0) are the second-order partial derivatives at x0\boldsymbol{x}_0; drag the point to see them change.

Key Results

  • If the relevant mixed partial derivatives are continuous near a point, then their order may be exchanged.
  • Second-order partial derivatives form the entries of the Hessian matrix for scalar-valued functions.
  • Second-order information is the basic input for local curvature and extrema tests.

Examples

  • For f(x,y)=x2+xy+y2f(x,y)=x^2+xy+y^2, ∂2f∂y∂x=1\frac{\partial^2 f}{\partial y\partial x}=1 and ∂2f∂x∂y=1\frac{\partial^2 f}{\partial x\partial y}=1.
  • For f(x,y)=x2y3f(x,y)=x^2y^3, ∂2f∂x∂y=6xy2\frac{\partial^2 f}{\partial x\partial y}=6xy^2 and ∂2f∂y2=6x2y\frac{\partial^2 f}{\partial y^2}=6x^2y.

Prerequisites

Sources

Aliases: second partial derivative, second-order mixed derivative, mixed partial Keywords: mixed partial derivative, cross partial derivative Symbols: partial, real numbers, subset

Last updated: 17 October 2026 21:53