Higher-Order Partial Derivative¶

Definition¶

A higher-order partial derivative is obtained by differentiating a partial derivative again with respect to one or more variables. For example, for $f\colon A\subseteq\mathbb{R}^n\to\mathbb{R}$ $R$ ,

f_{x_i x_j x_k}(\boldsymbol{a})= \frac{\partial}{\partial x_i} \left( \frac{\partial}{\partial x_j} \left( \frac{\partial f}{\partial x_k} \right) \right)(\boldsymbol{a})

is a third-order partial derivative, when all indicated derivatives exist. In this notation the derivative written closest to $f$ (here $\partial/\partial x_k$ ) is taken first; many texts instead read subscripts from left to right, so that $f_{xy}=(f_x)_y\text{.}$ For $C^3$ functions both readings give the same value.

Interpretation¶

Higher-order partial derivatives measure how earlier rates of change themselves change as one moves in chosen coordinate directions. They play the role that $f''$ and $f'''$ play for a function of one variable (higher-order derivative).

Tree of partial derivatives of f(x, y): two first-order partials, four second-order partials and eight third-order partials, each obtained by differentiating the level above in x or in y.

Each arrow differentiates once more, in $x$ to the left and in $y$ to the right, so the rows hold the $2$ , $4$ and $8$ partial derivatives of order $1$ , $2$ and $3$ . In $n$ variables there are $n^k$ ordered partials of order $k$ . As in the Definition, the new variable is written on the left, so $f_{yx}$ is obtained from $f_x$ .

Texts that read subscripts from left to right write these labels in reverse order. For $C^k$ functions the two readings agree, because then the order of differentiation does not matter (higher-order symmetry of partials). The eight third-order partials of a $C^3$ function of $(x,y)$ then take at most four values, $f_{xxx}$ , $f_{xxy}$ , $f_{xyy}$ and $f_{yyy}$ . Without continuity the order can matter: for $f(x,y)=xy(x^2-y^2)/(x^2+y^2)$ with $f(0,0)=0$ $0$ , the two mixed second partials at the origin are $1$ and $-1$ .

The second-order partials of a scalar function form its Hessian matrix, the third-order ones fill the third-order derivative tensor, and both supply the higher terms of multivariate Taylor approximations.

A common slip is to read $f_{xx}$ as $(f_x)^2$ . For $f(x,y)=x^2y^3$ from the Examples, $f_x=2xy^3$ $2 x y^{3}$ , so $f_{xx}=2y^3$ $2 y^{3}$ , while $(f_x)^2=4x^2y^6$ $4 x^{2} y^{6}$ .

Key Results¶

Second-order partial derivatives form the Hessian matrix when they exist.
Third-order partial derivatives can be arranged in a three-dimensional array.
Continuity assumptions are usually needed before changing the order of differentiation.

Examples¶

For $f(x,y)=x^2y^3$ $x^{2} y^{3}$ , $f_{xy}=6xy^2$ and $f_{xyy}=12xy$ $12 x y$ .
For $f(x,y,z)=xyz^2$ $x y z^{2}$ , $f_{xyz}=2z$ and $f_{xzz}=2y$ $2 y$ .

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 2, Lecture 1

Aliases: higher partial derivative Keywords: third-order partial derivative, iterated partial derivative Symbols: partial, real numbers, subset

Last updated: 20 October 2026 03:02