16.18 min read

Second Partial Derivatives

The first partial derivatives $f_x$ and $f_y$ are themselves functions of $(x, y)$ . Nothing stops us from differentiating them with respect to $x$ or $y$ again — these are the second partial derivatives.

For $f(x, y)$ , there are four second partials: differentiate $f_x$ with respect to $x$ or $y$ , and differentiate $f_y$ with respect to $x$ or $y$ . We write them as $f_{xx}$ , $f_{xy}$ , $f_{yx}$ , $f_{yy}$ .

The "mixed" partials $f_{xy}$ and $f_{yx}$ both differentiate once in $x$ and once in $y$ , just in different orders. As we will see shortly, they are usually equal.

For a function $f(\mathbf{x})$ on $\mathbb{R}^n$ , there are $n^2$ second partials $\frac{\partial^2 f}{\partial x_i \partial x_j}$ , which we can organize into a matrix.

Formal View

Definition 16.1 — Second Partial Derivatives

For

f(x, y)

with partial derivative functions

f_x

and

f_y

, the four second partial derivatives at

(x_0, y_0)

are:

f_{xx} = \frac{\partial^2 f}{\partial x^2}, \quad f_{xy} = \frac{\partial^2 f}{\partial y \partial x} = (f_x)_y, \quad f_{yx} = \frac{\partial^2 f}{\partial x \partial y} = (f_y)_x, \quad f_{yy} = \frac{\partial^2 f}{\partial y^2}

Definition 16.2 — Twice Differentiable ($D^2$)

A function

f(\mathbf{x})

\mathbb{R}^n

is twice differentiable at

\mathbf{x}_0

if all

n

first partial derivative functions exist in a neighborhood and are themselves differentiable at

\mathbf{x}_0

. The class of everywhere twice differentiable functions is

D^2

If first partials are differentiable they are certainly continuous, giving the hierarchy: $C^0 \supsetneq D^1 \supsetneq C^1 \supsetneq D^2 \supsetneq C^2$ .

Why This Matters

Second derivatives tell us about curvature — whether a function bends upward (bowl), downward (dome), or twists (saddle). This is what the second derivative test is built on.

Curvature of neural network loss landscapes (flat vs. sharp minima)
Newton's method uses second derivatives for faster convergence than gradient descent
Structural engineering: beam bending involves second derivatives of displacement
Image processing: Laplacian ( $f_{xx} + f_{yy}$ ) detects edges and blobs

Learning Resources

Second partial derivatives

Khan Academy

Clear introduction to computing second partial derivatives with examples.

9 min

Second derivatives and the Hessian matrix

Steve Brunton

Connects second partials to the Hessian and its role in optimization.

14 min

Quiz

Question 1

For $f(x,y) = x^2 y + y^3$ , what is $f_{xy}$ ?

Question 2

How many second partial derivatives does $f(x, y, z)$ have?

Question 3

What does $f_{yx}$ mean?

Question 4

If $f \in C^1$ (continuously differentiable), then $f \in D^2$ .

Common Mistakes

Confusing $f_{xy}$ and $f_{yx}$ — $f_{xy} = (f_x)_y$ means differentiate first by $x$ , then by $y$ .
Thinking that $n$ variables always give $2n$ second partials — there are $n^2$ (including mixed partials).
Assuming second partial derivative functions exist just because the function is smooth-looking.