Higher-Order Partial Derivatives
Differentiating the partials again — notation, equality, the Hessian, and PDE verification
1. The Bookkeeping: Second- and Higher-Order Partials
1.1 Four Second-Order Partials in Two Variables
Given a function $f(x,y)$, each first-order partial ($f_x$ and $f_y$) is itself a function of $x$ and $y$. We can therefore differentiate each of them again with respect to either variable. This gives us four second-order partial derivatives:
| Name | Subscript Notation | Leibniz Notation | Meaning |
|---|---|---|---|
| Pure (in $x$) | $f_{xx}$ | $\dfrac{\partial^2 f}{\partial x^2}$ | Differentiate w.r.t. $x$, then $x$ again |
| Mixed | $f_{xy}$ | $\dfrac{\partial^2 f}{\partial y\,\partial x}$ | Differentiate w.r.t. $x$ first, then $y$ |
| Mixed | $f_{yx}$ | $\dfrac{\partial^2 f}{\partial x\,\partial y}$ | Differentiate w.r.t. $y$ first, then $x$ |
| Pure (in $y$) | $f_{yy}$ | $\dfrac{\partial^2 f}{\partial y^2}$ | Differentiate w.r.t. $y$, then $y$ again |
The Notation Trap — Read This Carefully!
The two notation traditions read the order of differentiation in opposite directions:
- Subscript notation reads left → right: $f_{xy}$ means "first differentiate with respect to $x$, then with respect to $y$."
- Leibniz notation reads right → left (from the denominator): $\dfrac{\partial^2 f}{\partial y\,\partial x}$ means "first differentiate with respect to $x$ (rightmost), then with respect to $y$ (leftmost)."
So $f_{xy}$ and $\dfrac{\partial^2 f}{\partial y\,\partial x}$ denote the same operation. The traditions read in opposite directions — name this trap and you will never fall into it.