Partial Derivatives

The derivative of a one-variable function measures how the output responds to a change in its single input. A function of two variables has two inputs, and each can change on its own. The partial derivative isolates one of them: hold the other fixed, and the problem collapses to ordinary single-variable differentiation.

Definition

Fix $y = b$ . Then $g (x) = f (x, b)$ is a function of the one variable $x$ , and its derivative at $x = a$ measures how $f$ responds to a change in $x$ alone while $y$ stays at $b$ .

Each is an ordinary derivative in disguise: $f_{x} (a, b) = g^{'} (a)$ where $g (x) = f (x, b)$ , and $f_{y} (a, b) = G^{'} (b)$ where $G (y) = f (a, y)$ . The numerator in $f_{x}$ moves only the first slot; the numerator in $f_{y}$ moves only the second.

This gives the computational rule, using only single-variable differentiation.

Notation

Several notations coexist; all mean the same thing. For $z = f (x, y)$ ,

f_{x} = \frac{\partial f}{\partial x} = \frac{\partial z}{\partial x} = \partial_{x} f = D_{1} f, f_{y} = \frac{\partial f}{\partial y} = \frac{\partial z}{\partial y} = \partial_{y} f = D_{2} f .

The rounded $\partial$ distinguishes a partial derivative from the straight $d$ of a single-variable derivative. Unlike $d y / d x$ , the symbol $\partial z / \partial x$ is not a genuine ratio of differentials; it is one indivisible piece of notation, and the closest analogue of a ratio is the total differential.

Geometric interpretation

The graph of $z = f (x, y)$ is a surface $S$ . Fix $y = b$ : the vertical plane $y = b$ meets $S$ in a curve $C_{1}$ , namely $z = f (x, b) = g (x)$ . The slope of the tangent to $C_{1}$ at the point $P = (a, b, f (a, b))$ is exactly $g^{'} (a) = f_{x} (a, b)$ . Symmetrically, the plane $x = a$ cuts $S$ in a curve $C_{2}$ whose tangent has slope $f_{y} (a, b)$ .

The plane

y = b

slices the surface in the trace

C_{1}

; viewed head-on, the slope of its tangent at

P

is the partial derivative

f_{x} (a, b)

A heat-index table illustrates the two partials. If $I = f (T, H)$ is the heat index (perceived temperature) as a function of actual temperature $T$ and relative humidity $H$ , then $f_{T} (96, 70)$ is the rate at which perceived temperature rises per degree of actual temperature at fixed humidity, and $f_{H} (96, 70)$ is its rate of rise per percentage point of humidity at fixed temperature. Estimated from a table by symmetric difference quotients, $f_{T} (96, 70) \approx 3.75$ and $f_{H} (96, 70) \approx 0.9$ .¹

The two first partials are slopes in orthogonal directions at the same point:

f_{x}

along the

x

-axis,

f_{y}

along the

y

-axis.

Functions of more variables

Partial derivatives extend to any number of inputs: differentiate with respect to one variable, holding all others fixed. For $f (x, y, z)$ ,

f_{z} (x, y, z) = h \to 0 lim \frac{f ( x , y , z + h ) - f ( x , y , z )}{h},

and $f_{x}, f_{y}$ are defined the same way. Nothing about the freeze-and-differentiate rule changes; there are simply more variables to hold constant.

Higher-order partial derivatives

Because $f_{x}$ and $f_{y}$ are themselves functions of $x$ and $y$ , they can be differentiated again. This produces four second partial derivatives.

Symbol	Meaning	Also written
$f_{xx}$	differentiate by $x$ , then $x$	$\partial^{2} f / \partial x^{2}$
$f_{x y}$	differentiate by $x$ , then $y$	$\partial^{2} f / \partial y \partial x$
$f_{y x}$	differentiate by $y$ , then $x$	$\partial^{2} f / \partial x \partial y$
$f_{y y}$	differentiate by $y$ , then $y$	$\partial^{2} f / \partial y^{2}$

The two with distinct subscripts, $f_{x y}$ and $f_{y x}$ , are the mixed partials. Note the order convention: $f_{x y} = (f_{x})_{y}$ means differentiate by $x$ first, then $y$ — the subscript order and the $\partial$ -notation order run opposite to each other.

Clairaut's Theorem

When the mixed partials are continuous — the case for every polynomial, rational, exponential, trigonometric, or logarithmic combination away from singularities — the order of differentiation does not matter. This cuts the four second partials effectively to three, and the saving compounds: for third-order and higher derivatives, any two orderings with the same tally of $x$ 's and $y$ 's agree, so $f_{x y y} = f_{y x y} = f_{y y x}$ whenever these are continuous.

Clairaut's Theorem: differentiating by

x

then

y

lands on the same function as differentiating by

y

then

x

, when the results are continuous.

Partial differential equations

Second partials appear throughout physics as partial differential equations, relations a function must satisfy at every point. Two carry most of classical field theory.

Laplace's equation $f_{xx} + f_{y y} = 0$ . Its solutions, the harmonic functions, describe steady-state temperature, electrostatic potential in charge-free regions, and incompressible flow.
The wave equation $u_{tt} = c^{2} u_{xx}$ . It governs a vibrating string, a drumhead, sound, and light, relating the acceleration $u_{tt}$ of a displacement to its spatial curvature $u_{xx}$ through the wave speed $c$ .

A solution

u (x, t) = F (x - c t)

of the wave equation is a fixed profile sliding to the right at speed

c

; the two snapshots are at times

t_{1} < t_{2}

The sign of a second partial reports curvature in one direction: $f_{xx} > 0$ means the $x$ -trace bends upward (concave up), $f_{xx} < 0$ means it bends down. Combining the two directions is how the Second Derivatives Test classifies a critical point as a peak, a valley, or a saddle.

The sign of

f_{xx}

is the concavity of the

x

-trace:

f_{xx} > 0

bends the trace upward,

f_{xx} < 0

bends it downward.

Stewart, §14.3 — the heat-index example, with $f_{T}$ and $f_{H}$ estimated from a table of values by averaging forward and backward difference quotients. ↩