Partial Derivatives

Every scientist knows the reflex: change one thing at a time. If you water one tomato plant more and move it into the sun, and it thrives, you have learned nothing — was it the water or the light? So you clamp everything else and wiggle a single knob. Partial derivatives are that reflex, made into calculus.

A function of one variable has one knob, and its derivative measures the one available rate of change. But a surface z = f(x, y) is a landscape: stand on a hillside and the slope depends on which way you face. Walking due east (the x-direction) might take you gently uphill while due north (the y-direction) drops you off a cliff. "The slope of the hill" is not one number — so we start with the two simplest directions and measure each with the other clamped.

The partial derivative of f with respect to x freezes y at whatever value it has, wiggles x alone, and takes the usual limit:

\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x + h,\, y) - f(x,\, y)}{h}.

Look closely at the numerator: y appears in both terms, unchanged. Only x moved. The curly \partial — pronounced "partial", or affectionately "del" or "dabba" depending on whose lecture you're in — is just a d with a warning label: other variables exist, and I am holding them still. Likewise \partial f / \partial y freezes x and wiggles y.

The notation zoo. All of these mean exactly the same thing, and you will meet all of them:

\frac{\partial f}{\partial x} \;=\; f_x \;=\; \partial_x f \;=\; D_x f,

with f_x(a, b) or \left.\tfrac{\partial f}{\partial x}\right|_{(a,b)} for the value at a specific point. Leibniz's fraction-style \partial f/\partial x is best when you need to say which variable moved; the subscript f_x is best when partials start piling up (f_{xy} is far kinder than \tfrac{\partial^2 f}{\partial y\, \partial x}). Use whichever is clearest — they are one concept in four costumes.

The whole technique: treat the other letter as a number

Here is the good news: you already know how to compute every partial derivative. No new rules. To find f_x, read y as if it were the number 7 and differentiate with the one-variable toolkit you own. Watch the discipline in action on

f(x, y) = x^2 y^3 + 3x - y.

Computing f_x — mentally replace every y with "some constant":

f_x = 2xy^3 + 3.

Computing f_y — now the roles swap, and every x is the frozen one:

f_y = 3x^2 y^2 - 1.

Notice how the same term plays two different parts: 3x was alive in the x-derivative and dead weight in the y-derivative. Before differentiating any term, ask one question: does the moving variable actually appear here? If not, the term is a constant and its derivative is zero — no matter how many other letters it contains.

When the frozen variable hides inside

Freezing y does not mean y disappears — it means y behaves like a fixed coefficient wherever it sits, even inside a product or an exponent. Take

g(x, y) = x\, e^{xy}.

For g_x, both factors contain x, so the product rule applies — exactly as in one variable:

g_x = \underbrace{1 \cdot e^{xy}}_{(x)'\, e^{xy}} + \underbrace{x \cdot e^{xy} \cdot y}_{x\,(e^{xy})'} = e^{xy}(1 + xy).

The second piece used the chain rule: the derivative of e^{xy} in x is e^{xy} times the derivative of the exponent xy, and with y frozen that inner derivative is just y.

For g_y, the leading x is a mere constant factor — no product rule needed this time:

g_y = x \cdot e^{xy} \cdot x = x^2 e^{xy}.

Same function, two quite different-looking partials. That asymmetry is normal: the landscape genuinely is steeper in some directions than others.

Three traps catch nearly every newcomer:

Both partials, then partials of partials

Time for the full workout. Take

f(x, y) = x^2 y + \sin(xy).

Step 1 — \partial f/\partial x, treating y as constant. The first term x^2 y has constant factor y, so it differentiates to 2xy. For \sin(xy) the chain rule gives \cos(xy) times the derivative of the inside xy with respect to x, which is y:

f_x = 2xy + y\cos(xy).

Step 2 — \partial f/\partial y, treating x as constant. Now x^2 y has constant factor x^2, differentiating to x^2. For \sin(xy) the inside xy differentiates to x in y:

f_y = x^2 + x\cos(xy).

Step 3 — go to second order. A partial is itself a function of x and y, so we can differentiate again — four ways: f_{xx}, f_{yy}, and the two mixed partials. Here f_{xy} means "first x, then y": differentiate f_x = 2xy + y\cos(xy) with respect to y. The first term gives 2x. The second is a product y \cdot \cos(xy): the product rule gives \cos(xy) + y \cdot (-\sin(xy)) \cdot x, so

f_{xy} = 2x + \cos(xy) - xy\sin(xy).

Step 4 — the other order. Now f_{yx}: differentiate f_y = x^2 + x\cos(xy) with respect to x. The first term gives 2x. The second is x \cdot \cos(xy): the product rule gives \cos(xy) + x \cdot (-\sin(xy)) \cdot y, so

f_{yx} = 2x + \cos(xy) - xy\sin(xy).

Step 5 — compare. Two genuinely different computations — different starting expressions, different products — and yet the answers are letter-for-letter identical:

f_{xy} = f_{yx} = 2x + \cos(xy) - xy\sin(xy).

That was no accident. For any function whose second partials are continuous, the order of differentiation does not matter — a quiet, beautiful symmetry called Clairaut's theorem (also Schwarz's theorem; good theorems collect names).

The slope of a slice

Geometrically, fixing y = b slices the surface z = f(x, y) with the vertical plane y = b — a hot wire through the landscape. The cut edge is a one-variable curve z = f(x, b), and f_x(a, b) is nothing more exotic than its ordinary slope at x = a: the pitch you'd feel walking due east through the point (a, b).

Below is that slice for our function f(x, y) = x^2 y + \sin(xy). Slide b and the whole slice-curve morphs — each height of the cutting plane reveals a different profile of the hill. Slide a and the tangent line rolls along the curve; its slope at every moment is exactly f_x(a, b) = 2ab + b\cos(ab). Try b = 0: the slice flattens to a dead-level line and f_x is zero everywhere along it. Then push b negative and watch the parabola flip downhill.

That flat slice is a cut through a whole landscape. Here is the same z = f(x, y) as a rotatable surface — drag it to view the hill from any angle. The shaded sheet is the cutting plane y = b; nudge the very same b slider above and watch it sweep through the surface, while the bold curve where it bites is exactly the profile drawn flat below. The marked dot sits at (a, b), where f_x measures the surface's east–west (the x-direction) steepness.

Clairaut's theorem has a hypothesis — continuity of the second partials — and it earns its keep. The textbook counterexample is

f(x, y) = \frac{xy(x^2 - y^2)}{x^2 + y^2} \quad (f(0,0) = 0).

Away from the origin it is a perfectly ordinary rational function, but a careful computation of the mixed partials at the origin gives

f_{xy}(0, 0) = -1 \ne 1 = f_{yx}(0, 0).

The two orders genuinely disagree. The escape clause is precisely Clairaut's hypothesis: here f_{xy} is discontinuous at the origin, so the theorem never applied. The moral is the recurring one of multivariable calculus — the symmetric, well-behaved answer is the rule, but only when a continuity condition holds, and the pathologies hide exactly where it fails.

Here is why Clairaut's symmetry deserves a small gasp. f_{xy} asks: as I move north, how does the eastward slope change? And f_{yx} asks the mirror question: as I move east, how does the northward slope change? On the face of it these are different surveys of the terrain — one tracks an east-facing clinometer while walking north, the other a north-facing clinometer while walking east. Clairaut says every smooth landscape returns the same number either way: both measure the one intrinsic twist of the surface at that point, the amount by which it wrings like a wet towel.

The symmetry quietly runs whole subjects. In thermodynamics it becomes the Maxwell relations — swap the order of two partials of the energy and equations connecting entropy, pressure, volume and temperature fall out for free. In economics, where "marginal anything" is a partial derivative in disguise (marginal cost = ∂cost/∂quantity, holding wages, rent and everything else fixed), Clairaut says the marginal effect of labour on the productivity of capital equals the marginal effect of capital on the productivity of labour. One theorem, many costumes — much like the \partial itself.

A partial derivative you can feel: the ideal gas

Real formulas almost never have one input, which is why \partial appears on every whiteboard in physics, chemistry, engineering and economics. Take the ideal gas law, solved for pressure:

P(V, T) = \frac{nRT}{V},

with n (amount of gas) and R (the gas constant) fixed numbers. Two knobs, two experiments, two partials:

\frac{\partial P}{\partial T} = \frac{nR}{V}, \qquad \frac{\partial P}{\partial V} = -\frac{nRT}{V^2}.

Each line is a lab procedure. \partial P/\partial T: weld the container shut (volume clamped) and warm it — pressure climbs at the steady rate nR/V per degree. That is why a sealed aerosol can in a bonfire is a bad idea: fixed V, rising T, and the partial derivative does the rest. \partial P/\partial V: hold the temperature steady and squeeze — the minus sign says pressure moves opposite to volume, and the V^2 downstairs says the push-back stiffens sharply as the gas gets small, which is exactly what your thumb feels over a bicycle pump.

The habit to build: whenever you compute a partial of a real formula, say out loud what was held fixed. "Pressure rises with temperature at constant volume" is physics; "pressure rises with temperature" alone is ambiguous, because heating a gas that is also free to expand (a hot-air balloon, say) needn't raise its pressure at all. The \partial is precisely the bookkeeping of that at-constant-what clause. The same discipline prices mortgages (\partial(\text{payment})/\partial(\text{rate}) at fixed loan and term — steep, as anyone refinancing knows) and runs weather models, where temperature depends on three space coordinates and time, and the forecast is stitched together from partials in each.

From here the story opens out: the two partials assemble into a single vector, the gradient, and slopes in every direction — not just east and north — follow from directional derivatives.

See it explained