Partial derivatives

The formula

The partial derivative of a function f(x, y, ...) with respect to one variable is just the ordinary derivative taken while every other variable is frozen:

∂f/∂x = lim (h → 0) [ f(x + h, y) − f(x, y) ] / h

Reading: change x by a hair, leave y alone, and see how fast f responds per unit of that change. The curly ∂ is the flag that says "other variables are being held constant."

Stack all the first partials of a scalar function into a vector and you get the gradient:

∇f = ( ∂f/∂x , ∂f/∂y , ∂f/∂z )

Reading: the gradient points in the direction f climbs fastest, and its length is how steep that climb is.

Take a partial of a partial and you get a second-order or mixed partial:

∂²f/∂x∂y = ∂/∂x ( ∂f/∂y )

Reading: differentiate with respect to y first, then x. The order is written right-to-left in Leibniz notation and, under the right conditions, doesn't matter (more on that below).

The one every metrologist carries around is the law of propagation of uncertainty — the reason engineers meet partials at all:

u_c²(y) = Σ ( ∂f/∂x_i )² · u²(x_i)      (inputs uncorrelated)

Reading: the combined variance of the output is a sum of squares, each term being one input's uncertainty scaled by how sensitive the output is to that input. Those ∂f/∂x_i terms have a name in the trade: sensitivity coefficients, usually written c_i.

Where you meet it

Uncertainty budgets on the calibration bench. You build a measurement model y = f(x_1, x_2, ...) — say power from P = V²/R. The GUM (JCGM 100) tells you the combined uncertainty is a root-sum-square of each input's uncertainty times its sensitivity coefficient ∂f/∂x_i. Take V = 12 V, R = 100 Ω, so P = 1.44 W. The partials are ∂P/∂V = 2V/R = 0.24 W/V and ∂P/∂R = −V²/R² = −0.0144 W/Ω. With u(V) = 0.1 V and u(R) = 1 Ω, the combined standard uncertainty is u_c = sqrt( (0.24·0.1)² + (0.0144·1)² ) = 0.028 W. The voltage term dominates — the partials told you where to spend money on a better meter.

Any model with more than one knob at a design review. A link budget, a nozzle sizing spreadsheet, a beam-deflection calc. When someone asks "what if the mass is 5% heavy?" they're asking for a partial derivative times a perturbation. The sensitivity of range to burnout velocity, of deflection to wall thickness, of gain to spacing — all partials evaluated at the nominal design point.

Control and estimation on the test stand. Linearizing a nonlinear plant around an operating point is filling out a Jacobian — a matrix of partials, ∂f_i/∂x_j. Every extended Kalman filter, every small-signal model, every gain-scheduled controller is built on that matrix.

Thermodynamics and field problems. Heat, diffusion, and wave equations are written in partials because temperature and pressure depend on both position and time. ∂T/∂t = α ∂²T/∂x² is the heat equation, and it says nothing until you know which variable is held where.

How it works

The whole idea is one-variable-at-a-time. Freeze everything but one input and you're back to freshman calculus. That simplicity is also the trap: a partial derivative only tells you about motion along one axis. It says nothing about diagonal moves or about what happens when two inputs change together. In an uncertainty budget that distinction is the difference between the uncorrelated formula above and the full one, which adds cross terms 2 · (∂f/∂x_i)(∂f/∂x_j) · u(x_i, x_j) for correlated inputs. Ignore correlation when it's real and your budget is wrong — sometimes badly.

The sensitivity coefficient carries units and a sign, and both matter. ∂P/∂R above is negative: raise the resistance, drop the power. When you square it for the variance, the sign vanishes — but if you're reasoning about which way an error pushes the result, keep it. A common bench mistake is treating a sensitivity coefficient as dimensionless. It isn't. ∂P/∂V is in watts per volt; it converts a volt of input uncertainty into watts of output uncertainty. Get the units right or the RSS is nonsense.

Mixed partials usually commute — until they don't. Clairaut's theorem (also called Schwarz's theorem) says that if the second partials are continuous near a point, then ∂²f/∂x∂y = ∂²f/∂y∂x. Order doesn't matter, which is why you rarely think about it. But continuity is a real condition, not boilerplate. The textbook counterexample is f = xy(x²−y²)/(x²+y²) with f(0,0)=0. Work the limit definition carefully at the origin and you get ∂²f/∂x∂y = +1 but ∂²f/∂y∂x = −1. The mixed partials disagree because they aren't continuous there. For smooth engineering models this never bites, but it's a reminder that "obvious" symmetries rest on assumptions.

The deeper limit is linearity. A partial derivative is a local, first-order picture: it's the slope of a tangent plane, valid for small nudges. The GUM propagation formula is a truncated Taylor series — it assumes the model is roughly linear over the span of your input uncertainties. When the model curves hard, or the uncertainties are large, that assumption fails and you reach for second-order terms or a Monte Carlo run (the GUM Supplement 1 route) instead. The partial is exact; the budget built on it is an approximation, and you should know which regime you're in.

History

The curly ∂ did not spring from Leibniz alongside the ordinary d. It arrived later, out of the tangle of eighteenth-century work on functions of several variables, where mathematicians kept needing to say "the derivative with respect to this one, holding the rest fixed" and had no clean symbol for it. The mark itself first shows up in 1770, in a memoir on partial differential equations by Antoine-Nicolas de Condorcet, who used ∂ to distinguish a partial difference from a total one ^[1]^[2].

The person who gave it the modern fractional shape was Adrien-Marie Legendre. In 1786, in work on the calculus of variations, he wrote partials in the form ∂u/∂x that any engineer would recognize today ^[1]^[2]. But Legendre didn't stick with it — he abandoned the symbol in later writing, and for decades the notation drifted, with different authors using different marks and plenty of room for confusion between partial and total derivatives.

The ∂ was rescued and made standard by Carl Gustav Jacob Jacobi. In his 1841 paper on functional determinants — the work that also gave us the Jacobian matrix — Jacobi adopted ∂ deliberately, reserving the ordinary d for total differentials and ∂ for partial ones, and used it consistently enough that the convention finally stuck ^[1]^[2]. That split — d for total, ∂ for partial — is the one still taught and still printed in the GUM and every uncertainty standard downstream of it.

Related tools

/tools/link-budget
/tools/beam-deflection
/tools/stress-strain
/tools/tsiolkovsky-delta-v
/tools/standard-atmosphere

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →