Partial Derivatives

The limit definition

For a function f(x, y) of two variables, the partial derivative with respect to x at the point (x, y) is the limit:

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

Compare this with the single-variable derivative:

df/dx = lim_{h → 0} [f(x + h) − f(x)] / h

The forms are identical. The difference is what the function is — a multivariable f has more arguments, but during the limit we vary only one of them, holding y (and any other variables) frozen. Practically, you compute partial derivatives by treating the held variables as constants and differentiating with the usual single-variable rules.

The partial with respect to y is defined symmetrically:

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

For functions of three or more variables, every partial derivative ∂f/∂x_i has the same form: vary x_i while every other variable stays fixed.

Geometric interpretation

Plot z = f(x, y) as a surface in three-dimensional space. Slicing the surface with the plane y = y₀ (a vertical plane parallel to the xz-plane through y = y₀) produces a curve. That curve is the graph of g(x) = f(x, y₀) — a single-variable function of x. Its slope at x = x₀ is the partial derivative ∂f/∂x at (x₀, y₀).

Slicing with x = x₀ produces another curve, whose slope at y₀ is ∂f/∂y. Each partial derivative is a slope along one coordinate direction at the chosen point. Sliding the slicing plane around the surface gives the partial derivative as a function — the original surface's "directional slopes" along the x and y axes.

Worked example — a simple computation

Let f(x, y) = x² y + 3xy² + sin(x). Compute the partials.

Treat y as a constant. Then:
∂f/∂x = 2xy + 3y² + cos(x)

Treat x as a constant. Then:
∂f/∂y = x² + 6xy

Each derivative obeys the same single-variable rules — power rule, sum rule, the derivatives of sin, ln, exp — but the "constant" you ignore changes depending on which variable you are differentiating against. Notice that ∂(sin x)/∂y = 0 because sin(x) is treated as a constant when y is the variable.

Higher-order and mixed partials

Partial derivatives of partial derivatives:

∂²f/∂x²  =  ∂/∂x (∂f/∂x)
∂²f/∂y²  =  ∂/∂y (∂f/∂y)
∂²f/∂x∂y =  ∂/∂x (∂f/∂y)        ("mixed partial")
∂²f/∂y∂x =  ∂/∂y (∂f/∂x)        (the other mixed partial)

Clairaut's theorem (also called Schwarz's theorem on the equality of mixed partials) says ∂²f/∂x∂y = ∂²f/∂y∂x whenever both exist and are continuous near the point. For any function you can write down using elementary operations (polynomial, trig, exp, log, ...) the mixed partials are equal. Pathological counterexamples exist where the second partials are discontinuous, but they require deliberate construction.

For our running example f(x, y) = x²y + 3xy² + sin(x):

∂²f/∂x∂y = ∂/∂x (x² + 6xy) = 2x + 6y
∂²f/∂y∂x = ∂/∂y (2xy + 3y² + cos x) = 2x + 6y         ✓ equal

Partial vs total derivative — key distinctions

	Partial derivative ∂f/∂x	Total derivative df/dx
Domain	Multivariable function f	Composition along a path
Other variables	Held fixed	Allowed to depend on x
Notation	∂f/∂x, f_x	df/dx, sometimes Df
Value at a point	Single number per direction	Single number (chain-rule sum)
Gradient relation	Components of ∇f	∇f · velocity
Captures	Direct dependence on one variable	Direct + indirect dependence on x
Equation example	Heat equation: ∂u/∂t = α ∇²u	Lagrangian mechanics: dL/dt
Common in	PDEs, vector calculus, optimisation	Thermodynamics, economics, dynamics

The most important takeaway: partial derivatives ignore indirect chains of dependence. If u and v both depend on time, ∂(u + v)/∂t treats v as a frozen variable; d(u + v)/dt accounts for v's own time dependence.

The multivariable chain rule

If z = f(x, y) and x = g(t), y = h(t), how does z change with t? Both x and y now depend on t, so the change in z has two contributions:

dz/dt = (∂f/∂x) (dx/dt) + (∂f/∂y) (dy/dt)

Each term is "rate of change in one direction × velocity in that direction." The formula generalises: for f(x₁, x₂, …, x_n) with each x_i = g_i(t),

df/dt = Σ_i (∂f/∂x_i) (dx_i/dt) = ∇f · (velocity vector)

This is precisely how the directional derivative arises (∇f · û), and how backpropagation in neural networks chains together gradients across layers.

The heat equation — partial derivatives in space and time

One of the most influential equations of mathematical physics is the heat (or diffusion) equation:

∂u/∂t = α ∇²u = α (∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²)

Here u(x, y, z, t) is the temperature at position (x, y, z) at time t, and α is the thermal diffusivity. Why partials and not total derivatives? Because temperature really depends on both space and time. The left side asks: how fast does temperature change at a fixed location? The right side asks: how does the temperature at a point compare to the spatial average around it?

If u at a point is lower than the average of u in a small neighborhood, ∇²u > 0 and ∂u/∂t > 0 — the point heats up. If u is higher than the average, ∇²u < 0 and the point cools down. Partial derivatives in space encode "how spatially un-flat is the temperature?"; the partial in time encodes "how fast is the temperature evolving?" The same equation governs particle diffusion, the smoothing of images, and the smoothing of probability distributions in stochastic processes.

Partial derivatives in economics — marginals

In economics, partial derivatives are called marginals. If U(x₁, x₂, ...) is a utility function, ∂U/∂x_i is the marginal utility of good i. If C(K, L) is the cost as a function of capital K and labour L, ∂C/∂K is the marginal cost of capital. Production functions f(K, L) — the Cobb–Douglas form A K^α L^β is canonical — have partials that economists call marginal product of capital and marginal product of labour.

The principle of equal marginal utility per dollar — ∂U/∂x_i divided by price p_i is the same across all goods i at the optimum — is one of microeconomics' core results. It is just Lagrange multiplier optimisation in disguise, and partial derivatives carry the action.

Where partial derivatives appear

• Partial differential equations. The heat equation, wave equation, Laplace equation, Schrödinger equation, Navier–Stokes, Maxwell's equations — every PDE is built from partial derivatives.
• Optimisation. Critical points of f(x, y, ...) satisfy ∂f/∂x = ∂f/∂y = ... = 0. The Hessian (matrix of second partials) classifies them as max, min, or saddle.
• Vector calculus. Gradient, divergence, curl, Laplacian — all are expressions in partial derivatives.
• Machine learning. Backpropagation computes partials of the loss with respect to every parameter via the chain rule.
• Physics. Lagrangian and Hamiltonian mechanics use ∂L/∂q and ∂L/∂q̇ to derive Euler–Lagrange equations. Thermodynamic state functions (U, H, F, G) are filled with partials at constant pressure, constant volume, etc.
• Engineering. Stress, strain, flux, current density — every continuum-mechanics quantity satisfies field equations written in partial derivatives.
• Economics and finance. Marginal utility, marginal cost, the Greeks of options pricing (Δ, Γ, Θ, ν, ρ are partials of option price).

Common mistakes

• Mixing up partial and total derivatives. Especially common in thermodynamics. ∂U/∂T at constant volume is one quantity; at constant pressure another; dU/dT depends on what is changing as you change T.
• Forgetting to specify what is held fixed. A partial derivative is meaningless without saying which variables are constant. Thermodynamics writes (∂U/∂V)_T to mean "at constant T," because the same notation in a different problem could mean "at constant pressure."
• Treating ∂x as if it were a number. Unlike Leibniz's d, the symbol ∂ is not a literal differential — you cannot "cancel" the ∂y in (∂z/∂y)(∂y/∂x). The total derivative chain rule survives this game; the partial chain rule does not in general.
• Differentiating the wrong variable. When y depends on x, ∂f/∂x and ∂f/∂x|_y vary by the chain-rule term (∂f/∂y)(dy/dx). Sketch the dependency graph before computing.
• Concluding differentiability from existence of partials. All partials existing at a point does not imply f is differentiable there. You need continuity of the partials (or a more careful linearity argument).
• Forgetting metric factors in non-Cartesian coordinates. ∂/∂θ in polar/cylindrical/spherical coordinates is a derivative per radian, not per metre. Convert with arc-length scales before comparing to physical rates.
• Confusing ∂ with d for one-variable functions. If f depends on x alone, ∂f/∂x and df/dx coincide — but writing ∂ where d is conventional looks wrong and signals a misunderstanding of the multivariable distinction.