Total Differential: Definition, Approximation, Chain Rule

Suppose the volume of a gas depends on its pressure and its temperature. If both change a little at the same time, how much does the volume change? Adding the two separate effects together is not a guess — calculus gives an exact tool for it: the total differential.

For a function z = f(x, y), the total differential dz combines the tiny changes dx and dy into one formula: dz = fx dx + fy dy. It is the multivariable version of the single-variable approximation dy = f′(x) dx. It lets you estimate small changes quickly, bound measurement errors, and — as a bonus — it is the stepping stone to the chain rule for several variables.

Key Concepts

• Total differential: dz = (∂f/∂x) dx + (∂f/∂y) dy — the linear part of the change in f.
• dx, dy are small increments (like Δx, Δy) and are independent of each other.
• True change: Δz = f(x + Δx, y + Δy) − f(x, y). The differential dz approximates Δz when dx and dy are small.
• Linear approximation (tangent plane): L(x, y) = f(a, b) + fx(a, b)(x − a) + fy(a, b)(y − b).
• Differentiability: dz is a good approximation only when f is differentiable (its partials are continuous near the point).

Formulas

• dz = fx(x, y) dx + fy(x, y) dy
• Δz ≈ dz for small dx, dy; the approximation is f(x + dx, y + dy) ≈ f(x, y) + dz.
• Error analysis: if z is computed from measured x and y, the worst-case error in z is about |fx| · (error in x) + |fy| · (error in y).
• Three variables: dw = fx dx + fy dy + fz dz.

Worked Example

Use the total differential to estimate f(2.05, 2.98) for f(x, y) = x²y, starting from the nearby point (2, 3).

1. Partials at (2, 3). fx = 2xy, so fx(2, 3) = 2 · 2 · 3 = 12. fy = x², so fy(2, 3) = 4.
2. Increments. dx = 2.05 − 2 = 0.05; dy = 2.98 − 3 = −0.02.
3. Differential. dz = 12(0.05) + 4(−0.02) = 0.60 − 0.08 = 0.52.
4. Approximation. f(2.05, 2.98) ≈ f(2, 3) + dz = 12 + 0.52 = 12.52.
5. Check against the true value. (2.05)² × 2.98 = 4.2025 × 2.98 = 12.52345. The true change is Δz = 0.52345, so the differential’s estimate 0.52 was off by only about 0.003.

Common Misconceptions

• “dz is the exact change.” No — dz is the linear approximation; Δz is the exact change. They agree only in the limit of tiny increments.
• “You can always just add the separate one-variable effects.” That is precisely what the formula does, but it is valid only when f is differentiable at the point.
• “dx and dy must be equal.” They are fully independent — each can be positive, negative, or zero, as in the example above.

Real-World Applications

• Measurement error: if x and y are measured with small errors, dz bounds the resulting error in any computed quantity z = f(x, y).
• Sensitivity analysis: the sizes of fx and fy reveal which input a design or experiment is most sensitive to.
• Economics: the total differential of a production function shows how output responds when several inputs change at once.
• Chain rule: dividing dz = fx dx + fy dy by dt gives the multivariable chain rule dz/dt = fx dx/dt + fy dy/dt.

The total differential dz = fx dx + fy dy packages every small change into one linear formula. Use it to estimate values near a known point, to bound errors from imperfect measurements, and as the foundation for the chain rule — just remember it is an approximation, excellent for small increments and exact only in the limit.