Differentials and Estimating Change

To estimate how a small change in input changes the output, use differentials calculus. The differential dy = f'(x)dx gives the change along the tangent line, while Δy is the actual change on the curve. The difference between them is the error, which grows predictably with the square of the input change. This is the core of error propagation and measurement uncertainty.

You do not need the exact function value. You need a quick, reliable estimate of how much the output shifts when the input shifts by a tiny amount. That is what differentials are for.

dy is the change in y along the tangent line at a point. If you know f(a) and f'(a), then for a small change dx in x, the estimated output change is dy = f'(a)dx. Δy is the actual change in the function: Δy = f(a+dx) − f(a).

For example, with y = x³ at x = 1 and dx = 0.1: dy = 3(1)²(0.1) = 0.3; Δy = (1.1)³ − 1 = 0.331. The error is 0.031, about 9% of Δy. For y = √x at x = 4, dx = 0.1: dy = (1/(2√4))(0.1) = 0.025; Δy = √4.1 − 2 ≈ 0.6% of Δy. The approximation works best when the second derivative is small.

Draw this: a curve, its tangent line at a, a horizontal step dx, and the vertical rise dy (on the line) and Δy (on the curve). The gap between dy and Δy is the error. For functions with a small second derivative near a, that gap stays tiny. For highly curved functions, it grows fast.

The linear approximation formula L(x) = f(a) + f'(a)(x−a) is the tangent line itself. The differential dy = f'(a)dx is just the change part of that line, isolated from the starting value f(a). The two ideas are the same object seen from different angles.

Taylor's Inequality (n=1)

The error |f(x) − L(x)| is bounded by (M/2)|x−a|², where M is the maximum of |f''| on the interval between a and x. For f(x) = e^x at a = 0, M = e^0.1 ≈ 1.10517, so the bound is (1.10517/2)(0.1)² ≈ 0.005526; the actual error is 0.00517. For f(x) = sin x at a = 0, M = 1, bound = 0.005; actual error is 0.000167. The bound can be loose, but it guarantees the error stays under that ceiling.

Overestimate Or Underestimate?

If f''(x) > 0 (concave up) on the interval, the tangent line lies below the curve, so L(x) is an underestimate. If f''(x) < 0 (concave down), L(x) is an overestimate. For f(x) = √x, f'' < 0, so the linearization overestimates, L(4.1) = 2.025 vs actual 2.02485. For f(x) = e^x, f'' > 0, so L(0.1) = 1.1 vs actual 1.10517, an underestimate. Concavity tells you direction, not magnitude.

Propagated error differentials answer: if I measure the radius of a circle with a small uncertainty, how uncertain is the area? For a circle of radius r, A = πr². If r = 10 cm with a measurement error dr = 0.1 cm, the propagated error in area is dA = 2πr dr = 2π(10)(0.1) = 2π ≈ 6.28 cm².The differential estimate is off by about 0.04 cm².

Sphere Volume: A Worked Example

For a sphere, V = (4/3)πr³. Suppose a lab measures a ball bearing's radius as 5 mm with a possible error of 0.05 mm. The volume is about 523.6 mm³. The propagated error using differentials: dV = 4πr² dr = 4π(25)(0.05) = 5π ≈ 15.71 mm³. The relative error in volume is dV/V = (4πr² dr) / ((4/3)πr³) = 3 dr/r = 3(0.05/5) = 0.03, or 3%.1% of the actual change, a tight enough estimate for most lab work.

The failure case: if dr is not tiny relative to r, this method breaks.27 mm³, the estimate is off by about 1.56 mm³. Always check that dr is small before using differentials.

Relative error differentials express the size of the error relative to the measurement. For the sphere above, the relative error in volume is 3 dr/r, which is three times the relative error in radius. This is a general rule: for a power function y = kxⁿ, the relative error in y is n times the relative error in x.

Percentage error is relative error × 100%. For the sphere, the percentage error in volume is 3% when the radius error is 1.67%, and 9% when the radius error is 3%. The factor n (here, 3) is the sensitivity multiplier.

For a product or quotient, the relative errors add. For a sum or difference, the absolute errors add. These rules let you chain differentials through multi-step calculations without recomputing from scratch each time.

• Using the wrong point of tangency. Picking a point far from your input guarantees large error. Always choose a where you know f(a) and f'(a) exactly and a is close to the x you care about.
• Forgetting to use radians. Small-angle approximations like sin θ ≈ θ only work in radians. In degrees, sin 1° ≈ 0.0175 while 1° = 0.0175 rad, the numbers look similar but the formula fails for larger angles.
• Confusing dy and Δy. dy is the tangent-line change; Δy is the real change. Use dy only to estimate, never as the exact value. The gap is the error.
• Assuming the error bound is the actual error. Taylor's inequality gives a maximum, not the true error. The actual error is often much smaller. For sin x at a = 0, the bound is 0.005 but the actual error is 0.000167.
• Overusing the approximation. A 10% relative change in input can produce a 30% relative error in output for cubic relationships. The method works only for small dx.
• Mixing up overestimate and underestimate. The direction depends on f'' sign. f'' > 0 means underestimate; f'' < 0 means overestimate. Check concavity on the whole interval, not just at a.

The single most common failure: using a = 0 for a function not defined at 0, like ln x. The linearization does not exist there. Check differentiability before you start.

Use differentials for quick, hand-calculated estimates of how measurement errors propagate through formulas. They suit AP Calculus AB/BC and Calculus I students required to compute linearizations by hand and interpret over/underestimates using concavity. They also suit Physics students using small-angle approximations (sin θ ≈ θ, tan θ ≈ θ) in pendulum and optics derivations. Calculus III students extend the idea to tangent-plane approximations for functions of two variables.

Skip differentials if you need exact function values, numerical root-finding (Newton's method), or higher-order approximations (Taylor polynomials beyond n = 1). Those cases need a Taylor series calculator or a root-finding resource, not a tangent-line estimate. Also skip if your measurement error is large relative to the measurement, the differential estimate will be misleadingly small.

What goes wrong most often: using differentials without checking that the function is differentiable at the point and that the input change is truly small. Do both checks first, or the estimate is worthless.