The derivative dxdy is defined as a limit—a single object, not a fraction. But Leibniz notation invites the question: can dy and dx stand on their own? Differentials answer yes, with precise meaning. The symbol dx represents an independent small change in x, and dy=f′(x)dx represents the corresponding change in y predicted by the tangent line.
This formalism does more than justify notation. It provides a direct tool for approximating function values near a known point, estimating how errors in measurement propagate through calculations, and explaining why Leibniz notation behaves like fraction arithmetic in the chain rule and in integration.
The Differential dx
The differential dx is an independent variable. It represents a change in x—an increment away from a given value. Unlike the h or Δx in the limit definition, dx is not required to approach zero. It is a finite quantity that can be positive, negative, or zero.
There is no formula for dx—it is chosen freely. Choosing dx=0.1 means considering what happens when x shifts by 0.1. Choosing dx=−2 means shifting x by −2. The differential dx sets the scale for the approximation that follows.
The notation is consistent with Leibniz notation for the derivative. When dy/dx is treated as a ratio of differentials rather than a limit symbol, the algebraic manipulations that make Leibniz notation powerful—cancellation in the chain rule, separation in differential equations—become formally valid rather than merely suggestive.
The Differential dy
Given y=f(x) where f is differentiable, the differential of y is defined as
dy=f′(x)⋅dx
The differential dy depends on two things: the point x (which determines the slope f′(x)) and the increment dx (which determines the scale). For fixed x, dy is a linear function of dx—doubling dx doubles dy.
Geometrically, dy is the vertical change along the tangent line at x when the horizontal position shifts by dx. The tangent line rises (or falls) at rate f′(x), so a horizontal shift of dx produces a vertical shift of f′(x)⋅dx.
For a linear function f(x)=mx+b, the differential dy=m⋅dx equals the actual change in f exactly, because the tangent line to a linear function is the function itself. For nonlinear functions, dy is an approximation—exact in the limit as dx→0, and increasingly approximate as dx grows.
dy Versus Δy
Two quantities measure the change in y when x changes by dx:
Δy=f(x+dx)−f(x)(actual change)
dy=f′(x)⋅dx(tangent line estimate)
The actual change Δy follows the curve. The differential dy follows the tangent line. The difference Δy−dy is the error introduced by the linear approximation.
For small dx, the error Δy−dy is much smaller than dx itself. Precisely, limdx→0dxΔy−dy=0—the error shrinks faster than the increment. This is what makes the tangent line a good approximation near the point of tangency.
For larger dx, the gap between Δy and dy widens. The tangent line is a local tool: reliable near x, increasingly unreliable far from it. The curvature of f—governed by the second derivative—determines how quickly the approximation deteriorates.
Linear Approximation
The differential formula rearranges into an approximation for function values:
f(x+dx)≈f(x)+f′(x)⋅dx
This is the linearization of f at x. Given a known value f(x) and the slope f′(x), the tangent line estimates f at nearby points.
To approximate 4.03: take f(x)=x, x=4, dx=0.03. Then f(4)=2 and f′(x)=2x1, so f′(4)=41. The estimate is 2+41(0.03)=2.0075. The actual value is 4.03≈2.00749...—the approximation is accurate to five decimal places.
The linearization can also be written as L(x)=f(a)+f′(a)(x−a), the tangent line at x=a used as a function. For x near a, L(x)≈f(x). The quality of the approximation depends on two factors: the size of ∣x−a∣ and the magnitude of f′′ near a, which controls how quickly the curve diverges from its tangent.
Error Estimation
If a quantity x is measured with error dx, the computed value f(x) inherits an error approximately equal to the differential:
absolute error in f≈∣dy∣=∣f′(x)∣⋅∣dx∣
The derivative acts as an error amplification factor. Where ∣f′(x)∣ is large, small measurement errors in x produce large errors in f(x). Where ∣f′(x)∣ is small, errors are suppressed.
The relative error normalizes by the function value:
∣f(x)∣∣dy∣=∣f(x)∣∣f′(x)∣⋅∣dx∣
The ratio f(x)f′(x)=dxd[ln∣f(x)∣] is the logarithmic derivative. Relative error in the output equals the logarithmic derivative times the absolute error in the input.
For f(x)=xn: f(x)f′(x)=xn, so the relative error in xn is n times the relative error in x. Squaring a measurement doubles its relative error. Cubing triples it. This scaling rule is a standard tool in experimental science for propagating uncertainties through power-law relationships.
Differentials of Multiple Variables
When a formula involves several measured quantities, each contributes to the total error. If z=f(x,y), the total differential is
dz=∂x∂fdx+∂y∂fdy
Each partial derivative weights the contribution of the corresponding variable's error. The total differential estimates how z responds to simultaneous small changes in all inputs.
For the area of a rectangle A=lw: dA=wdl+ldw. If l=10±0.1 and w=5±0.05, then dA=5(0.1)+10(0.05)=1.0. The area is approximately 50±1.0.
For the volume of a cylinder V=πr2h: dV=2πrhdr+πr2dh. The radius error is amplified by 2πrh and the height error by πr2. Since r enters as a square, its error contributes more heavily—consistent with the power-law scaling of relative error.
This extension uses partial derivatives, which generalize the single-variable derivative to functions of several variables. The total differential remains a linear approximation, now in multiple dimensions.
Differentials and Leibniz Notation
Differentials retroactively justify the algebraic behavior of Leibniz notation.
The chain rule states dxdy=dudy⋅dxdu. In differential form: dy=dudydu and du=dxdudx. Substituting the second into the first gives dy=dudy⋅dxdudx, which is dy=dxdydx. The intermediate variable du cancels as though these were fractions—and with differentials, they are.
Integration notation ∫f(x)dx also uses the differential dx meaningfully. The substitution rule u=g(x), du=g′(x)dx replaces dx with an expression involving du—a literal change of variable in the differential. The notation is not merely symbolic; it reflects the algebraic structure of differentials.
Separation of variables in differential equations—writing dxdy=g(x)h(y) as h(y)dy=g(x)dx and integrating both sides—depends on treating dy and dx as independent objects that can be rearranged. Differentials make this manipulation rigorous rather than heuristic.