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.
Key Terms
Differential— dx is a free increment; dy=f′(x)⋅dx is the tangent-line response
Derivative— dy/dx separated into two independent quantities
Partial Derivative— extends differentials to functions of several variables
Tangent Line— the differential dy follows the tangent line, not the curve
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
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.
Pitfall: Dropping the dx
Writing dy=3x2 for y=x3 leaves out dx and gives dy=12 at x=2 for any step. With dx=0.1, dy=3(2)2(0.1)=1.2, close to the actual Δy=1.261.
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.
RULE
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:
dy Versus Δy
Δy=f(x+dx)−f(x)(actual change)
dy Versus Δy
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.
THEOREM
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.
dy and Δy side by side
f(x) = x² at x = 1 with dx = 1. Along the tangent line, whose slope is f′(1) = 2, the run of 1 produces a rise of dy = 2 (navy column). Along the curve, the same run produces Δy = f(2) − f(1) = 3 (blue column). The amber piece on top is the difference Δy − dy = 1, the error of the tangent-line estimate. A smaller dx would shrink the error much faster than it shrinks dx.
For f(x) = x² the error is exactly (dx)²: halve dx and the error drops to a quarter.
Aspect
dy (tangent line)
Δy (actual curve)
Definition
dy = f'(x) · dx
Δy = f(x + dx) − f(x)
Geometric meaning
vertical change along the tangent line
vertical change along the curve itself
Computation
one slope evaluation × increment
requires evaluating f at two points
Small dx behavior
accurate estimate of Δy
nearly equal to dy; error shrinks faster than dx
Large dx behavior
drifts away from Δy; error scales with f''(x)·(dx)²
diverges from dy as curvature accumulates
Typical use
quick approximation, error propagation
exact change when both function values are known
Differential Notation
Notation
Differential Notation
Four marks, and the whole subject turns on telling two pairs apart: d from Δ, and dy from Δy.
dxdy as a derivative belongs to Derivatives. Here it is re-read as an actual quotient.
Δx · Δy
delta x, delta y
Actual change. Δx=x2−x1, and Δy=f(x+Δx)−f(x) is what the function really does over that step. Exact, no approximation. Capital delta only — the keyboard has Δ under Greek letters.
CasesΔ shrinking toward zero is the whole content of the limit definition of the derivative. In numerical work it stays finite and is the step size.
Also writtenh for Δx in limit definitions, where the delta is cumbersome inside a fraction.
Do not confuseLowercase δ is not this. It is the tolerance in an epsilon-delta argument, or a variation in the calculus of variations — a quantity you choose, not a change you measure.
dx
dee x
The differential of the independent variable — an increment you assign. dx=Δx exactly, by definition. It is not infinitely small, and nothing here requires it to be.
CasesWritten large in error estimation, tiny in approximation, symbolic under an integral sign. Same object each time; only the size of the number changes.
Also written∂x once other variables are held fixed.
Do not confuseThe d is not a factor multiplying x, and dx is not "d times x". It is one symbol. The substitution rule in integration manipulates it algebraically, which is what makes the reading tempting and wrong.
Same glyph elsewhereThe same d opens dxd as an operator on higher-order derivatives, where it is not a quantity at all.
dy
dee y
The differential of the dependent variable: dy=f′(x)dx. Not the actual change in y — the change predicted by the tangent line. It is defined, never measured.
CasesOn a curve, dy=Δy and the gap is the approximation error. On a straight line they are equal for every step size, which is the only case where the distinction disappears.
Also writtendf when the function is named rather than its output.
Do not confuseΔy is what happened; dy is what the tangent predicted. The section above measures the gap. Treating them as interchangeable is the commonest error with differentials, and it silently discards the entire error term.
dxdy
dee y by dee x
Everywhere else this is one indivisible symbol. Here, and only here, it is a genuine quotient of two defined quantities — which is what licenses the algebra in the chain rule and in separation of variables.
CasesDivide dy=f′(x)dx through by dx and the derivative falls out. That step is legitimate on this page and illegitimate on Derivatives, where dy and dx have not been defined as objects.
Also writtendf/dx, dxdf — the operator form, which deliberately breaks the quotient reading.
Do not confuseThe permission is local. dx2d2y is not a quotient of d2y and dx2 — second-order differentials do not cancel the way first-order ones do, and the notation gives no warning.
Near the marked point the curve and its tangent line are nearly indistinguishable, and the line’s slope, 1.89, is the f′(a) of the formula: moving dx along the line raises it by 1.89 dx, which is dy. The gap between line and curve, visible only farther out, is the error the differential ignores. Move the point and watch the approximating line follow on the tangent line at a point tool.
Linear approximation is therefore nothing more than reading the tangent line instead of the curve.
Linear Approximation
The differential formula rearranges into an approximation for function values:
This is the linearization of f at a. Given a known value f(a) and the slope f′(a), 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 agrees to four decimal places, with an error of about 0.000014.
Pitfall: Reporting the change as the value
Stopping at dy=41(0.03)=0.0075 gives only the change in f, not 4.03. The estimate adds the known value: f(4)+dy=2+0.0075=2.0075.
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.
Near a known point, the tangent line turns a hard function value into one multiplication and one addition.
Error Estimation
If a quantity x is measured with error dx, the computed value f(x) inherits an error approximately equal to the differential:
Error Estimation
absolute error in f≈∣dy∣=∣f′(x)∣⋅∣dx∣
Fact: Kepler's barrels sat at the maximum
In 1615 Kepler found that Austrian wine barrels were built close to the shape that holds the most wine for a given diagonal rod measurement. Near that maximum the volume barely responds to the shape: a barrel 10% off in its height-to-diameter ratio holds less than 1% less wine.
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 extreme case is a point where f′(x)=0, such as the top of a maximum. There the differential predicts no change at all: to first order, an error in x does not reach f(x). The change that remains is proportional to (dx)2, far smaller than dx itself when dx is small.
The relative error normalizes by the function value:
Error Estimation
relative error in f≈∣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 is approximately the absolute value of 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.
Pitfall: Doubling the absolute error
For x=10±0.1, taking the error in x2 as 2(0.1)=0.2 applies the rule to the wrong error. The rule doubles the relative error, from 1% to 2%, so ∣dy∣=2(10)(0.1)=2 and x2=100±2.
In practice the rule is applied in three steps. A measured value comes with an absolute uncertainty, a ± figure such as ±0.1, and dividing it by the value turns it into a relative error. The rule scales that relative error by ∣n∣, and multiplying the result by the computed value xn turns it back into a ± figure for the output. For a cube, n=3, and the same steps give the uncertainty in a volume computed from a measured edge.
The derivative is the exchange rate between input error and output error; for a power, relative errors scale by the exponent.
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
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.
Pitfall: Adding the input errors unweighted
Adding the errors directly gives dA=0.1+0.05=0.15 and A≈50±0.15. Each error has to be multiplied by its partial derivative: dA=5(0.1)+10(0.05)=1.0, so A≈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.
rearranging dy and dx as independent objects is rigorous
With differentials defined, the fraction-like moves of Leibniz notation are genuine algebra, not lucky shorthand.
Higher-Order Differentials
If dx is held constant (treated as a fixed increment), the second differential of y is
Higher-Order Differentials
d2y=f′′(x)⋅(dx)2
The notation d2y/dx2 for the second derivative arises from dividing both sides by (dx)2:
Higher-Order Differentials
dx2d2y=(dx)2d2y=f′′(x)
This explains the Leibniz notation for higher derivatives. The superscript on d2y counts the number of times the differential operator d has been applied. The square on dx2 means (dx)2—the differential dx appearing as a factor twice—not the differential d(x2).
Higher-order differentials extend the approximation. The second-order estimate of the change in y is
Higher-Order Differentials
Δy≈f′(x)dx+21f′′(x)(dx)2
Fact: Einstein's energy contains 21mv2
In relativity a body's energy is E(v)=1−v2/c2mc2, with E′(0)=0 and E′′(0)=m. The second-order estimate at v=0 gives E≈mc2+21mv2: the rest energy plus Newton's kinetic energy.
This is the beginning of the Taylor expansion written in differential notation: f(x+dx)≈f(x)+f′(x)dx+21f′′(x)(dx)2+⋯. Each additional term uses a higher-order derivative and a higher power of dx, improving the approximation at the cost of more computation.
When f′(x)=0, the first term drops out and the second-order term is the whole estimate: Δy≈21f′′(x)(dx)2. The change then grows with the square of the step rather than with the step itself, and the flat tangent line, which predicts no change at all, misses it entirely.
Notation
Meaning
How it arises
d²y
second differential of y; equals f''(x)·(dx)²
applying the differential operator d twice with dx held constant
dx²
(dx)² — the differential dx appearing as a factor twice
The single formula dy=f′(x)dx underlies every application of differentials covered above. The table below collects those applications side by side, pairing each with the formula it specializes to, a worked example from the page, and the situation where it earns its place over alternatives. Together, these capture the practical payoff of treating dx and dy as independent quantities rather than parts of an unbreakable dy/dx symbol.
Use
Formula
Example
Where it shines
Linear approximation
f(x + dx) ≈ f(x) + f'(x) dx
√4.03 ≈ 2 + ¼(0.03) = 2.0075
quick estimates near a known value
Absolute error
|dy| ≈ |f'(x)| · |dx|
error in xⁿ is n·x^(n−1)·dx
propagating a single measurement uncertainty
Relative error
|dy|/|f(x)| ≈ |f'(x)/f(x)| · |dx|
xⁿ multiplies relative error by |n|
power-law uncertainty scaling
Multivariable propagation
dz = (∂f/∂x) dx + (∂f/∂y) dy
rectangle: dA = w dl + l dw
combining errors from several inputs
Notation algebra
dy and dx as independent quantities
chain rule cancellation, ∫…dx substitution
working symbolically with derivatives
Second-order refinement
Δy ≈ f'(x) dx + ½f''(x)(dx)²
first two Taylor terms
improving accuracy beyond the tangent line
Common Pitfalls
Four mistakes cause most wrong answers when differentials are used for estimates and error bounds. Each is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.
Dropping the dx
The differential dy is written down as the derivative alone, without the factor dx. The result no longer depends on the step, so it cannot describe a change.
Wrong working: for y=x3, dy=3x2. At x=2 this gives dy=12, for a step of 0.1 and for a step of 0.001 alike.
Right working: dy=f′(x)dx=3x2dx. At x=2 with dx=0.1, dy=12(0.1)=1.2. The actual change is Δy=2.13−23=9.261−8=1.261, close to 1.2 and nowhere near 12.
Why it happens: the derivative is computed first and stands on its own in prime notation; the differential is the same expression with dx attached, and the factor is easy to leave off.
How to check: dy must scale with dx. Halving the step should halve dy; if the answer contains no dx, nothing changes, and the factor is missing.
Reporting the change as the value
In a linear approximation, the differential dy=f′(a)dx is computed and given as the approximate value of f, although it is only the change from the known value f(a).
Wrong working: for 4.03 with a=4 and dx=0.03, dy=41(0.03)=0.0075, so 4.03≈0.0075.
Right working: 4.03≈f(4)+f′(4)dx=2+0.0075=2.0075.
Why it happens: the differential is the step that takes the work, and once it is done the problem feels finished; the known value f(a) was found at the start and is forgotten by the end.
How to check: the estimate must be close to the known value f(a), because x is close to a. 4.03 has to be a little more than 4=2.
Doubling the absolute error
The power rule for errors, that xn carries ∣n∣ times the relative error of x, is applied to the absolute error, the ± figure of the measurement.
Wrong working: x=10±0.1, so x2=100±2(0.1)=100±0.2.
Right working: the relative error of x is 100.1, which is 1%. Squaring doubles it to 2%, and 2% of 100 is 2, so x2=100±2. The differential gives the same: ∣dy∣=2x∣dx∣=2(10)(0.1)=2.
Why it happens: "squaring doubles the error" is remembered without the word relative, and the ± figure is the error that is written down.
How to check: square the largest possible value. 10.12=102.01, about 2 above 100, not 0.2.
Adding the input errors unweighted
For a quantity that depends on several measurements, the input errors are added as they are, without the partial derivatives that turn each one into a change of the result.
Wrong working: for A=lw with l=10±0.1 and w=5±0.05, dA=0.1+0.05=0.15, so A≈50±0.15.
Right working: dA=∂l∂Adl+∂w∂Adw=wdl+ldw=5(0.1)+10(0.05)=1.0, so A≈50±1.0.
Why it happens: the errors in length and width are both lengths, so adding them looks natural. But an error dl in the length adds or removes a strip of size w×dl, so it changes the area by w times as much.
How to check: compute the area at the largest values. (10.1)(5.05)=51.005, about 1 above 50, not 0.15.
Differentials FAQ
Does dx mean d times x?
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No — dx is a single symbol, not a product. The d is not a factor and cannot be cancelled or divided out on its own. The misreading is tempting because dx really does behave algebraically: substitution in integration rearranges it, and dy/dx genuinely becomes a quotient once differentials are defined. What gets rearranged is dx as a whole.Read more →
When is dy exactly equal to Δy?
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Only when f is linear. For f(x) = mx + b the tangent line is the function itself, so dy = m·dx matches the actual change f(x + dx) − f(x) for every increment, however large. For any curved function the two differ, and the gap Δy − dy is the approximation error, widening as dx grows.Read more →
How accurate is a linear approximation?
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Two things control it: how far you move from the base point, and how sharply the curve bends there. The error grows with the distance from the base point and with the size of f″ nearby, since f″ measures how fast the curve pulls away from its tangent. Approximating √4.03 from x = 4 agrees to four decimal places; a much larger step would not be.Read more →
Why does squaring a measurement double its relative error?
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Because for f(x) = xⁿ the logarithmic derivative f′(x)/f(x) equals n/x, so the relative error in the output is |n| times the relative error in the input. Squaring gives n = 2, cubing gives n = 3. A 1% error in a measured radius becomes a 2% error in its square and a 3% error in its cube.Read more →
How do you get a better approximation than the tangent line?
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Add the second-order term: Δy ≈ f′(x)dx + ½f″(x)(dx)². The tangent line matches the curve's value and slope at the base point; the quadratic term also matches its bending, so it tracks the curve further out. Continuing this way — each term using one more derivative and one more power of dx — produces the Taylor expansion.Read more →