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) \cdot 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

See All Calculus Definitions →

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

\Delta x

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

-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

Differential

dy = f'(x)\, dx

Learn more about this formula: Differential →

Browse all derivative formulas →

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) \cdot dx

.

For a linear function

f(x) = mx + b

, the differential

dy = m \cdot 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 \to 0

, and increasingly approximate as

dx

grows.

dy Versus Δy

Two quantities measure the change in

y

when

x

changes by

dx

\Delta y = f(x + dx) - f(x) \qquad \text{(actual change)}

dy = f'(x) \cdot dx \qquad \text{(tangent line estimate)}

The actual change

\Delta y

follows the curve. The differential

dy

follows the tangent line. The difference

\Delta y - dy

is the error introduced by the linear approximation.

For small

dx

, the error

\Delta y - dy

is much smaller than

dx

itself. Precisely,

\lim_{dx \to 0} \frac{\Delta y - dy}{dx} = 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

\Delta y

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.

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

\Delta

, and

dy

from

\Delta y

\frac{dy}{dx}

as a derivative belongs to Derivatives. Here it is re-read as an actual quotient.

\Delta x

\Delta y

delta x, delta y

Actual change.

\Delta x = x_2 - x_1

, and

\Delta y = f(x + \Delta x) - f(x)

is what the function really does over that step. Exact, no approximation. Capital delta only — the keyboard has

\Delta

under Greek letters.

Cases

\Delta

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 written

h

for

\Delta x

in limit definitions, where the delta is cumbersome inside a fraction.

Do not confuseLowercase

\delta

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 = \Delta x

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

\partial 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

\frac{d}{dx}

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 \neq \Delta y

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 written

df

when the function is named rather than its output.

Do not confuse

\Delta y

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.

\frac{dy}{dx}

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 written

df/dx

\frac{d}{dx}f

— the operator form, which deliberately breaks the quotient reading.

Do not confuseThe permission is local.

\frac{d^2y}{dx^2}

is not a quotient of

d^2y

and

dx^2

— second-order differentials do not cancel the way first-order ones do, and the notation gives no warning.

All calculus symbols →↑ Derivatives

Linear Approximation

The differential formula rearranges into an approximation for function values:

Linear Approximation

f(x) \approx f(a) + f'(a)(x - a) \qquad \Delta y \approx f'(a)\, \Delta x

Learn more about this formula: Linear Approximation →

Browse all derivative formulas →

This is the linearization of

f

x

. Given a known value

f(x)

and the slope

f'(x)

, the tangent line estimates

f

at nearby points.

To approximate

\sqrt{4.03}

: take

f(x) = \sqrt{x}

x = 4

dx = 0.03

. Then

f(4) = 2

and

f'(x) = \frac{1}{2\sqrt{x}}

, so

f'(4) = \frac{1}{4}

. The estimate is

2 + \frac{1}{4}(0.03) = 2.0075

. The actual value is

\sqrt{4.03} \approx 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) \approx 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:

\text{absolute error in } f \approx |dy| = |f'(x)| \cdot |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:

\frac{|dy|}{|f(x)|} = \frac{|f'(x)|}{|f(x)|} \cdot |dx|

The ratio

\frac{f'(x)}{f(x)} = \frac{d}{dx}[\ln|f(x)|]

is the logarithmic derivative. Relative error in the output equals the logarithmic derivative times the absolute error in the input.

Logarithmic Derivative

\frac{d}{dx}[\ln f(x)] = \frac{f'(x)}{f(x)}

Learn more about this formula: Logarithmic Derivative →

Browse all derivative formulas →

For

f(x) = x^n

\frac{f'(x)}{f(x)} = \frac{n}{x}

, so the relative error in

x^n

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

Total Differential

dz = \frac{\partial z}{\partial x}\, dx + \frac{\partial z}{\partial y}\, dy

Learn more about this formula: Total Differential →

Browse all derivative formulas →

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 = w\,dl + l\,dw

. If

l = 10 \pm 0.1

and

w = 5 \pm 0.05

, then

dA = 5(0.1) + 10(0.05) = 1.0

. The area is approximately

50 \pm 1.0

.

For the volume of a cylinder

V = \pi r^2 h

dV = 2\pi r h\,dr + \pi r^2\,dh

. The radius error is amplified by

2\pi r h

and the height error by

\pi r^2

. 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

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

. In differential form:

dy = \frac{dy}{du}\,du

and

du = \frac{du}{dx}\,dx

. Substituting the second into the first gives

dy = \frac{dy}{du} \cdot \frac{du}{dx}\,dx

, which is

dy = \frac{dy}{dx}\,dx

. The intermediate variable

du

cancels as though these were fractions—and with differentials, they are.

Integration notation

\int 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

\frac{dy}{dx} = g(x)h(y)

\frac{dy}{h(y)} = 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.

See All Calculus Symbols and Notations →

Operation	Differential manipulation	What it justifies
Chain rule	dy = (dy/du) du and du = (du/dx) dx → dy = (dy/du)(du/dx) dx	du cancels algebraically, not merely suggestively
Substitution in integration	u = g(x), du = g'(x) dx — replace dx in the integrand	∫f(x)dx is a literal change of variable, not symbolic shorthand
Separation of variables	dy/dx = g(x)h(y) → dy/h(y) = g(x) dx, integrate both sides	rearranging dy and dx as independent objects is rigorous

Higher-Order Differentials

dx

is held constant (treated as a fixed increment), the second differential of

y

d^2 y = f''(x) \cdot (dx)^2

The notation

d^2 y / dx^2

for the second derivative arises from dividing both sides by

(dx)^2

\frac{d^2 y}{dx^2} = \frac{d^2 y}{(dx)^2} = f''(x)

This explains the Leibniz notation for higher derivatives. The superscript on

d^2 y

counts the number of times the differential operator

d

has been applied. The square on

dx^2

reflects that

dx

appears as a factor twice—not that

dx

is squared in a power sense.

Higher-order differentials extend the approximation. The second-order estimate of the change in

y

\Delta y \approx f'(x)\,dx + \frac{1}{2}f''(x)(dx)^2

This is the beginning of the Taylor expansion written in differential notation:

f(x + dx) \approx f(x) + f'(x)\,dx + \frac{1}{2}f''(x)(dx)^2 + \cdots

. Each additional term uses a higher-order derivative and a higher power of

dx

, improving the approximation at the cost of more computation.

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	NOT dx raised to a power in any new sense; just dx·dx
d²y / dx²	second derivative f''(x)	dividing d²y = f''(x)·(dx)² by (dx)²

Summary: What Differentials Are Good For

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

Differentials FAQ

Does dx mean d times x?

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?

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?

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 is accurate to five decimal places; a much larger step would not be.Read more →

Why does squaring a measurement double its relative error?

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?

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 →