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Differentials



The Differential dx
The Differential dy
dy Versus Δy
Linear Approximation
Error Estimation
Differentials of Multiple Variables
Differentials and Leibniz Notation
Higher-Order Differentials
Summary: What Differentials Are Good For



Separating dy and dx


The derivative dydx\frac{dy}{dx} is defined as a limit—a single object, not a fraction. But Leibniz notation invites the question: can dydy and dxdx stand on their own? Differentials answer yes, with precise meaning. The symbol dxdx represents an independent small change in xx, and dy=f′(x) dxdy = f'(x)\,dx represents the corresponding change in yy 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— dxdx is a free increment; dy=f′(x)⋅dxdy = f'(x) \cdot dx is the tangent-line response
Derivative— dy/dxdy/dx separated into two independent quantities
Partial Derivative— extends differentials to functions of several variables
Tangent Line— the differential dydy follows the tangent line, not the curve

See All Calculus Definitions →


The Differential dy


Given y=f(x)y = f(x) where ff is differentiable, the differential of yy is defined as

Differential
dy=f′(x) dxdy = f'(x)\, dx
Learn more about this formula: Differential →


The differential dydy depends on two things: the point xx (which determines the slope f′(x)f'(x)) and the increment dxdx (which determines the scale). For fixed xx, dydy is a linear function of dxdx—doubling dxdx doubles dydy.

Geometrically, dydy is the vertical change along the tangent line at xx when the horizontal position shifts by dxdx. The tangent line rises (or falls) at rate f′(x)f'(x), so a horizontal shift of dxdx produces a vertical shift of f′(x)⋅dxf'(x) \cdot dx.

For a linear function f(x)=mx+bf(x) = mx + b, the differential dy=m⋅dxdy = m \cdot dx equals the actual change in ff exactly, because the tangent line to a linear function is the function itself. For nonlinear functions, dydy is an approximation—exact in the limit as dx→0dx \to 0, and increasingly approximate as dxdx grows.

dy Versus Δy


Two quantities measure the change in yy when xx changes by dxdx:

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


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


The actual change Δy\Delta y follows the curve. The differential dydy follows the tangent line. The difference Δy−dy\Delta y - dy is the error introduced by the linear approximation.

For small dxdx, the error Δy−dy\Delta y - dy is much smaller than dxdx itself. Precisely, lim⁡dx→0Δy−dydx=0\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 dxdx, the gap between Δy\Delta y and dydy widens. The tangent line is a local tool: reliable near xx, increasingly unreliable far from it. The curvature of ff—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

Linear Approximation


The differential formula rearranges into an approximation for function values:

Linear Approximation
f(x)≈f(a)+f′(a)(x−a)Δy≈f′(a) Δxf(x) \approx f(a) + f'(a)(x - a) \qquad \Delta y \approx f'(a)\, \Delta x
Learn more about this formula: Linear Approximation →


This is the linearization of ff at xx. Given a known value f(x)f(x) and the slope f′(x)f'(x), the tangent line estimates ff at nearby points.

To approximate 4.03\sqrt{4.03}: take f(x)=xf(x) = \sqrt{x}, x=4x = 4, dx=0.03dx = 0.03. Then f(4)=2f(4) = 2 and f′(x)=12xf'(x) = \frac{1}{2\sqrt{x}}, so f′(4)=14f'(4) = \frac{1}{4}. The estimate is 2+14(0.03)=2.00752 + \frac{1}{4}(0.03) = 2.0075. The actual value is 4.03≈2.00749...\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)L(x) = f(a) + f'(a)(x - a), the tangent line at x=ax = a used as a function. For xx near aa, L(x)≈f(x)L(x) \approx f(x). The quality of the approximation depends on two factors: the size of ∣x−a∣|x - a| and the magnitude of f′′f'' near aa, which controls how quickly the curve diverges from its tangent.

Error Estimation


If a quantity xx is measured with error dxdx, the computed value f(x)f(x) inherits an error approximately equal to the differential:

absolute error in f≈∣dy∣=∣f′(x)∣⋅∣dx∣\text{absolute error in } f \approx |dy| = |f'(x)| \cdot |dx|


The derivative acts as an error amplification factor. Where ∣f′(x)∣|f'(x)| is large, small measurement errors in xx produce large errors in f(x)f(x). Where ∣f′(x)∣|f'(x)| is small, errors are suppressed.

The relative error normalizes by the function value:

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


The ratio f′(x)f(x)=ddx[ln⁡∣f(x)∣]\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
ddx[ln⁡f(x)]=f′(x)f(x)\frac{d}{dx}[\ln f(x)] = \frac{f'(x)}{f(x)}
Learn more about this formula: Logarithmic Derivative →


For f(x)=xnf(x) = x^n: f′(x)f(x)=nx\frac{f'(x)}{f(x)} = \frac{n}{x}, so the relative error in xnx^n is nn times the relative error in xx. 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)z = f(x, y), the total differential is

Total Differential
dz=∂z∂x dx+∂z∂y dydz = \frac{\partial z}{\partial x}\, dx + \frac{\partial z}{\partial y}\, dy
Learn more about this formula: Total Differential →


Each partial derivative weights the contribution of the corresponding variable's error. The total differential estimates how zz responds to simultaneous small changes in all inputs.

For the area of a rectangle A=lwA = lw: dA=w dl+l dwdA = w\,dl + l\,dw. If l=10±0.1l = 10 \pm 0.1 and w=5±0.05w = 5 \pm 0.05, then dA=5(0.1)+10(0.05)=1.0dA = 5(0.1) + 10(0.05) = 1.0. The area is approximately 50±1.050 \pm 1.0.

For the volume of a cylinder V=πr2hV = \pi r^2 h: dV=2πrh dr+πr2 dhdV = 2\pi r h\,dr + \pi r^2\,dh. The radius error is amplified by 2πrh2\pi r h and the height error by πr2\pi r^2. Since rr 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 dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}. In differential form: dy=dydu dudy = \frac{dy}{du}\,du and du=dudx dxdu = \frac{du}{dx}\,dx. Substituting the second into the first gives dy=dydu⋅dudx dxdy = \frac{dy}{du} \cdot \frac{du}{dx}\,dx, which is dy=dydx dxdy = \frac{dy}{dx}\,dx. The intermediate variable dudu cancels as though these were fractions—and with differentials, they are.

Integration notation ∫f(x) dx\int f(x)\,dx also uses the differential dxdx meaningfully. The substitution rule u=g(x)u = g(x), du=g′(x) dxdu = g'(x)\,dx replaces dxdx with an expression involving dudu—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 dydx=g(x)h(y)\frac{dy}{dx} = g(x)h(y) as dyh(y)=g(x) dx\frac{dy}{h(y)} = g(x)\,dx and integrating both sides—depends on treating dydy and dxdx 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


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

d2y=f′′(x)⋅(dx)2d^2 y = f''(x) \cdot (dx)^2


The notation d2y/dx2d^2 y / dx^2 for the second derivative arises from dividing both sides by (dx)2(dx)^2:

d2ydx2=d2y(dx)2=f′′(x)\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 d2yd^2 y counts the number of times the differential operator dd has been applied. The square on dx2dx^2 reflects that dxdx appears as a factor twice—not that dxdx is squared in a power sense.

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

Δy≈f′(x) dx+12f′′(x)(dx)2\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)≈f(x)+f′(x) dx+12f′′(x)(dx)2+⋯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 dxdx, 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) dxdy = 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 dxdx and dydy as independent quantities rather than parts of an unbreakable dy/dxdy/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