What Is a Differential in Mathematics?

A differential is the infinitesimal (linearly approximated) change in a function’s value resulting from a small change in its input. In one variable, if y = f(x) is differentiable, its differential is dy = f′(x) dx; in several variables, the total differential is df = ∑i (∂f/∂x_i) dx_i. This captures the best linear approximation to a function near a point and underpins core ideas in calculus, error analysis, coordinate changes, and modern differential geometry.

Two Complementary Viewpoints

Mathematicians and scientists use “differential” in closely related but distinct ways. The following list outlines the main interpretations you are likely to encounter.

• Linear approximation (derivative as a map): At a point a, the differential Df(a) is a linear map that sends an input increment h to the leading-order output change Df(a)[h]. In one dimension this is dy ≈ f′(a) dx.
• Leibniz/infinitesimal notation: Treat dx and dy as infinitesimal quantities with dy = f′(x) dx, enabling algebraic manipulation in computations (heuristically justified by limits).
• Differential forms: For scalar-valued f, df is a 1-form (an element of the cotangent space). In coordinates, df = ∑i (∂f/∂x_i) dx_i, where the dx_i are basis 1-forms.

These views agree operationally: each describes the same first-order (linear) behavior of a differentiable function near a point, framed for different purposes (calculus, computation, or geometry).

Single-Variable Calculus: dy = f′(x) dx

In one dimension, the differential dy gives the linearized change in y when x changes by a small amount dx. It satisfies dy = f′(x) dx, so Δy ≈ dy when Δx is small. This is the backbone of tangent-line approximations, sensitivity analysis, and error estimates.

Illustrative example

Let y = x². Then dy = 2x dx. Near x = 3, a small increment dx = 0.01 gives dy ≈ 2·3·0.01 = 0.06, so y increases by about 0.06. The exact change is (3.01)² − 9 = 0.0601; the difference 0.0001 is a higher-order (quadratic) term ignored by the linear approximation.

Multivariable Functions: The Total Differential

For f(x₁, …, x_n), the total differential aggregates first-order effects from each variable:

df = ∑i (∂f/∂x_i) dx_i.

This formula is the multivariable analog of dy = f′(x) dx and is central in optimizing multivariable functions, propagating measurement errors, and changing coordinates.

Example: Computing df

If f(x, y) = x²y + sin y, then df = (2xy) dx + (x² + cos y) dy. For small (dx, dy), the change in f is approximated by substituting those increments into df.

How to Use Differentials for Approximation

The following steps summarize how differentials support quick, first-order estimates of changes in outputs based on small input changes.

1. Confirm differentiability of the function near the point of interest.
2. Compute the derivative (1D) or gradient/partial derivatives (multi-D).
3. Form the differential (dy = f′ dx or df = ∑i f_{x_i} dx_i).
4. Insert the small input changes (dx or dx_i) to estimate the output change (dy or df).
5. Assess whether higher-order terms might be significant for your application.

Following these steps yields fast, reliable first-order estimates; if accuracy demands, refine with higher-order terms or numerical methods.

Error Propagation and Units

Differentials quantify how measurement uncertainty in inputs affects uncertainty in outputs.

For y = f(x), an input uncertainty δx induces output uncertainty |δy| ≈ |f′(a)| |δx|. In multiple variables, |δf| ≈ |∑i (∂f/∂x_i) δx_i|, often summarized using gradients and norms. Differentials carry physical units: if x is in meters and f is in square meters, then df has units of area, consistent with dimensional analysis.

Change of Variables and “dx” in Integrals

In integration, the symbol dx plays a dual role: it signals the variable of integration and, under change of variables, contributes the Jacobian factor that rescales “volume.”

Key connections appear in these settings:

• Substitution in one dimension: u = g(x) with du = g′(x) dx; integrals transform via ∫ f(g(x)) g′(x) dx = ∫ f(u) du.
• Multivariable change of variables: dV transforms by |det J|, where J is the Jacobian of the transformation; symbolically, dx dy = |det J| du dv.
• Line/surface integrals: The element ds or dS arises from differentials of parametrizations and encodes geometric scaling.

While Leibniz-style manipulations with differentials are intuitive, the rigorous foundation comes from the substitution theorem and Jacobian determinant, which formalize how measures transform.

Differentials as 1-Forms and the Exterior Derivative

In differential geometry, df is a 1-form: a linear functional on tangent vectors. If x = (x₁, …, x_n), the dx_i form a basis of 1-forms, and df = ∑i f_{x_i} dx_i. The exterior derivative d generalizes this, sending k-forms to (k+1)-forms and obeying d² = 0. This viewpoint unifies many integration theorems (Green’s, Stokes’, divergence theorem) into a single framework.

Common Uses of Differentials

Differentials appear across mathematics, science, and engineering in the following ways.

• Linear approximations and tangent lines/planes.
• Uncertainty propagation and sensitivity analysis.
• Change of variables in integrals (Jacobian factors).
• Formulation of exact differential equations M dx + N dy = 0.
• Geometric calculus via differential forms and Stokes’ theorem.

Across these applications, the central idea is the same: differentials capture the leading-order behavior of change.

Caveats and Good Practice

The following points help avoid common mistakes when working with differentials.

• Do not confuse Δ (finite change) with d (infinitesimal/linearized change); Δf ≈ df only for sufficiently small inputs.
• Check differentiability; corners or discontinuities invalidate the linear approximation.
• Mind higher-order terms when changes are not tiny; linear models can under- or overestimate.
• Track units; differentials inherit units, aiding physical interpretation and consistency checks.
• In integrals, treat “dx” consistently: it indicates the variable of integration and transforms with the Jacobian under substitution.

Keeping these cautions in view ensures that differential-based calculations remain both accurate and interpretable.

Summary

A differential is the first-order, linearized change of a function with respect to small changes in its inputs: dy = f′(x) dx in one variable and df = ∑i (∂f/∂x_i) dx_i in several variables. It can be seen as a linear map (the derivative), an infinitesimal calculus tool (Leibniz notation), or a 1-form (in geometry). This concept enables approximations, error propagation, coordinate changes in integration, and the formulation of differential equations, all grounded in the idea that smooth functions are locally linear to first order.

What is the differential of 2x?

2
To find the derivative of 2x, we can use a well-known formula to make it a very simple process. The formula for the derivative of cx, where c is a constant, is given in the following image. Since the derivative of cx is c, it follows that the derivative of 2x is 2.

What is an example of a differential?

x2 + y2xy and xy + yx are examples of homogenous differential equations. y + x(dy/dx) = 0 is a homogenous differential equation of degree 1. x4 + y4(dy/dx) = 0 is a homogenous differential equation of degree 4.

What is meant by differential in maths?

Introduction. The term differential is used nonrigorously in calculus to refer to an infinitesimal (“infinitely small”) change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x).

How do you calculate a differential?

To find the differential dy for a function y = f(x), you first find the derivative of the function, f'(x), and then multiply it by dx, resulting in the formula dy = f'(x)dx. For a multivariate function f(x, y), the total differential df is found by calculating the partial derivatives with respect to each variable and summing their products with the corresponding differentials: df = (∂f/∂x)dx + (∂f/∂y)dy. 
For a single-variable function (y = f(x)) 

1. Find the derivative: Calculate the derivative of the function y with respect to x, which is denoted as f'(x) or dy/dx.
2. Multiply by dx: Multiply the resulting derivative by dx to get the differential dy.

Example:
For the function y = sin(x): 

1. Derivative: f'(x) = cos(x).
2. Differential: dy = cos(x) dx.

For a multi-variable function (f(x, y)) 

1. Find partial derivatives: Opens in new tabCalculate the partial derivative of the function with respect to x (treating y as a constant) and the partial derivative with respect to y (treating x as a constant).
2. Form the total differential: Opens in new tabAdd the product of the partial derivative with respect to x and dx to the product of the partial derivative with respect to y and dy.

Example:
For the function f(x, y) = x²y³: 

1. Partial derivatives:
   • ∂f/∂x = 2xy³ (treating y³ as a constant).
   • ∂f/∂y = x²(3y²) = 3x²y² (treating x² as a constant).
2. Total differential: df = 2xy³ dx + 3x²y² dy.