Differential of a function

Part of a series of articles about
Calculus
${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds
Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions (Heaviside's method) Changing order Reduction formulae Differentiating under the integral sign Risch algorithm
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence generalized Stokes Helmholtz decomposition
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Advanced Calculus on Euclidean space Generalized functions Limit of distributions
Specialized Fractional Malliavin Stochastic Variations
Miscellanea Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis
v t e

In calculus, the differential represents the principal part of the change in a function ${\displaystyle y=f(x)}$ with respect to changes in the independent variable. The differential ${\displaystyle dy}$ is defined by $${\displaystyle dy=f'(x)\,dx,}$$ where ${\displaystyle f'(x)}$ is the derivative of f with respect to ${\displaystyle x}$, and ${\displaystyle dx}$ is an additional real variable (so that ${\displaystyle dy}$ is a function of ${\displaystyle x}$ and ${\displaystyle dx}$). The notation is such that the equation

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

holds, where the derivative is represented in the Leibniz notation ${\displaystyle dy/dx}$, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes

$${\displaystyle df(x)=f'(x)\,dx.}$$

The precise meaning of the variables ${\displaystyle dy}$ and ${\displaystyle dx}$ depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. Traditionally, the variables ${\displaystyle dx}$ and ${\displaystyle dy}$ are considered to be very small (infinitesimal), and this interpretation is made rigorous in non-standard analysis.