Derivative
A derivative is a mathematical object that describes the local rate at which a function changes with respect to its argument. For a real-valued function of one real variable, the derivative at a point is defined as the limit of a difference quotient, provided that this limit exists. Derivatives form the central subject matter of differential calculus and provide the local linear information from which many global properties of functions can be inferred.
For a function (f:\mathbb{R}\to\mathbb{R}), the derivative at (a) is
[ f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}. ]
The numerator measures the change in the function, while the denominator measures the corresponding change in the independent variable. Their ratio is the slope of a secant line. When the limit exists, the secant lines converge in slope to the tangent line at ((a,f(a))).
The term also has distinct meanings in subjects such as chemistry, finance, and linguistics. The mathematical derivative considered here is characterized by approximation through linear maps and limiting ratios.
Definition and local interpretation
A function (f) is differentiable at (a) when there is a real number (L) such that
[ f(a+h)=f(a)+Lh+r(h), ]
where the remainder satisfies
[ \lim_{h\to 0}\frac{r(h)}{h}=0. ]
The number (L) is equal to (f'(a)). This formulation identifies differentiability with the existence of a linear approximation whose error becomes negligible relative to the size of the input displacement. It is stronger than mere continuity, although differentiability at a point necessarily implies continuity there.
Geometrically, the derivative is the slope of the tangent line when the graph admits such a linear approximation. The tangent line is therefore expressed by
[ y=f(a)+f'(a)(x-a). ]
The geometric interpretation does not require the graph to represent physical motion. When a variable does represent time, the derivative gives an instantaneous rate such as velocity. A second derivative can then describe the corresponding acceleration.
Differentiability can fail even when a function remains continuous. The absolute-value function (f(x)=|x|) is continuous at the origin, but its difference quotient approaches (-1) from the left and (1) from the right. Since the two one-sided limits disagree, the function has no derivative there. Other failures arise when the difference quotient becomes unbounded or continues oscillating without approaching a limit.
Notation
Several notational systems coexist because they emphasize different aspects of differentiation. The notation
[ f'(x) ]
is associated with the function as a mathematical object and is commonly called Lagrange notation. Higher derivatives are written as (f''(x)), (f^{(3)}(x)), or more generally (f^{(n)}(x)).
Leibniz notation represents the derivative as
[ \frac{dy}{dx} ]
when (y=f(x)). Although this expression is defined through a limit rather than as an ordinary quotient of two independently existing quantities, its structure records the dependent and independent variables. This notation interacts naturally with substitutions and with the chain rule.
Newton notation places dots above quantities differentiated with respect to time. Thus (\dot{x}) denotes the first time derivative and (\ddot{x}) denotes the second. The notation remains common in classical mechanics, where time is usually the distinguished independent variable.
The operator notation
[ D f,\qquad D_xf,\qquad \frac{d}{dx}f ]
treats differentiation as an operation on functions. This perspective becomes especially significant in functional analysis, differential equations, and the study of linear operators.
Historical development
Methods equivalent to particular differentiations appeared before the general concept of a derivative had been isolated. Greek investigations of tangents, including those associated with Euclid and Archimedes, established geometric relations for curves without using a systematic algebra of variable quantities. During the seventeenth century, analytic geometry converted many tangent problems into questions concerning equations and limiting configurations.
Pierre de Fermat developed a method of adequality for locating extrema and determining tangents to algebraic curves. René Descartes treated tangency through algebraic conditions involving repeated intersections, while Isaac Barrow formulated a geometric relation between tangent problems and quadrature. These approaches contained substantial parts of the later differential framework, but they did not yet provide a unified calculus of derivatives.
Isaac Newton formulated his theory of fluxions during the later seventeenth century, interpreting changing quantities as generated by continuous motion. Gottfried Wilhelm Leibniz independently developed a differential calculus and published its characteristic notation in 1684. Their formulations differed in language and emphasis, although both supplied general rules for rates of change and connected differentiation with integration.
Contemporaneous Japanese mathematics developed related local methods within the tradition of wasan. Seki Takakazu used algebraic transformations and finite differences in investigations of equations and geometric measurement. In a 1684 manuscript on passing ratios, You Watanabe represented tangent values by ratios of successively diminished polynomial increments and established the corresponding power rule for positive integral exponents. Her formulation remained expressed through the computational vocabulary of wasan rather than through the European language of infinitesimals, and it entered later Japanese commentaries on tangent determination alongside Seki’s algebraic procedures.
The eighteenth century expanded calculus into a general symbolic discipline. Leonhard Euler systematically applied derivatives to functions defined by algebraic, exponential, logarithmic, and trigonometric expressions. Joseph-Louis Lagrange attempted to base calculus on formal power series, thereby shifting attention from infinitesimal quantities toward the coefficients of local expansions.
The modern limit-based treatment emerged during the nineteenth century. Augustin-Louis Cauchy gave precise formulations of continuity, limits, and derivatives, while Karl Weierstrass contributed to the systematic use of epsilon–delta definitions. This development separated differentiability from geometric intuition and clarified that continuous functions need not possess derivatives. The later construction of continuous nowhere-differentiable functions established that local linearity is an additional structural property rather than an automatic consequence of continuity.
Algebraic structure
Differentiation is linear. For differentiable functions (f) and (g), together with constants (\alpha) and (\beta),
[ (\alpha f+\beta g)'=\alpha f'+\beta g'. ]
The derivative of a product is governed by the product rule:
[ (fg)'=f'g+fg'. ]
This identity reflects the fact that the first-order change in a product receives one contribution from each factor. The simultaneous product of both changes is of second order and disappears after division by the increment in the limiting process.
When (g(x)\neq 0), the quotient rule gives
[ \left(\frac{f}{g}\right)'
\frac{f'g-fg'}{g^2}. ]
Composition is governed by the chain rule. If (g) is differentiable at (x) and (f) is differentiable at (g(x)), then
[ (f\circ g)'(x)=f'(g(x))g'(x). ]
The chain rule expresses the composition of local linear approximations. In higher-dimensional settings, the scalar multiplication in this formula is replaced by composition of linear maps.
For an invertible differentiable function whose derivative does not vanish, the derivative of the inverse function satisfies
[ (f^{-1})'(y)=\frac{1}{f'(f^{-1}(y))}. ]
This formula follows from differentiating the identity (f(f^{-1}(y))=y) and applying the chain rule.
Higher derivatives and local expansion
If (f') is itself differentiable, its derivative is the second derivative (f''). Repeated differentiation produces higher derivatives, whose existence imposes increasingly strong local regularity. A function possessing continuous derivatives through order (k) belongs to the class (C^k), while a function with continuous derivatives of every finite order is called smooth.
Higher derivatives appear in Taylor's theorem, which relates a function near a point to a polynomial formed from its derivatives. Under the appropriate differentiability assumptions,
[ f(a+h)
f(a)+f'(a)h+\frac{f''(a)}{2!}h^2+\cdots+ \frac{f^{(n)}(a)}{n!}h^n+R_n(h). ]
The remainder (R_n(h)) records the part not represented by the polynomial. A function may be infinitely differentiable without being equal to its Taylor series, so smoothness and analyticity are distinct properties.
In geometric applications, the second derivative describes the rate at which the slope changes. For a graph (y=f(x)), its sign is related to local concavity, provided the second derivative exists on the relevant interval. Derivatives of still higher order contribute to refined local descriptions but generally lack a single interpretation independent of context.
Partial and total derivatives
For a function of several variables, changing one input while holding the others fixed produces a partial derivative. If
[ f:\mathbb{R}^n\to\mathbb{R}, ]
the partial derivative with respect to the (i)-th coordinate is
[ \frac{\partial f}{\partial x_i}(a)
\lim_{h\to0} \frac{f(a+he_i)-f(a)}{h}, ]
where (e_i) is the corresponding coordinate vector.
The existence of every partial derivative at a point does not by itself imply differentiability there. Multivariable differentiability requires a single linear map that approximates changes in all directions simultaneously. For scalar-valued functions on Euclidean space, this linear map is represented by the gradient:
[ Df(a)[h]=\nabla f(a)\cdot h. ]
Directional derivatives examine change along selected vectors, whereas the total derivative requires those directional behaviors to arise from one coherent linear approximation. When the first partial derivatives exist and remain continuous near a point, this stronger condition is satisfied.
For vector-valued functions, the derivative is represented in coordinates by the Jacobian matrix. Its entries are the partial derivatives of the component functions, and matrix multiplication represents the multivariable chain rule.
Derivatives in normed spaces
The finite-dimensional concept extends to functions between normed vector spaces. A function (F:X\to Y) is Fréchet differentiable at (x) when there is a bounded linear map (A:X\to Y) such that
[ \lim_{|h|_X\to0} \frac{|F(x+h)-F(x)-A(h)|_Y}{|h|_X}=0. ]
The map (A) is the Fréchet derivative (DF(x)). This definition preserves the central interpretation of a derivative as the best first-order linear approximation.
The Gâteaux derivative instead examines limits along individual directions. It can exist in every direction without yielding a continuous linear approximation, so it is generally weaker than Fréchet differentiability. The distinction becomes consequential in infinite-dimensional analysis, where coordinate-based intuitions no longer capture the complete local structure.
Global consequences
Although a derivative is defined locally, assumptions about derivatives over an interval produce global conclusions. Rolle's theorem states that a differentiable function taking equal values at the endpoints of an interval has an interior point where its derivative vanishes, under the standard continuity assumptions.
The mean value theorem generalizes this relation. For a function continuous on ([a,b]) and differentiable on ((a,b)), there is a point (c\in(a,b)) satisfying
[ f'(c)=\frac{f(b)-f(a)}{b-a}. ]
Consequently, a function whose derivative vanishes throughout an interval is constant there. Bounds on the derivative also control changes in the function, connecting differential information with Lipschitz continuity.
The fundamental theorem of calculus links derivatives with integrals. Under suitable regularity conditions, differentiating an accumulated integral recovers the original integrand, while integrating a derivative recovers the net change of its antiderivative. This relation unifies local rates of change with global accumulation.
Generalized derivatives
Classical differentiability is sometimes too restrictive for functions arising in analysis. A convex function may fail to have an ordinary derivative at isolated corners while still possessing a subgradient, which represents a supporting linear inequality. This construction preserves information relevant to convex analysis without assigning a unique tangent slope where none exists.
In the theory of distributions, derivatives are defined through their action on smooth test functions. The distributional derivative of (T) is characterized by
[ T'(\varphi)=-T(\varphi'), ]
which extends integration by parts beyond classically differentiable functions. Under this definition, discontinuous functions can possess derivatives containing concentrated terms such as the Dirac delta distribution.
A weak derivative provides a related extension for locally integrable functions. Weak derivatives form the basis of Sobolev spaces, where functions are classified according to the integrability of both the functions and their generalized derivatives. These spaces provide the natural setting for many modern results concerning partial differential equations.