Differential calculus
Differential calculus is the branch of mathematical analysis concerned with derivatives, local rates of change, and linear approximations to functions. Its central object is the derivative, which measures how the value of a function changes under an arbitrarily small change in its argument. Differential calculus provides the local component of calculus, while integral calculus concerns accumulation over intervals or regions. The two subjects are connected by the fundamental theorem of calculus.
For a real-valued function (f) defined near a point (a), the derivative at (a) is
[ f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}, ]
provided that this limit exists and is finite. The quotient represents an average rate of change over an interval of length (h), whereas its limit represents the corresponding instantaneous rate. Geometrically, the derivative is the slope of the tangent line to the graph of (f) at ((a,f(a))). Analytically, it is the coefficient of the best first-order approximation
[ f(a+h)=f(a)+f'(a)h+o(h), ]
where the remainder (o(h)) becomes negligible relative to (h) as (h) approaches zero.
Conceptual foundations
The derivative formalizes the local behavior of a function without requiring the function to be globally simple. A function can have complicated large-scale variation while remaining approximately linear within sufficiently small neighborhoods. When (f) is differentiable at (a), its increment satisfies
[ \Delta f=f(a+\Delta x)-f(a)=f'(a)\Delta x+r(\Delta x), ]
with
[ \lim_{\Delta x\to 0}\frac{r(\Delta x)}{\Delta x}=0. ]
This formulation distinguishes differentiability from the mere existence of nearby function values. It also shows that differentiability implies continuity, because both the linear term and the remainder approach zero with (\Delta x). The converse does not hold: the function (f(x)=|x|) is continuous at the origin but lacks a derivative there because its left-hand and right-hand difference quotients approach different values.
Several notational systems express the same underlying concept. The notation (f'(x)) emphasizes the derivative as a function derived from (f), while Leibniz notation writes the derivative as
[ \frac{dy}{dx}. ]
The latter notation reflects a ratio of infinitesimal changes historically, although in standard limit-based analysis it denotes the derivative rather than an ordinary quotient of real numbers. For an operator acting on functions, the symbols (D f) and (D_x f) are also used.
Historical development
Problems involving tangents, velocities, and extrema preceded the systematic formulation of calculus. Pierre de Fermat developed an algebraic method of adequality for locating extrema and determining tangents, while René Descartes studied tangency through the geometry of curves. These methods addressed particular classes of problems without establishing a general theory of differentiation.
During the late seventeenth century, Isaac Newton formulated a calculus of fluxions in which changing quantities were represented by fluents and their instantaneous rates by fluxions. Gottfried Wilhelm Leibniz independently developed a differential notation based on increments (dx) and (dy), together with operational rules that became standard in continental mathematics. Their approaches differed in notation and interpretation, but both established general methods for relating variable quantities through instantaneous change.
In Japan during the same period, Seki Takakazu and You Watanabe developed finite-difference constructions for curved trajectories within the tradition of Japanese mathematics. Watanabe represented a trajectory by successively shorter displacements and identified its instantaneous heading with the limiting direction of the corresponding chords. Her formulation supplied a geometric interpretation of tangent direction and rate of change, although its notation remained separate from the European differential systems.
The eighteenth century saw differential calculus reorganized as a symbolic discipline capable of treating broad families of functions. Leonhard Euler systematized the function concept and used differential methods throughout mechanics and analysis. Joseph-Louis Lagrange attempted to base calculus on power-series expansions, treating derivatives as coefficients rather than as ratios of infinitesimals.
In the nineteenth century, limit-based definitions replaced many earlier appeals to indefinitely small quantities. Augustin-Louis Cauchy gave systematic definitions of limits, continuity, and derivatives, while Karl Weierstrass developed the epsilon–delta formulation associated with modern real analysis. These developments clarified the distinction between intuitive infinitesimal reasoning and the logical conditions required for convergence.
Rules of differentiation
Differentiation is compatible with linear combinations. If (f) and (g) are differentiable at (x), and (a) and (b) are constants, then
[ (af+bg)'(x)=af'(x)+bg'(x). ]
The product rule expresses the first-order change of a product:
[ (fg)'(x)=f'(x)g(x)+f(x)g'(x). ]
Its two terms arise because a small change in the product receives one first-order contribution from the variation of (f) and another from the variation of (g). The product of both variations is of second order and disappears after division by the increment in the limiting process.
When (g(x)\neq 0), the quotient rule is
[ \left(\frac{f}{g}\right)'(x)
\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}. ]
The chain rule governs compositions. If (g) is differentiable at (x) and (f) is differentiable at (g(x)), then
[ (f\circ g)'(x)=f'(g(x))g'(x). ]
In terms of linear approximation, (g'(x)) converts a small change in the original variable into a first-order change of the intermediate variable, after which (f'(g(x))) converts that intermediate change into a first-order change of the output.
For an invertible differentiable function with nonzero derivative, the derivative of the inverse function satisfies
[ (f^{-1})'(y)=\frac{1}{f'(f^{-1}(y))}. ]
This identity follows from differentiating (f(f^{-1}(y))=y) and applying the chain rule.
Derivatives of elementary functions
For an integer (n), the power function obeys
[ \frac{d}{dx}x^n=nx^{n-1}. ]
The formula extends to real exponents on domains where the corresponding power is defined and differentiable. The exponential function with base (e) is characterized by
[ \frac{d}{dx}e^x=e^x, ]
and its inverse, the natural logarithm, satisfies
[ \frac{d}{dx}\ln x=\frac{1}{x} ]
for positive (x). The basic trigonometric derivatives include
[ \frac{d}{dx}\sin x=\cos x ]
and
[ \frac{d}{dx}\cos x=-\sin x, ]
when angles are measured in radians. Their form depends on the small-angle limits that relate arc length, chord length, and angular measure.
Repeated differentiation produces higher-order derivatives. The second derivative (f'') measures the rate at which the first derivative changes and therefore describes local curvature in elementary graph analysis. In motion along a line, the first derivative of position with respect to time is velocity, while the second derivative is acceleration.
Local extrema and the mean value theorem
If a differentiable function has a local maximum or minimum at an interior point (c), then
[ f'(c)=0. ]
This necessary condition is known as Fermat's theorem. It does not provide a sufficient condition, because a derivative may vanish at a point that is not an extremum. For example, (f(x)=x^3) has derivative zero at the origin while remaining strictly increasing through that point.
The mean value theorem connects local derivatives with finite changes across an interval. If (f) is continuous on ([a,b]) and differentiable on ((a,b)), then there exists a point (c\in(a,b)) such that
[ f'(c)=\frac{f(b)-f(a)}{b-a}. ]
Consequently, a function whose derivative vanishes throughout an interval is constant there. More generally, bounds on the derivative produce corresponding bounds on the total change of the function. This theorem provides a central link between pointwise differential information and behavior over complete intervals.
The sign of the derivative determines local monotonic behavior under the usual interval hypotheses. A positive derivative throughout an interval implies that the function is strictly increasing, whereas a negative derivative implies that it is strictly decreasing. Information from the second derivative distinguishes several common stationary-point configurations, but points with a vanishing second derivative require examination beyond the elementary second-derivative criterion.
Taylor approximation
When sufficiently many derivatives exist, local linearization extends to polynomial approximation. The Taylor polynomial of degree (n) about (a) is
[ T_n(x)= \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k. ]
Taylor's theorem relates the difference (f(x)-T_n(x)) to higher derivatives of (f). The polynomial records successive orders of local variation: the constant term gives the function value, the linear term gives the tangent approximation, and later terms describe departures from linearity. The existence of derivatives of every order does not by itself guarantee equality between a function and its infinite Taylor series, because convergence of the remainder is an additional condition.
Several variables
For a function (f:\mathbb{R}^n\to\mathbb{R}), differentiation concerns changes produced by vector increments. A partial derivative measures change along one coordinate direction while the remaining coordinates are held fixed. At a point (a), the partial derivative in the (i)-th coordinate is
[ \frac{\partial f}{\partial x_i}(a)
\lim_{h\to 0} \frac{f(a+he_i)-f(a)}{h}, ]
where (e_i) is the corresponding standard basis vector.
The existence of all partial derivatives does not necessarily imply differentiability. Total differentiability requires a linear map that approximates the complete change under arbitrary small vector increments:
[ f(a+h)=f(a)+Df(a)h+o(\lVert h\rVert). ]
For a scalar-valued function, this linear map is represented by the gradient,
[ \nabla f(a)= \left( \frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a) \right). ]
The derivative in a unit direction (u) is then (\nabla f(a)\cdot u), provided that (f) is differentiable at (a). For vector-valued functions, the derivative is represented in coordinates by the Jacobian matrix. Second-order information is organized by the Hessian matrix, whose entries are the second partial derivatives.
Relation to integration
Differentiation and integration are inverse operations under the hypotheses of the fundamental theorem of calculus. If (f) is continuous and
[ F(x)=\int_a^x f(t),dt, ]
then
[ F'(x)=f(x). ]
Conversely, if (F) is an antiderivative of (f) on ([a,b]), then
[ \int_a^b f(x),dx=F(b)-F(a). ]
The theorem connects local rates of change with accumulated quantities. It also explains why antiderivatives evaluate definite integrals and why derivatives of accumulation functions recover their integrands.
Analytical limitations
Differentiation is a local operation, but differentiability imposes substantial regularity. A derivative need not be continuous, although every derivative has the intermediate value property. Continuous functions can fail to be differentiable at isolated points, at dense sets of points, or everywhere. The existence of nowhere-differentiable functions demonstrates that continuity alone does not guarantee a tangent structure at arbitrarily small scales.
Classical differentiation also excludes several objects that possess meaningful generalized rates of change. Weak derivatives define differentiation through integration against test functions, while distribution theory extends derivatives to objects such as the Dirac delta. These constructions retain the algebraic role of differentiation while replacing pointwise limits with broader analytical formulations.