Differential calculus

The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.

In mathematics, differential calculus is a subfield of calculus[1] concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve.[2]

The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.

Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.

Differentiation has applications to nearly all quantitative disciplines. For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. The derivative of the momentum of a body with respect to time equals the force applied to the body; rearranging this derivative statement leads to the famous F = ma equation associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories.

Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra.


The tangent line at (x,f(x))
The derivative at different points of a differentiable function

Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, there is a corresponding value of y. This relationship can be written as y = f(x). If f(x) is the equation for a straight line (called a linear equation), then there are two real numbers m and b such that y = mx + b. In this "slope-intercept form", the term m is called the slope and can be determined from the formula:

where the symbol Δ (the uppercase form of the Greek letter delta) is an abbreviation for "change in". It follows that Δy = m Δx.

A general function is not a line, so it does not have a slope. Geometrically, the derivative of f at the point x = a is the slope of the tangent line to the function f at the point a (see figure). This is often denoted f ′(a) in Lagrange's notation or dy/dx|x = a in Leibniz's notation. Since the derivative is the slope of the linear approximation to f at the point a, the derivative (together with the value of f at a) determines the best linear approximation, or linearization, of f near the point a.

If every point a in the domain of f has a derivative, there is a function that sends every point a to the derivative of f at a. For example, if f(x) = x2, then the derivative function f ′(x) = dy/dx = 2x.

A closely related notion is the differential of a function. When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once. Taking the best linear approximation in a single direction determines a partial derivative, which is usually denoted y/x. The linearization of f in all directions at once is called the total derivative.

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Simple English: Differential calculus
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