Differential calculus

Source: Wikipedia. Pages: 87. Chapters: Adjoint equation, Automatic differentiation, Benjamin-Bona-Mahony equation, Binomial differential equation, Checkpointing scheme, Darboux derivative, Deal.II, Derivative of a constant, Difference quotient, Differentiable function, Differential (mathematics), Differential algebraic equation, Differential calculus over commutative algebras, Differential coefficient, Differential of a function, Differentiation in Fréchet spaces, Differentiation of trigonometric functions, Differentiation rules, Differentiation under the integral sign, Directional derivative, Domain-straightening theorem, Euler-Poisson-Darboux equation, Faà di Bruno's formula, Fermat's theorem (stationary points), Flat function, Functional derivative, Fundamental increment lemma, Generalizations of the derivative, Gradient, Hyperbolic angle, Implicit function, Infinitely near point, Inflection point, Institutiones calculi differentialis, Jacobian matrix and determinant, Laplace transform applied to differential equations, Left and right derivative, Leibniz's notation, Limit (mathematics), Linearity of differentiation, Linearization, Linear approximation, Lin-Tsien equation, Logarithmic derivative, Logarithmic differentiation, Method of Fluxions, Metric derivative, Newton's notation, Notation for differentiation, Numerical differentiation, Ordinary differential equation, Parametric derivative, Q-derivative, Quantum calculus, Reduced derivative, Second derivative, Second derivative test, Semi-differentiability, Skew gradient, Symmetrically continuous function, Symmetric derivative, Time derivative, Time evolution of integrals, Total derivative. Excerpt: In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. Informally, the derivative is the ratio of the infinitesimal change of the output over the infinitesimal change of the input producing that change of output. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function. The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus. At each point, the derivative of is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where a green line appears, negative where a red line appears, and zero where a black line appears.Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more pr...