Linear operators in calculus

thumb|300px|A definite integral of a function can be represented as the signed area of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of f(x) between a and b is the yellow (−) area subtracted from the blue (+) area|alt=Definite integral example

In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. The derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a d

thumb|300px|The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high).

500px|thumb|upright=1.75|alt= A vector field with diverging vectors, a vector field with converging vectors, and a vector field with parallel vectors that neither diverge nor converge|The divergence of different vector fields. The divergence of vectors from point (x,y) equals the sum of the partial derivative-with-respect-to-x of the x-component and the partial derivative-with-respect-to-y of the y-component at that point: \nabla\!\cdot(\mathbf{V}(x,y)) = \frac{\partial\, {V_x(x,y){\partial{x+\frac{\partial\, {V_y(x,y){\partial{y

thumb|The slope field of F(x) = \frac{x^3}{3} - \frac{x^2}{2} - x + C, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant .

differential operator describing the rotation at a point in a 3D vector field

divergence of the gradient

notion in calculus

discrete analog of a derivative

mathematical operation

In mathematics, in the area of combinatorics and quantum calculus, the '''q-derivative, or Jackson derivative', is a q''-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration. For other forms of q-derivative, see .

In fractional calculus, an area of mathematical analysis, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by \mathbb{D}^q f is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several definitions of the differintegral.

the inverse of a finite difference