In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input.
Leibniz notation, named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas prime notation is written by adding a prime mark.
The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.
It can be calculated in terms of the partial derivatives with respect to the independent variables.
For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function
[8] In order to make this intuition rigorous, a system of rules for manipulating infinitesimal quantities is required.
[9] The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.
The application of hyperreal numbers to the foundations of calculus is called nonstandard analysis.
This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the
denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real.
Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist.
Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points.
Informally, this means that hardly any random continuous functions have a derivative at even one point.
[18] This derivative can alternately be treated as the application of a differential operator to a function,
[19] Unlike some alternatives, Leibniz notation involves explicit specification of the variable for differentiation, in the denominator, which removes ambiguity when working with multiple interrelated quantities.
The derivative of a composed function can be expressed using the chain rule: if
[20] Another common notation for differentiation is by using the prime mark in the symbol of a function
This notation is used exclusively for derivatives with respect to time or arc length.
[25] However, the dot notation becomes unmanageable for high-order derivatives (of order 4 or more) and cannot deal with multiple independent variables.
Another notation is D-notation, which represents the differential operator by the symbol
[26] To indicate a partial derivative, the variable differentiated by is indicated with a subscript, for example given the function
Higher partial derivatives can be indicated by superscripts or multiple subscripts, e.g.
Higher order derivatives are the result of differentiating a function repeatedly.
Suppose that a function represents the position of an object at the time.
Partial derivatives are used in vector calculus and differential geometry.
To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".
is differentiable at every point in some domain, then the gradient is a vector-valued function
, no single directional derivative can give a complete picture of the behavior of
The total derivative gives a complete picture by considering all directions at once.