The logarithmic derivative of a function is defined as the derivative of the logarithm of a function. For example, the digamma function is defined as the logarithmic derivative of the gamma function,
Faà di Bruno's formula gives an explicit equation for the th derivative of the composition . If and are functions for which all necessary derivatives are defined, then(1)where and the sum is over all partitions of , i.e., values of , ..., such that(2)(Roman 1980).It can also be expressed in terms of Bell polynomial as(3)(M. Alekseyev, pers. comm., Nov. 3, 2006).Faà di Bruno's formula can be cast in a framework that is a special case of a Hopf algebra (Figueroa and Gracia-Bondía 2005).The first few derivatives for symbolic and are given by(4)(5)(6)
A smooth function is a function that has continuous derivatives up to some desired order over some domain. A function can therefore be said to be smooth over a restricted interval such as or . The number of continuous derivatives necessary for a function to be considered smooth depends on the problem at hand, and may vary from two to infinity. A function for which all orders of derivatives are continuous is called a C-infty-function.
A smooth curve is a curve which is a smooth function, where the word "curve" is interpreted in the analytic geometry context. In particular, a smooth curve is a continuous map from a one-dimensional space to an -dimensional space which on its domain has continuous derivatives up to a desired order.
The directional derivative is the rate at which the function changes at a point in the direction . It is a vector form of the usual derivative, and can be defined as(1)(2)where is called "nabla" or "del" and denotes a unit vector.The directional derivative is also often written in the notation(3)(4)where denotes a unit vector in any given direction and denotes a partial derivative.Let be a unit vector in Cartesian coordinates, so(5)then(6)
A vector derivative is a derivative taken with respect to a vector field. Vector derivatives are extremely important in physics, where they arise throughout fluid mechanics, electricity and magnetism, elasticity, and many other areas of theoretical and applied physics.The following table summarizes the names and notations for various vector derivatives.symbolvector derivativegradientLaplacian or vector Laplacian or directional derivativedivergencecurlconvective derivativeVector derivatives can be combined in different ways, producing sets of identities that are also very important in physics.Vector derivative identities involving the curl include(1)(2)(3)(4)(5)In Cartesian coordinates(6)(7)In spherical coordinates,(8)(9)(10)Vector derivative identities involving the divergenceinclude(11)(12)(13)(14)(15)In Cartesian coordinates,(16)(17)(18)(19)(20)(21)In spherical coordinates,(22)(23)(24)(25)(26)(27)By..
The operator representing the computation of a derivative,(1)sometimes also called the Newton-Leibniz operator. The second derivative is then denoted , the third , etc. The integral is denoted .The differential operator satisfies the identity(2)where is a Hermite polynomial (Arfken 1985, p. 718), where the first few cases are given explicitly by(3)(4)(5)(6)(7)(8)The symbol can be used to denote the operator(9)(Bailey 1935, p. 8). A fundamental identity for this operator is given by(10)where is a Stirling number of the second kind (Roman 1984, p. 144), giving(11)(12)(13)(14)and so on (OEIS A008277). Special cases include(15)(16)(17)A shifted version of the identity is given by(18)(Roman 1984, p. 146).
A fractional derivative of order 1/2. The semiderivative of is given byso the semiderivative of the constant function is given by
The second fundamental theorem of calculus holds for a continuous function on an open interval and any point in , and states that if is defined by the integral (antiderivative)thenat each point in , where is the derivative of .
The Leibniz integral rule gives a formula for differentiation of a definiteintegral whose limits are functions of the differential variable,(1)It is sometimes known as differentiation under the integral sign.This rule can be used to evaluate certain unusual definite integrals such as(2)(3)for (Woods 1926).Feynman (1997, pp. 69-72) recalled seeing the method in Woods (1926) and remarked "So because I was self-taught using that book, I had peculiar methods for doing integrals," and "I used that one damn tool again and again."
A real function is said to be differentiable at a point if its derivative exists at that point. The notion of differentiability can also be extended to complex functions (leading to the Cauchy-Riemann equations and the theory of holomorphic functions), although a few additional subtleties arise in complex differentiability that are not present in the real case.Amazingly, there exist continuous functions which are nowhere differentiable. Two examples are the Blancmange function and Weierstrass function. Hermite (1893) is said to have opined, "I turn away with fright and horror from this lamentable evil of functions which do not have derivatives" (Kline 1990, p. 973).
Suppose is a function of that is twice differentiable at a stationary point . 1. If , then has a local minimum at . 2. If , then has a local maximum at . The extremum test gives slightly more general conditions under which a function with is a maximum or minimum.If is a two-dimensional function that has a local extremum at a point and has continuous partial derivatives at this point, then and . The second partial derivatives test classifies the point as a local maximum or local minimum.Define the second derivative testdiscriminant as(1)(2)Then 1. If and , the point is a local minimum. 2. If and , the point is a local maximum. 3. If , the point is a saddle point. 4. If , higher order tests must be used.
A generalization of the product rule for expressingarbitrary-order derivatives of products of functions,where is a binomial coefficient. This can also be written explicitly as(Roman 1980), where is the differential operator.
Implicit differentiation is the procedure of differentiating an implicit equation with respect to the desired variable while treating the other variables as unspecified functions of .For example, the implicit equation(1)can be solved for(2)and differentiated directly to yield(3)Differentiating implicitly instead gives (4)(5)(6)(7)Plugging in verifies that this approach gives the same result as before.Implicit differentiation is especially useful when is needed, but it is difficult or inconvenient to solve for in terms of .
The first fundamental theorem of calculus states that, if is continuous on the closed interval and is the indefinite integral of on , then(1)This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.The second fundamental theorem of calculus holds for a continuous function on an open interval and any point in , and states that if is defined by(2)then(3)at each point in .The fundamental theorem of calculus along curves states that if has a continuous indefinite integral in a region containing a parameterized curve for , then(4)
If is differentiable at the point and is differentiable at the point , then is differentiable at . Furthermore, let and , then(1)There are a number of related results that also go under the name of "chain rules." For example, if , , and , then(2)The "general" chain rule applies to two sets of functions(3)(4)(5)and(6)(7)(8)Defining the Jacobi rotation matrix by(9)and similarly for and , then gives(10)In differential form, this becomes(11)(Kaplan 1984).
A related rates problem is the determination of the rate at which a function defined in terms of other functions changes.Related rates problems can be solved by computing derivatives for appropriate combinationsof functions using rules such as the chain rule(1)(for and ),product rule(2)quotient rule(3)sum rule(4)or power rule(5)
The pathological function(originally defined for ) that is continuous but differentiable only on a set of points of measure zero. The plots above show for (red), 3 (green), and 4 (blue).The function was published by Weierstrass but, according to lectures and writings by Kronecker and Weierstrass, Riemann seems to have claimed already in 1861 that the function is not differentiable on a set dense in the reals. However, Ullrich (1997) indicates that there is insufficient evidence to decide whether Riemann actually bothered to give a detailed proof for this claim. du Bois-Reymond (1875) stated without proof that every interval of contains points at which does not have a finite derivative, and Hardy (1916) proved that it does not have a finite derivative at any irrational and some of the rational points. Gerver (1970) and Smith (1972) subsequently proved that has a finite derivative (namely, 1/2) at the set of points where and are integers. Gerver..
Numerical differentiation is the process of finding the numerical value of a derivative of a given function at a given point. In general, numerical differentiation is more difficult than numerical integration. This is because while numerical integration requires only good continuity properties of the function being integrated, numerical differentiation requires more complicated properties such as Lipschitz classes. Numerical differentiation is implemented as ND[f, x, x0, Scale -> scale] in the Wolfram Language package NumericalCalculus` .There are many applications where derivatives need to be computed numerically. Thesimplest approach simply uses the definition of the derivativefor some small numerical value of .
The derivative of a function represents an infinitesimalchange in the function with respect to one of its variables.The "simple" derivative of a function with respect to a variable is denoted either or(1)often written in-line as . When derivatives are taken with respect to time, they are often denoted using Newton's overdot notation for fluxions,(2)The "d-ism" of Leibniz's eventually won the notation battle against the "dotage" of Newton's fluxion notation (P. Ion, pers. comm., Aug. 18, 2006).When a derivative is taken times, the notation or(3)is used, with(4)etc., the corresponding fluxion notation.When a function depends on more than one variable, a partial derivative(5)can be used to specify the derivative with respect to one or more variables.The derivative of a function with respect to the variable is defined as(6)but may also be calculated more symmetrically as(7)provided the..
The derivative identity (1)(2)(3)(4)where denotes the derivative of . The Leibniz identity extends the product rule to higher-order derivatives.
The first fundamental theorem of calculus states that, if is continuous on the closed interval and is the indefinite integral of on , thenThis result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.
Suppose is continuous at a stationary point . 1. If on an open interval extending left from and on an open interval extending right from , then has a local maximum (possibly a global maximum) at . 2. If on an open interval extending left from and on an open interval extending right from , then has a local minimum (possibly a global minimum) at . 3. If has the same sign on an open interval extending left from and on an open interval extending right from , then has an inflection point at .
Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation.(1)The above partial derivative is sometimes denoted for brevity.Partial derivatives can also be taken with respect to multiple variables, as denoted for examples(2)(3)(4)Such partial derivatives involving more than one variable are called mixedpartial derivatives.For a "nice" two-dimensional function (i.e., one for which , , , , exist and are continuous in a neighborhood ), then(5)More generally, for "nice" functions, mixed partial derivatives must be equal regardless of the order in which the differentiation is performed, so it also is true that(6)If the continuity requirement for mixed partials is dropped, it is possible to construct functions for which mixed partials are not equal. An example is the function(7)which has and (Wagon 1991). This..
There are at least two meanings of the term "total derivative" in mathematics.The first is as an alternate term for the convectivederivative.The total derivative is the derivative with respect to of the function that depends on the variable not only directly but also via the intermediate variables . It can be calculated using the formulaThe total derivative of a function with respect to is implemented in the Wolfram Language as Dt[f[t, x, y, ...], t].
The Blancmange function, also called the Takagi fractal curve (Peitgen and Saupe 1988), is a pathological continuous function which is nowhere differentiable. Its name derives from the resemblance of its first iteration to the shape of the dessert commonly made with milk or cream and sugar thickened with gelatin.The iterations towards the continuous function are batrachions resembling the Hofstadter-Conway $10,000 sequence. The first six iterations are illustrated below. The th iteration contains points, where , and can be obtained by setting , lettingand looping over to 1 by steps of and to by steps of .
Let be the revenue for a production , the cost, and the profit. Thenand the marginal profit for the th unit is defined bywhere , , and are the derivatives of , , and , respectively.
The term "pathological" is used in mathematics to refer to an example specifically cooked up to violate certain almost universally valid properties. Pathological problems often provide interesting examples of counterintuitive behavior, as well as serving as an excellent illustration of why very detailed conditions of applicability are required in order for many mathematical statements to be universally true.For example, the pathological Weierstrass and Blancmange functions are examples of a continuous function that is nowhere differentiable, a possibility that many students of calculus find quite surprising.In 1899, Poincaré remarked on the proliferation of pathological functions, "Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less..