Wolfram Mathematica Derivative

[Karren Bangura]

Itshould be noted that the above definitions refer to "real" derivatives, i.e., derivatives which are restricted to directions along the real axis. However, this restriction is artificial, and derivatives are most naturally defined in the complex plane, where they are sometimes explicitly referred to as complex derivatives. In order for complex derivatives to exist, the same result must be obtained for derivatives taken in any direction in the complex plane. Somewhat surprisingly, almost all of the important functions in mathematics satisfy this property, which is equivalent to saying that they satisfy the Cauchy-Riemann equations.

These considerations can lead to confusion for students because elementary calculus texts commonly consider only "real" derivatives, never alluding to the existence of complex derivatives, variables, or functions. For example, textbook examples to the contrary, the "derivative" (read: complex derivative) of the absolute value function does not exist because at every point in the complex plane, the value of the derivative depends on the direction in which the derivative is taken (so the Cauchy-Riemann equations cannot and do not hold). However, the real derivative (i.e., restricting the derivative to directions along the real axis) can be defined for points other than as

As a result of the fact that computer algebra languages and programs such as the Wolfram Language generically deal with complex variables (i.e., the definition of derivative always means complex derivative), correctly returns unevaluated by such software.

Note that in order for the limit to exist, both and must exist and be equal, so the function must be continuous. However, continuity is a necessary but not sufficient condition for differentiability. Since some discontinuous functions can be integrated, in a sense there are "more" functions which can be integrated than differentiated. In a letter to Stieltjes, Hermite wrote, "I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives."

A three-dimensional generalization of the derivative to an arbitrary direction is known as the directional derivative. In general, derivatives are mathematical objects which exist between smooth functions on manifolds. In this formalism, derivatives are usually assembled into "tangent maps."

Performing numerical differentiation is in many ways 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.

Derivatives of functions play a fundamental role in calculus and its applications. In particular, they can be used to study the geometry of curves, solve optimization problems and formulate differential equations that provide mathematical models in areas such as physics, chemistry, biology and finance. The function D computes derivatives of various types in the Wolfram Language and is one of the most-used functions in the system. My aim in writing this post is to introduce you to the exciting new features for D in Version 11.1, starting with a brief history of derivatives.

Starting from the derivative of a function, one can compute derivatives of higher orders to gain further insight into the physical phenomenon described by the function. For example, suppose that the position s(t) of a particle moving along a straight line at time t is defined as follows.

Then, the velocity and the acceleration of the particle are given by its first and second derivatives, respectively. The higher derivatives too can be computed easily using D; they also have special names, which can be seen in the following computation.

An immediate application of the above closed form would be to compute higher-order derivatives of functions with blinding speed. D itself uses this method to compute the billionth derivative of Sin in a flash, using Version 11.1.

The Wolfram Language has a rich variety of mathematical functions, starting from elementary functions such as Power to advanced special functions such as EllipticE. The nth derivatives for many of these functions can be computed in closed form using D in Version 11.1. The following table captures the beauty and complexity of these formulas, each of which encodes all the information required to compute higher derivatives of a given function.

Some of the entries in the table are rather simple. For example, the first entry states that all the derivatives of the exponential function are equal to the function itself, which generalizes the following result from basic calculus.

The familiar sum, product and chain rules of calculus generalize very nicely to the case of nth derivatives. The sum rule is the easiest, and simply states that the nth derivative of a sum is the sum of the nth derivatives.

The special functions in the Wolfram Language typically occur in families, with different members of each family labeled by integers or other parameters. For example, there is one function BesselJ[n,z] for each integer n. The first four members of this family are pictured below (the sinusoidal character of Bessel functions helps in the modelling of circular membranes).

It turns out that the derivatives of BesselJ[n,z] can be expressed in terms of other Bessel functions from the same family. While earlier versions did make some use of these relationships, Version 11.1 exploits them more fully to return compact answers for examples such as the following, which generated 210=1024 instances of BesselJ in earlier releases!

The functions considered so far are differentiable in the sense that they have derivatives for all values of the variable. The absolute value function provides a standard example of a non-differentiable function, since it does not have a derivative at the origin. Unfortunately, the built-in Abs function is defined for complex values, and hence does not have a derivative at any point. Version 11.1 overcomes this limitation by introducing RealAbs, which agrees with Abs for real values, as seen in the following plot.

This real absolute value function is continuous and only mildly non-differentiable, but in 1872, Karl Weierstrass stunned the mathematical world by introducing a fractal function that is continuous at every point but differentiable nowhere. Version 11.1 introduces several fractal curves of this type, which are named after their discoverers. Approximations for a few of these curves are pictured here.

From the above, we see that KroneckerDelta[i, j] is 1 if its components i and j are equal, and is equal to 0 otherwise. As a result, it allows us to sift through all the terms in the following sum and select, say, the third term f(3) from it.

Along with the improvements for the functionality of D, Version 11.1 also includes a major documentation update for this important function. In particular, the reference page now includes many application examples of the types encountered in a typical college calculus course. These examples are based on a large collection of more than 5,000 textbook exercises that were solved by a group of talented interns using the Wolfram Language during the summer of 2016. Some of the graphics from these examples are shown here. You can click anywhere inside each of the three following graphics to view their corresponding examples in the online documentation.

D is a venerable function that has been available since Version 1.0 (1988). We hope that the enhancements for this function in Version 11.1 will make it even more appealing to users at all levels. Any comments or feedback about the new features are very welcome.

A derivative is a mathematical concept that represents the rate of change of a function at a particular point. It is important because it allows us to analyze and understand the behavior of a function, such as its slope and curvature, which have real-world applications in fields like physics, economics, and engineering.

To evaluate a derivative at a point in Mathematica, you can use the built-in function "D" and specify the function and the point at which you want to evaluate the derivative. For example, to evaluate the derivative of f(x) at x=3, you would use the command "D[f[x],x]/.x->3".

Yes, Mathematica has the ability to compute derivatives of complex functions using the same syntax as for real functions. However, it is important to note that the result may also be complex, so it is necessary to use appropriate functions to visualize or manipulate the result.

To plot the derivative of a function in Mathematica, you can use the "Plot" function and specify the derivative as the function to be plotted. For example, to plot the derivative of f(x), you would use the command "Plot[D[f[x],x], x, a, b]", where "a" and "b" are the range of values for the x-axis.

Yes, Mathematica has built-in functions such as "Solve" and "NSolve" that can find the critical points and inflection points of a function. You can also use the "FindRoot" function to numerically find these points. Additionally, you can use the "Plot" function to visualize these points on a graph.

The derivative is a fundamental tool of calculus that quantifies the sensitivity of 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. For this reason, 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.[1] The process of finding a derivative is called differentiation.

There are multiple different notations for differentiation, two of the most commonly used being Leibniz notation and prime notation. 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. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. 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.

3a8082e126