question: What are derivatives in Calculus and where could one actually use this in the real world?
comment: Derivatives in calculus are formulas for the rate of change of a function. If I'm walking 1 m/s in one second, 2 m/s in the next, 3 m/s in the third, one could describe my speed overtime as 1*t where t is the number of seconds I walked. The derivative of this function (1) is my acceleration, or the rate at which my speed changes. Derivatives are used in ***a lot*** of places. Basically any problem that involves optimization, whether in engineering, economics, physics, statistics, or really anywhere involves taking a derivative. To understand why this is, consider the following example: The efficiency of an engine as a function of the speed at which it's operating. If you were to graph it it would probably look something like [this]( URL_0 ). If we wanted to find the speed at which the engine is most efficient we could take the derivative of that function and find where the derivative (again, the rate of change) is 0. This corresponds to when the graph changes directions. Since there's only one point where that happens (the max), we know we've found our optimal speed.
passages: It is noted that Farid and Simoncelli have derived first derivative coefficients which are more accurate compared to the ones provided above. However, the latter are consistent with the second derivative interpolator and, therefore, are better to use if both the first and second derivatives are sought. In the opposite case, when only the first derivative is desired, the optimal first derivative coefficients should be employed; more details can be found in their paper.
  
  Section::::Hast Derivatives.
   where denotes the Fourier transform.
  
  Section::::Fractional derivatives.:Other types.
  
  Classical fractional derivatives include:
  
  BULLET::::- Grünwald–Letnikov derivative
  
  BULLET::::- Sonin–Letnikov derivative
  
  BULLET::::- Liouville derivative
  
  BULLET::::- Caputo derivative
  
  BULLET::::- Hadamard derivative
  
  BULLET::::- Marchaud derivative
  
  BULLET::::- Riesz derivative
  
  BULLET::::- Riesz–Miller derivative
  
  BULLET::::- Miller–Ross derivative
  
  BULLET::::- Weyl derivative
  
  BULLET::::- Erdélyi–Kober derivative
  
  New fractional derivatives include:
  
  BULLET::::- Machado derivative (This derivative does not exist anywhere in the literature)
  
  BULLET::::- Chen–Machado derivative
  
  BULLET::::- Coimbra derivative
  
  BULLET::::- Katugampola derivative
  
  BULLET::::- Caputo–Katugampola derivative
  
  BULLET::::- Hilfer derivative
  
  BULLET::::- Hilfer–Katugampola derivative
  
  BULLET::::- Davidson derivative
  
  BULLET::::- Chen derivative
  
  BULLET::::- Caputo Fabrizio derivative
  
  BULLET::::- Atangana–Baleanu derivative
  
  BULLET::::- Pichaghchi derivative
  
  Section::::Generalizations.
  
  Section::::Generalizations.:Erdélyi–Kober operator.
   where is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.
  
  Section::::Fractional derivatives.:Atangana–Baleanu derivative.
   Farid and Simoncelli. propose to use a pair of kernels, one for interpolation and another for differentiation (compare to Sobel above). These kernels, of fixed sizes 5 x 5 and 7 x 7, are optimized so that the Fourier transform approximates their correct derivative relationship.
  
  In Matlab code the so called 5-tap filter is
  
  And the 7-tap filter is
  
  As an example the first order derivatives can be computed in the following using Matlab in order to perform the convolution
   Stability and control derivatives change as flight conditions change. The collection of stability and control derivatives as they change over a range of flight conditions is called an aero model. Aero models are used in engineering flight simulators to analyze stability, and in real-time flight simulators for training and entertainment.
  
  Section::::"Stability" derivative vs. "control" derivative.
   Motion graphs and derivatives
  
  In mechanics, the derivative of the position vs. time graph of an object is equal to the velocity of the object. In the International System of Units, the position of the moving object is measured in meters relative to the origin, while the time is measured in seconds. Placing position on the y-axis and time on the x-axis, the slope of the curve is given by:
   More generally, a similar computation shows that the derivative of the square function at is :
  
  Section::::Differentiation.:Continuity and differentiability.
   BULLET::::- Application in simulators for stability analysis: In a flight simulator, it is possible to "look up" new values for stability and control derivatives as conditions change. And so, the "linear approximations" aren't as great and stability can be assessed in maneuvers that span a greater range of flight conditions. Flight simulators used for analysis such as this are called "engineering simulators". The set of values for stability and control derivatives (as they change over various flight conditions) is called an aero model.
  
  Section::::Uses.:Use in flight simulators.
   Steerable filters can be used for computing derivatives Moreover, Savitzky and Golay propose a least-squares polynomial smoothing approach, which could be used for computing derivatives and Luo et al discuss this approach in further detail. Scharr shows how to create derivative filters by minimizing the error in the Fourier domain and Jähne et al discuss in more detail the principles of filter design, including derivative filters.
  
  Section::::External links.
  
  BULLET::::- derivative5.m Farid and Simoncelli: 5-Tap 1st and 2nd discrete derivatives.
  
  BULLET::::- derivative7.m Farid and Simoncelli: 7-Tap 1st and 2nd discrete derivatives
   BULLET::::- Derivative — The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
   BULLET::::- . Notes from a course held by Francesco Severi at the Istituto Nazionale di Alta Matematica (which at present bears his name), containing appendices of Enzo Martinelli, Giovanni Battista Rizza and Mario Benedicty. An English translation of the title reads as:-""Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome"".
   A similar fact also holds true for the velocity vs. time graph. The slope of a velocity vs. time graph is acceleration, this time, placing velocity on the y-axis and time on the x-axis. Again the slope of a line is change in formula_9 over change in formula_10:
   The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using "rules" for obtaining derivatives of more complicated functions from simpler ones.
  
  Section::::Rules of computation.:Rules for basic functions.
  
  Most derivative computations eventually require taking the derivative of some common functions. The following incomplete list gives some of the most frequently used functions of a single real variable and their derivatives.
  
  BULLET::::- "Derivatives of powers": if
   BULLET::::- . ""Elementary introduction to the theory of functions of complex variables with particular regard to integral representations"" (English translation of the title) are the notes form a course, published by the Accademia Nazionale dei Lincei, held by Martinelli when he was ""Professore Linceo"".
  
  BULLET::::- . A textbook on complex analysis including many historical notes on the subject.
   Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the time cancels out, and only displacement remains.)
  
  The same multiplication rule holds true for acceleration vs. time graphs. When acceleration is multiplied by time, velocity is obtained.
  
  Section::::Variable rates of change.
   The expressions given above apply only when the rate of change is constant or when only the average (mean) rate of change is required. If the velocity or positions change non-linearly over time, such as in the example shown in the figure, then differentiation provides the correct solution. Differentiation reduces the time-spans used above to be extremely small (infinitesimal) and gives a velocity or acceleration at each point on the graph rather than between a start and end point. The derivative forms of the above equations are
   These names for the first, second and third derivative of position are commonly used in basic kinematics. By extension, the higher-order derivatives can be computed in a similar fashion. Study of these higher-order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to accurately represent the displacement function as a sum of an infinite sequence, enabling several analytical techniques in engineering and physics.
  
  Section::::See also.
  
  BULLET::::- Affine space
  
  BULLET::::- Horizontal position
  
  BULLET::::- Six degrees of freedom
  
  BULLET::::- Line element
  
  BULLET::::- Parametric surface
  
  BULLET::::- Vertical position
  
  Section::::References.
   BULLET::::- . ""Areolar derivative and functions of bounded variation"" (free English translation of the title) is an important reference paper in the theory of areolar derivatives.
  
  BULLET::::- . ""Studies on essential singular points of analytic functions of two or more complex variables"" (English translation of the title) is an important paper in the theory of functions of several complex variables, where the problem of determining what kind of hypersurface can be the boundary of a domain of holomorphy.
   This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in divided by the change in varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let be a function, and fix a point in the domain of . is a point on the graph of the function. If is a number close to zero, then is a number close to . Therefore, is close to . The slope between these two points is
   Here formula_2 is the position of the object, and formula_3 is the time. Therefore, the slope of the curve gives the change in position divided by the change in time, which is the definition of the average velocity for that interval of time on the graph. If this interval is made to be infinitesimally small, such that formula_4 becomes formula_5 and formula_6 becomes formula_7, the result is the instantaneous velocity at time formula_3, or the derivative of the position with respect to time.
   From there, we can obtain an expression for velocity as the area under an acceleration vs. time graph. As above, this is done using the concept of the integral:
  
  Section::::Equation of motion.:Relationship to acceleration.:Constant acceleration.
  
  In the special case of constant acceleration, velocity can be studied using the suvat equations. By considering a as being equal to some arbitrary constant vector, it is trivial to show that
   Section::::Historical notes.:The work of Dimitrie Pompeiu in 1912 and 1913: a new formulation.
  
  According to , a new step in the definition of the concept was taken by Dimitrie Pompeiu: in the paper , given a complex valued differentiable function (in the sense of real analysis) of one complex variable formula_8 defined in the neighbourhood of a given point formula_9 he defines the areolar derivative as the following limit
   This generalization is useful, for example, if y("t") is the position vector of a particle at time "t"; then the derivative y′("t") is the velocity vector of the particle at time "t".
  
  Section::::In higher dimensions.:Partial derivatives.
  
  Suppose that "f" is a function that depends on more than one variable—for instance,
  
  "f" can be reinterpreted as a family of functions of one variable indexed by the other variables:
  
  In other words, every value of "x" chooses a function, denoted "f", which is a function of one real number. That is,
   BULLET::::- The q-derivative of a function is defined by the formula
   List of derivatives and integrals in alternative calculi
  
  There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi. Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea.
  
  The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). Interested readers are encouraged to improve the table by inserting citations for verification, and by inserting more functions and more calculi.
  
  Section::::Table.
  