Derivative | Wikipedia audio article
Channel: wikipedia tts
Date: 25/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Derivative


00:01:47 1 Differentiation
00:02:30 1.1 Notation
00:05:22 1.2 Rigorous definition
00:06:37 1.3 Definition over the hyperreals
00:12:09 1.4 Example
00:12:40 1.5 Continuity and differentiability
00:12:55 1.6 The derivative as a function
00:13:28 1.7 Higher derivatives
00:13:39 1.8 Inflection point
00:15:38 2 Notation (details)
00:16:36 2.1 Leibniz's notation
00:16:53 2.2 Lagrange's notation
00:18:37 2.3 Newton's notation
00:19:38 2.4 Euler's notation
00:20:07 3 Rules of computation
00:21:26 3.1 Rules for basic functions
00:23:35 3.2 Rules for combined functions
00:23:48 3.3 Computation example
00:28:12 4 In higher dimensions
00:29:13 4.1 Vector-valued functions
00:29:22 4.2 Partial derivatives
00:32:33 4.3 Directional derivatives
00:33:33 4.4 Total derivative, total differential and Jacobian matrix
00:35:07 5 Generalizations
00:36:10 6 History
00:37:31 7 See also
00:38:01 8 Notes
00:39:34 9 References
00:43:17 10 Bibliography
00:46:27 10.1 Print
00:48:32 10.2 Online books
00:48:38 11 External links
00:48:48 === Vector-valued functions
00:49:00 A vector-valued function y of a real variable sends real numbers to vectors in some vector space Rn.  A vector-valued function can be split up into its coordinate functions y1(t), y2(t), …, yn(t), meaning that y(t)
00:51:57 Partial derivatives
01:00:35 Directional derivatives
01:02:26 λu.  Substitute h
01:05:05 Total derivative, total differential and Jacobian matrix
01:21:56 Generalizations
01:24:48 History
01:25:20 See also



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SUMMARY
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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.
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.
Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables.  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 process of finding a derivative is called differentiation. The reverse process is called antidifferentiation.  The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.