Derivative Definition and 1000 Threads

In mathematics, 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 can 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 relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

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  1. S

    B Reconciling basis vector operators with partial derivative operators

    Ref. 'Core Principles of Special and General Relativity' by Luscombe. Apologies in advance for the super-long question, but it's necessary to show my thought process. Let ##\gamma:I\to M## be a smooth curve from an open interval ##I\subset\mathbb{R}## to a manifold ##M##, and let...
  2. T

    B Is Differentiation the Same as Finding the Derivative?

    I’m am on a path of trying to learn calculus which I should have done long ago. I am making some progress. But I would like to know this... I know what a derivative is. Is differentiation the process of finding a derivative? In other words, when I am finding the derivative can it be said...
  3. P

    B Derivative of a constant scalar field at a point

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  4. A

    I Derivative of a definite integral

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  5. leticia beira

    Finding the derivative of this trig function

    Para f (θ) = √3.cos² (θ) + sen (2θ), uma inclinação da reta tangente, uma função em θ = π / 6, é?
  6. SchroedingersLion

    A Partial / Total Derivative, Compositions

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  7. J

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  8. BvU

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  9. PeroK

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  10. C

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  11. cianfa72

    I About Covariant Derivative as a tensor

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  12. N

    B Second Derivative Question -- Help Understanding the Importance Please

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  13. M

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  14. O

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  15. Jorzef

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  16. ElectronicTeaCup

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  17. thaiqi

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  18. A

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  19. M

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  21. G

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  22. cemtu

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  23. S

    I Derivative of a complex function along different directions

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  24. R

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  25. O

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  26. A

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  27. S

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  29. T

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  30. Monoxdifly

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  31. 1

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  32. B

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  33. Saptarshi Sarkar

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  34. M

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  35. Arman777

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  36. D

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  37. M

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  38. bagasme

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  39. D

    I Ricci Tensor: Covariant Derivative & Its Significance

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  40. PeroK

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  41. Saptarshi Sarkar

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  42. S

    B Increasing and monotonically increasing: related to the first derivative

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  43. SchroedingersLion

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  44. B

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  45. Jatex

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  46. D

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  47. currently

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  48. Euge

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  49. F

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  50. polytheneman

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