What is Derivative: Definition and 1000 Discussions

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. H

    Index Notation, taking derivative

    Can anyone explain how to take the derivative of (Aδij),j? I know that since there is a repeating subscript I have to do the summation then take the derivative, but I am not sure how to go about that process because there are two subscripts (i and j) and that it is the Kronecker's Delta (not...
  2. D

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  3. W

    Help understanding equation involving a partial derivative

    Mod note: Moved from a homework section 1. Homework Statement N/A Homework Equations f(x + Δx,y) = f(x,y) + ∂f(x,y)/∂x*Δx The Attempt at a Solution Sorry this isn't really homework. We were given this equation today in order to derive the Taylor expansion formula in two variables and I'm not...
  4. terryds

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  5. T

    Are a vector and its derivative perpendicular at all times?

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  6. S

    Does derivative of wave function equal zero at infinity?

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  7. P

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  8. A

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  9. Mr Davis 97

    Derivative and function terminology

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  10. S

    Partial Derivative of x^2 on Manifold (M,g)

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  11. loops496

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  12. B

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  13. M

    Prove existence of open interval with non-zero derivative

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  14. Ravendark

    Is the Sign in the Covariant Derivative Important for Local Gauge Invariance?

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  15. Titan97

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  16. pellman

    What is a derivative in the distribution sense?

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  17. N

    Prove that the derivative of tan(2x) - cot(2x) equals....

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  18. J

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  19. Ravendark

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  20. N

    Confused on these two derivative problems

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  21. Y

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  22. U

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  23. ognik

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  24. N

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  26. haael

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  27. dhalilsim

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  28. Z

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  29. R

    Help finding the derivative for Faraday's equation?

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  30. hideelo

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  31. S

    Finding derivative of x^x using limit definition

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  32. S

    What is the partial derivative of f with respect to w?

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  33. B

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  34. K

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  35. C

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  36. F

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  37. B

    Derivative of the mixed metric tensor

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  38. MidgetDwarf

    Can relative maximum and minimum exist when the derivative....

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  39. E

    Derivative of best approximation

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  40. Joseph Nechleba

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  41. M

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  42. P

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  43. R

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  44. E

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  45. N

    Proof of derivative uniqueness

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  46. C

    Calculating Derivative of cos(xy)+ye^x Near (0,1) and Level Curve f(x,y)=f(0,1)

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  47. DavideGenoa

    Derivative of angular momentum

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  48. A

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  49. R

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  50. Chrono G. Xay

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