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. karush

    MHB How to Find the Second Derivative with Given Equation at a Specific Point?

    If $(x+2y)\cdot \dfrac{dy}{dx}=2x-y$ what is the value of $\dfrac{d^2y}{dx^2}$ at the point (3,0)? ok not sure of the next step but $\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$
  2. Arman777

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  3. Vyrkk

    A Covariant derivative and connection of a covector field

    I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors. What people usually do is take the covariant derivative of the covector acting on a vector, the result being a scalar Invoke a product rule to...
  4. M

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  5. V

    Finding the derivative and rate of change

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  6. Physyx

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    I am using the derivative of momentum (dp/dt) with Newton’s 3rd Law with the gravitational force of Earth. F - [Force of gravity on rocket] = dp/dt F - (G * m_e * m_r / r2 ) = v * dm/dt + ma F = Force created by fuel (at time t) G = Gravitational Constant m_e = Mass of Earth m_r = Mass of...
  7. NP04

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  8. karush

    MHB 3.3.04 AP Calculus Exam 2nd derivative

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  9. T

    I How Does the Dot Product of Vector Derivatives Relate to Their Original Vectors?

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  10. sviego

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  11. DiracPool

    B Derivative and integral of the exponential e^t

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  12. R

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  13. George Keeling

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  14. Another

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  15. Matt & Hugh play with a Brick and derive Centripetal Acceleration

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    • scottdave
    • Media item
    • accelaration centrifugal centripetal derivative vector
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  16. barryj

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  17. Math Amateur

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  18. barryj

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  19. V

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  20. Kisok

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  21. E

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  22. K

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  23. V

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  24. looseleaf

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  25. D

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  26. Pencilvester

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  27. haushofer

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  28. A

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  29. P

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  30. maistral

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  31. P

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  32. L

    I Understanding the definition of derivative

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  33. karush

    MHB 2.2.206 AP Calculus Practice question derivative of a composite sine

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  34. George Keeling

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  35. F

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  36. hilbert2

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  37. E

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  38. snoopies622

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  39. Robin04

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  40. berlinspeed

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  41. Mcp

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  42. A

    I Partial Derivative: Correct Formulation?

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  43. Physics lover

    Derivative of a Series: How to Solve for x^n/((x-a1)(x-a2)...(x-an)) in Calculus

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  44. R

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  45. A

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  46. sergiokapone

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  47. S

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  48. D

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  49. T

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  50. A

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