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. W

    Derivative of unit vector in spherical coords.

    Homework Statement Given ## d \vec r = dr \hat r + r d \theta \hat {\theta} + r \sin \theta d \phi \hat {\phi}.## Find ## d \hat r , d \hat {\theta} , d \hat {\phi}. ## Homework Equations I know that ## d \hat {e_j} = \omega^i_j \hat {e_i} ## and that ## \omega_{ij}=- \omega_{ji} ## and ## 0 =...
  2. davidge

    I Derivative with several terms in denominator

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  3. P

    I A directional, partial derivative of a scalar product?

    Let's say I have two vector fields a(x,y,z) and b(x,y,z). Let's say I have a scalar field f equal to a•b. I want to find a clean-looking, simple way to express the directional derivative of this dot product along a, considering only changes in b. Ideally, I would like to be able to express...
  4. bwest121

    I Question about Cross Product and Derivative

    Hi everyone, Given a vector-valued function ##\vec{A}##, how do I show that: $$\vec{\nabla} \times \left(\frac{\partial \vec{A}}{\partial x}\right) = \frac{\partial}{\partial x}(\vec{\nabla} \times \vec{A})$$ In other words, are the cross product and derivative commutative w/ each other? I...
  5. L

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  6. J

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  7. Z

    Finding the derivative of the radial vector r

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  8. I

    Kinematics, deriving equations.

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  9. Y

    MHB Derivative and Limit of an Exponential Function

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  10. binbagsss

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  11. G

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  12. H

    Intrinsic derivative of constant vector field along a curve

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  13. F

    A Diffeomorphisms & the Lie derivative

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  14. TheRainMan713

    Finding the nth derivative of a function

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  15. F

    First derivative 3 point forward difference formula

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  16. M

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  17. M

    I How is a vector a directional derivative?

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  18. Drakkith

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  19. D

    I Numerical Derivative Formula: X's Not Same Distance

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  20. M

    Material Derivative and Implicitly Given Variables for Velocity Calculation

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  21. karush

    MHB How Do You Differentiate $\frac{x \sqrt{x^2+1}}{(x+1)^{2/3}}$?

    $\tiny{242.2q.3}$ $\textsf{find the derivative}\\$ \begin{align} \displaystyle y&=\frac{x \, \sqrt[]{x^2+1}}{(x+1)^{2/3}} \\ \ln{y}&=\ln x + \frac{1}{2}\ln(x^2+1) - \frac{2}{3}\ln(x+1)\\ \end{align} $\textit{thot this would help but what next??}$
  22. karush

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  23. D

    I Covariant derivative of field strength tensor

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  24. B

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  25. L

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  26. Rectifier

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  27. K

    What is the Derivative of Inverse Secant and its Graph Representation?

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  28. W

    Derivative in spherical coordinates

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  29. K

    Derivative of an inverse function

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  30. P

    How to take the time derivative of a potential gradient ?

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  31. N

    Discontinuities in the time derivative of the magnetic field

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  32. S

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  33. J

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  34. O

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  35. W

    I Directional derivative: identity

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  36. J

    MHB How Does Water Depth Change as a Cone-Shaped Tank Empties?

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  37. S

    Feynman rules for derivative self-interactions

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  38. N

    MHB Find xf^7(x)''(0): Chain Rule Explanation

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  39. N

    MHB Derivative: Simplifying an Equation

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  40. F

    I What Is the Derivative of cos(x) with Respect to 1/x?

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  41. U

    MHB Find the derivative of the antiderivative

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  42. Adeel Ahmad

    Partial Derivatives: Solve Homework Quickly

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  43. H

    I Partial derivative of a total derivative

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  44. A

    I Are the functions for mixed derivative always equal?

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  45. T

    I Derivative of A Def. Integral Equals Another Def. Integral?

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  46. S

    Spatial derivative of Electric Field in Faraday's Law?

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  47. S

    A Gauge invariance and covariant derivative

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  48. T

    MHB What Is the Derivative of 4f(x) at x = 6?

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  49. T

    MHB Find Derivative: Solve G'(3) Problem

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  50. naima

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