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. G

    I Existence of Directional Derivative in Normed Linear Space

    Given a finite-dimensional normed linear space ##(L,\lVert \cdot \rVert)##, is there anything that suggests that at every point ##x_0 \in L##, there exists a direction ##\delta \in L## such that that ##\lVert x_0 + t\delta \rVert \geqslant \lVert x_0 \rVert## for all ##t \in \mathbb{R}##?
  2. Math Amateur

    MHB Directional Derivative Example .... SHifrin, Ch.3, Section 1, Example 3 ....

    I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Section 3.1 Partial Derivatives and Directional Derivatives ... I need some help with Example 3 in Chapter 3, Section 1 ... Example 3 in Chapter 3, Section 1 reads as follows:In the above text we read...
  3. LesterTU

    Expressing Covariant Derivative in Matrix Form

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  4. E

    A Lie derivative of vector field defined through integral curv

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  5. K

    I Derivative changes under CPT transformations

    Hello! Do the derivatives change sign under C, P or T transformation. For example, for the photon vector field we have, under C, ##A_\mu \to -A_\mu##. Do we also get ##\partial_\mu \to -\partial_\mu ##? And what about P and T? Thank you!
  6. barryj

    Purpose of the derivative of the inverse function

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  7. Mr Davis 97

    I Flipping the sign in the definition of derivative

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  8. Zack K

    Value of an implicit derivative

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  9. m4r35n357

    I Request for specific derivative data

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  10. betamu

    Find Directional Derivative at Given Point in Direction of Given Vector

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  11. Runei

    I Total Derivative of a Constrained System

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  12. Z

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  13. D

    I The total derivative of a wavefunction

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  14. M

    A How Does the Chain Rule Define the Differential of a Function on a Manifold?

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  15. D

    Derivative for a Galilean Tranformation

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  16. Prez Cannady

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  17. B

    Derivative of Cosine with unit vector

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  18. Matt Chu

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  19. D

    Derivative of expanded function wrt expanded variable?

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  20. L

    A Higher Order Derivative Test and Germs

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  21. P

    I Is the derivative of a discontinuity a delta function?

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  22. M

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  23. Krushnaraj Pandya

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  24. M

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    Homework Statement Assume that you want to the derivative of a vector V with respect to a component Zk, the derivative is then ∂ViZi/∂Zk=Zi∂Vi/∂Zk+Vi∂Zi/∂Zk = Zi∂Vi/∂Zk+ViΓmikZm Now why is it that I can change m to i and i to j in ViΓmikZm?
  25. K

    I Covariant Derivative Equivalence: Exploring an Intriguing Result

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  26. Q

    I Derivative of a Variation vs Variation of a Derivative

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  27. Q

    I What are the insights into the Total Derivative formula?

    I’ve always been confused by the formula for the Total Derivative of a function. $$\frac{df(u,v)}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f }{\partial u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\partial f}{\partial v}\frac{\mathrm{d}v }{\mathrm{d} x}$$ Any insight would be greatly...
  28. B

    I Differential equation from derivative of time dilation

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  29. K

    I Derivative of this function is injective everywhere

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  30. Phylosopher

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  31. S

    B Derivative of the Lorentz factor

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  32. I

    B Can you deduce ##\tan(\theta) = \frac {df} {dx}## from this graph?

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  33. R

    I "Undo" Second Derivative With Square Root?

    In my classical mechanics course, the professor did a bit of algebraic wizardry in a derivation for one of Kepler's Laws where a second derivative was simplified to a first derivative by taking the square root of both sides of the relation. It basically went something like this: \frac{d^2...
  34. I

    Relationship between force and potential energy

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  35. Peter Alexander

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  36. M

    How Is the Derivative of Basis Vectors Computed in Polar Coordinates?

    Homework Statement I am unsure as to how the partial derivative of the basis vector e_r with respect to theta is (1/r)e_theta in polar coordinates Homework EquationsThe Attempt at a Solution differentiating gives me -sin(theta)e_x+cos(theta)e_y however I'm not sure how to get 1/r.
  37. binbagsss

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  38. aphirst

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  39. K

    I Lie Derivative in Relativity: Examples & Uses

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  40. M

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  41. C

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  42. R

    Calculus derivatives word problem

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  43. K

    Determine the second derivative of a function

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  44. shintashi

    B When do we use which notation for Delta and Differentiation?

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  45. ADDA

    I What Happens When You Differentiate Euler's Formula?

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  46. S

    I What is term for DEQ that only has terms of a derivative?

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  47. EastWindBreaks

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  48. binbagsss

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  49. O

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  50. M

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