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. F

    I Derivative and Numerical approach

    Hello, I am dealing with a function f(x) involving various complicated trig functions. I took the first derivative of the function f(x), which is doable, and set it to zero to find the at which value x the function f(x) would be maximum. However, once I have the derivative of f(x), it does not...
  2. Math Amateur

    MHB The Directional Derivative .... in Scalar Fields and Vector Fields ....

    I need some guidance regarding the directional derivative ... Two books I am reading introduce the directional derivative somewhat differently ... these books are as follows: Theodore Shifrin: Multivariable Mathematics and Susan Jane Colley: Vector Calculus (Second Edition)Colley...
  3. Math Amateur

    MHB Reconciling Differences in Vector-Valued Function Derivatives

    I am trying to reconcile apparent differences between the definitions of the derivative of a vector-valued function f: U \rightarrow \mathbb{R}^n (where U \subset \mathbb{R}^m ) of a vector variable from two textbooks ... The textbooks are as follows: Andrew Browder: "Mathematical Analysis...
  4. B

    I How Do You Solve an nth Derivative Problem?

  5. Bishamonten

    Understanding functional derivative

    Homework Statement "The functional ## J[f] = \int [f(y)]^pφ(y)\, dy ## has a functional derivative with respect to ## f(x) ## given by: $$ \frac {δJ[f]} {δf(x)} = \lim_{ε \rightarrow 0} \frac 1 ε \left[ \int[f(y) + εδ(y-x)]^pφ(y)\, dy - \int [f(y)]^pφ(y)\, dy\right] $$ $$ =...
  6. Cathr

    I How to find the matrix of the derivative endomorphism?

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  7. yecko

    Derivative of directional vector

    Homework Statement Find the unit vectors along which the given functions below increase and decrease most rapidly at P0 . Then find the derivatives of the functions in these directions. Homework Equations solution: The Attempt at a Solution why are the derivatives’ values along these...
  8. O

    Derivative of two polynomials, one of them being squared

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  9. E

    Understanding the Derivative of r(dot): Step-by-Step Guide

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  10. R

    Definition of Momentum in terms of a partial derivative

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  11. Philosophaie

    I Partial Vector Derivative: Is This the Correct Derivative of B?

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  12. topsquark

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  13. mertcan

    A Covariant derivative only for tensor

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  14. G

    How can I calculate the derivative of this function?

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  15. M

    What is the derivative of a skew symmetric matrix?

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  16. D

    Intuition about derivative of x^2 at 0

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  17. K

    What is the Derivative as a Limit?

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  18. T

    The Divergence of a Polar Vector Function

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  19. Bill_Nye_Fan

    B Rule to integrate a function with respect to its derivative

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  20. S

    Pullback and exterior derivative

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  21. A

    Finding the approximate change in the perimeter of a circle

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  22. K

    Second derivative in parametric equations

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  23. K

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  24. K

    Derivative as a Limit: Finding dy/dx for y = √(2x+3)

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  25. K

    Derivative of a parametric equation

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  26. U

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  27. K

    Derivative of Secant: Find dy/dx

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  28. K

    Understanding the Derivative of y with Respect to x

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  29. EEristavi

    An Error Formula for Linearization (involving second Derivative)

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  30. D

    Partial Derivative Homework: Calculate ∂f/∂x

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  31. Abdul Wali

    MATLAB Matlab Derivative block analysis and filter design

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  32. M

    Mathematica Bessel function derivative in sum

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  33. Oats

    I Must functions really have interval domains for derivatives?

    Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so...
  34. weezy

    I Confusion in variation derivative

    This link shows us how to derive Hamilton's generalised principle starting from D'Alembert's principle. While I had no trouble understanding the derivation I am stuck on this particular step. I can't justify why ## \frac{d}{dt} \delta r_i = \delta [\frac{d}{dt}r_i] ##. This is because if I...
  35. Pushoam

    Derivative of angular velocity of rotating co. system

    What is time derivative of angular velocity ( measured w.r.t. an inertial frame ) of a rotating co. system w.r.t. the same rotating co. system? I think a person sitting in a closed rotating box feels the an object at rest w.r.t. him as rest. He doesn't observe the angular velocity of the...
  36. binbagsss

    Lie derivative vector fields, show the Leibniz rule holds

    Homework Statement Homework Equations ##V=V^u \partial_u ## I am a bit confused with the notation used for the Lie Derivative of a vector field written as the commutator expression: Not using the commutator expression I have: ## (L_vU)^u = V^u \partial_u U^v - U^u\partial_u V^v## (1)...
  37. H

    Find the equation of a tangent line to y = x^2?

    Homework Statement the line goes through (0, 3/2) and is orthogonal to a tangent line to the part of parabola y = x^2, x > 0 Homework EquationsThe Attempt at a Solution I have problems regarding finding the equation of tangent line to the part of parabola because the question not specifically...
  38. Saracen Rue

    B Second derivative differential equations in terms of y?

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  39. karush

    MHB 7.6.16 Find the derivative of y with respect to x

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  40. Vectronix

    B Derivative with the double cross product

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  41. binbagsss

    GR Lie Derivative of metric vanish <=> metric is independent

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  42. binbagsss

    GR - Lie Derivative of metric - Killing Equation

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  43. karush

    MHB 242.7x.25 Find the derivative of y with respect to x

    $\tiny{242.7x.25}$ $\textsf{Find the derivative of y with respect to x}$ \begin{align*}\displaystyle y&=8\ln{x}+\sqrt{1-x^2}\arccos{x} \\ y'&=\frac{8}{x}+? \end{align*} the first term was easy but the second😰
  44. lfdahl

    MHB Can you find $f^{(n)}(1)$ with the given conditions?

    Let $f(x)$ be a function satisfying \[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\] Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.
  45. V

    Calculus in the velocity and acceleration of satellites.

    Homework Statement I am working on a project dealing with the velocity and acceleration of satellites based on their distance from Earth. I was recommended to include some calculus in this project. Originally I thought I could just take the derivative of the orbital speed equation to find...
  46. Felix Gonzales

    I don't understand the derivation process here, help?

    Homework Statement I understand the derivation it showed that included the sin (15.7 in the image) I just don't understand the following (15.8 in the image). Does "t" get pulled out of the equation? If so what do we derive for then? Does it become 0? If so, it would remain 0 and sin(0) is just...
  47. K

    I Why no higher derivative in physics?

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  48. Jess Karakov

    Simplifying this derivative....

    Homework Statement Evaluate the derivative of the following function: f(w)= cos(sin^(-1)2w) Homework Equations Chain Rule The Attempt at a Solution I did just as the chain rule says where F'(w)= -[2sin(sin^(-1)2w)]/[sqrt(1-4w^(2)) but the book gave the answer as F'(w)=(-4w)/sqrt(1-4w^(2))...
  49. N

    SIFT is derivative of DoG needed for Hessian or just DoG?

    Wikipedia defines hessian of Difference of Gaussians as and earlier in the page uses D for difference of gaussians, So do i just need D(x,y) or do i need d/dx D(x,y) for the elements? If so how does one go about differentiating DoG? Any help appreciated
  50. E

    I What is the Result of this Partial Derivative

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