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. S

    A Definition of the Lie derivative

    Consider the Lie derivative of the vector field ##\bf{Y}## with respect to the vector field ##\bf{X}## on manifold ##M^{n}(x)## defined as ##\displaystyle{[\mathcal{L}_{\bf{X}}Y]_{x}:=\lim_{t\rightarrow 0} \frac{[{\bf{Y}}_{\phi_{t}x}-\phi_{t*}{\bf{Y}}_{x}]}{t}}## Now, I understand that...
  2. A

    How Do You Handle a Discontinuous Derivative in Calculus?

    Homework Statement http://prntscr.com/czcn8h Homework Equations n/a The Attempt at a Solution I know that if you derive x^2sin(1/x) you get -cos(1/x) + sin(1/x)(2x). But what do I do from here? If I use the limit definition, i'll end up getting something like h(sin(1/h)) after evaluating. I...
  3. Kanashii

    Finding the second derivative using central difference formula

    Homework Statement Develop aprogram that will determine the second derivative of pi(16 x^2 - y^4) at y=2 with step sizes of 0.1, 0.01, 0.001…. until the absolute error (numerical-analytical) converges to 0.00001. Use the 2nd order Central Difference Formula. User Input: y, tolerance Output: h...
  4. T

    A Backward finite differences on higher order derivative

    I am trying to solve a system of equations and have a question regarding the validity of my approach when implementing a fifth-order Cash-Karp Runge-Kutta (CKRK) embedded method with the method of lines. To give the questions some context, let me state the problem I am attempting to solve: $$...
  5. S

    A Square of the exterior derivative

    Is ##\text{d}^{2}=\text{d}\wedge\text{d}## a definition of the exterior algebra, or can it be derived from more fundamental mathematical statements?
  6. T

    A Computing first derivative based on second derivative

    I am trying to numerically solve a PDE, and just had a question as to the validity of a certain approach. For example, given the PDE: $$ \frac {\partial ^2 E}{\partial t^2} = - k\frac {\partial E}{\partial t} + c^2 \frac {\partial ^2 E}{\partial z^2} - c\frac {\partial E}{\partial z}$$ If I...
  7. MiLara

    I Why do some but not all derivatives have physical meaning?

    I know that taking the derivative of certain functions that explain physical phenomena can lead to another statement describing the physical system, the most famous being the derivatives of position. That is, position-->velocity-->acceleration-->jerk-->jounce...and taking any other further...
  8. U

    MHB Find the derivative using implicit differentiation (with inverse trig functions)

    Here is the question: This is the step I came to after taking the derivatives and doing some simplification: ^ I did the work myself on paper, I just couldn't type out the whole thing clearly so that anyone else can see what I'm referring too... so I used some online tool to show that...
  9. A

    I What is the derivative of the inverse secant function?

    Please refer to the below image (Example 5). Do anyone know how 5x^4 > 1 > 0?
  10. Q

    Second Derivative (Implicit Differentiation)

    Homework Statement Find y'' Homework Equations 9x^2 +y^2 = 9 The Attempt at a Solution y' 18x+2y(y')=0 y'=-18x/2y y'=9x/y For the second derivative, I get the correct answer (same as the book) up until the very last step. Here's where I'm left at: -9( (-9x^2 - y^2) / y^3 ) The book then...
  11. MiLara

    B The Significance of the Smallest Non-Zero Derivative for a Polynomial Function

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  12. toforfiltum

    Conflicting result in derivative of composite function

    Homework Statement Let $$f(x,y)=\begin{cases} \frac{x^2y}{x^2+y^2} \space & \text{if} \space(x,y)\neq(0,0)\\0 \space & \text{if} \space(x,y)=(0,0)\end{cases}$$ a) Use the definition of the partial derivative to find ##f_x(0,0)## and ##f_y(0,0)##. b) Let a be a nonzero constant and let...
  13. M

    Theoretic doubt about the definition of derivatives.

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  14. B

    Derivative of Position Vector at Specified Time

    Homework Statement My homework problem is a proof in orbital mechanics, but I'm not looking for specific help on that just yet, I'd like to work through it on my own. In doing so however, I'm having a hard time conceptualizing the idea of derivatives of vectors at a specified time. If r is a...
  15. Q

    Derivative of exponential function

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  16. Elnur Hajiyev

    A Hubble parameter vs Scale factor's derivative

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  17. toforfiltum

    Confused about partial derivative to function

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  18. DoobleD

    Partial or total derivative in Faraday's law

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  19. M

    I A-level differentiation/derivative dilemma

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  20. Drakkith

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  21. Drakkith

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  22. U

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  23. Pouyan

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  24. D

    I Lie derivative of a differential form

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  25. Z

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  26. CivilSigma

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  27. ChrisBrandsborg

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  28. W

    Is the derivative in my textbook correct here?

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  29. ibkev

    I Partial derivative used in Calc of Variation

    I'm working through the discussion of calculus of variations in Taylor's Classical Mechanics today. There's a step where partial differentiation is involved that I don't understand. Given: $$S(\alpha)=\int_{x_1}^{x_2} f(y+\alpha\eta, y'+\alpha\eta', x)\,dx$$ The goal is to determine ##y(x)##...
  30. karush

    MHB Calculating the Derivative of an Exponential Function with Logarithms?

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  31. O

    Derivative of Definite Integral Conundrum

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  32. O

    I How to logically derive the total derivative formula?

    Consider this equation: f(x(t),y(t))=2(x(t))^2+x(t)y(t)+y(t) One way to calculate df/dt is directly using the chain rule: \frac{df}{dt}=4x(t)\frac{dx}{dt}+\frac{dx}{dt}y(t)+\frac{dy}{dt}x(t)+\frac{dy}{dt} \frac{df}{dt}=(4x(t)+y(t))\frac{dx}{dt}+(x(t)+1)\frac{dy}{dt} Another way is by using...
  33. A

    I What is a partial derivative and how is it used in Schrodinger's equation?

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  34. T

    I Verifying derivative of multivariable integral equation

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  35. Dyatlov

    I Need help with a derivative solution

    Hello. We have the derivative of a function: d F(x)/dx. If we substitute x = au, how can I show that d F(x)/dx = (1/a) (dF/du) ?
  36. karush

    MHB Derivative of inverse tangent function

    Find the derivative of the function $f(y)$ $$f(y)=\tan^{-1}\left({8{y}^{3}+1}\right)$$
  37. A

    I Geodesic Equation: Lagrange Approximation Solution for Schwarzschild Metric

    Hello so if we have geodesic equation lagrange approximation solution: d/ds(mgμνdxν/ds)=m∂gμν∂xλdxμ/ds dxν/ds. So if we have schwarzschild metric (wich could be used to describe example sun) which is:ds2=(1-rs/r)dt2-(1-rs/r)-1dr2-r2[/SUP]-sin22. But that means that ∂gμν/∂xλ=0. So that means that...
  38. karush

    MHB How Do You Differentiate a Natural Logarithm Function Like This?

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  39. Boon

    A Why Use Taylor Expansion in Fréchet Derivative Derivation?

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  40. orion

    I Need help with derivative notation

    If I have a scalar function of a variable ##x## I can write the derivative as: ##f'(x)=\frac{df}{dx}##. Now suppose ##x## is no longer a single variable but a vector: ## x=(x^1, x^2, ..., x^n)##. Then of course we have for the derivative ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial...
  41. G

    MHB How to find a partial derivative from such a complicated object

    Hi guys. I have the following equilibrium equation from which I want to extract \d{\theta}{\nu} z+\theta m(\theta)[\,\frac{\int_{0}^{n^*} \,W(n)g(n)dn+h(n^{*})W(n^{*})G^{*}}{1-(1-h(n^{*}))G^{*}}-U\,]=-c+m(\theta)J'(0) Where \nu, z, c, n, r, \delta, \xi are parameters, m(.), w(.) and h(.)...
  42. perplexabot

    A Derivative of log of normal distribution

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  43. M

    MHB Why Does ∫ln(x) dx Include dx?

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  44. Jeffack

    A Create function which meets slope and point requirements

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  45. orion

    I Do derivative operators act on the manifold or in R^n?

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  46. redtree

    A Relationship between metric tensor and position vector

    Given the definition of the covariant basis (##Z_{i}##) as follows: $$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$ Then, the derivative of the covariant basis is as follows: $$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$ Which is also equal...
  47. T

    MHB Derivative of function with a natural log in the exponent

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  48. A

    MATLAB How to Calculate the Second Derivative of a Curve in Matlab?

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  49. DavideGenoa

    I Substitution in a Lebesgue integral

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  50. K

    Are Partial Derivatives Commutative for Functions of Multiple Variables?

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