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. T

    Derivative of Axial Resolution from Rayleigh's Limit

    I am currently studying optical microscope and discover that the axial resolution is limited as r(z) = 2pi / (NA)^2. However, while I got hints that it is due to the Rayleigh's limit, I can't derivative the equation using numerical method. It would be huge thanks if anyone can help me on the...
  2. K

    Derivative of lagrangian density

    i have a mathematical question which is quite similar to one asked before, still a bit different https://www.physicsforums.com/threads/derivative-of-first-term-in-lagrangian-density-for-real-k-g-theory.781472/the first term of KG-Lagrangian is: \frac{1}{2}(\partial^{\mu} \phi)(\partial_{\mu}...
  3. W

    Find stationary points of a two variable function involving

    Homework Statement Find all stationary points of the function G(x, y) = (x^3)*e^(−x^2−y^2) Homework Equations fx=0 and fy=0 The Attempt at a Solution Gx = 3x^2*e^(-x^2-y^2) +x^3(-2x)e^(-x^2-y^2) = e^(-x^2-y^2)(3x^2-2x^4) Gx = 0 implies 3x^2-2x^4=0 x^2(3-2x^2)=0 hence x =0 ,+or-...
  4. B

    Log Derivative of a Function: Find Derivatives of f(c,l) with Example"

    Homework Statement f(c,l) = log(c - ψ(1-l)^θ ) What is the derivative of this function wrt. l and c?Homework Equations I know that the derivative of log (x) = 1/x The Attempt at a Solution I got wrt c: 1/ c - ψ(1-l)θ and wrt l: θψ(1-l)^θ-1 / c - ψ(1-l)^θ
  5. AwesomeTrains

    Question about the derivation of the energy momentum tensor

    Hey I'm trying to follow the derivation given here: http://lampx.tugraz.at/~hadley/ss1/studentpresentations/Bloch08.pdf Homework Statement As it says in the pdf: "Based on Noether's theorem construct the energy-momentum tensor for classical electromagnetism from the above Lagrangian. L=-1/4...
  6. RaulTheUCSCSlug

    Directional Derivative of Lake Depth at Point (-1, 2) in Direction (4, 1)

    Having a melt down as I have done this problem twice now and my exam is tomorrow and I can't seem to figure it out anymore... ugh. 1. Homework Statement The depth of a lake at the point on the surface with coordinates (x, y ) is given by D(x, y ) = 100−4x 2 −y 2 . a) If a boat at the point (−1...
  7. I

    Time Derivative of Force: What is the Missing Variable?

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  8. NATURE.M

    Derivative of Log Likelihood Function

    So looking through my notes I can't seem to understand how to get from one step to the next. I have attached a screenshot of the 2 lines I'm very confused about. Thanks. BTW: The equations are for the log likelihood in a mixture of gaussians model EDIT: To elaborate I am particularly...
  9. Nader AbdlGhani

    What is the Higher Derivative of a function?

    I'm relatively new to calculus and I have a new chapter in my study which is on the Implicit Function, Implicit Differentiation and Higher Derivatives of a function, the problem is I don't understand the meaning of a 2nd or 3rd or whatever the higher derivative of a function is, what I know is...
  10. S

    Derivative of unit step function

    Homework Statement Show that δ(x-x') = d/dx Θ(x-x') Homework Equations ∫ f(x') δ(x-x') dx' = f(x) Θ(x-x') vanishes if x-x' is negative and 1 if x-x' is positive The Attempt at a Solution I saw a relation of the δ function but I don't know why is it like that. Integral of δ(x-x') from -∞ to x...
  11. N

    Relation between affine connection and covariant derivative

    I now study general relativity and have a few questions regarding the mathematical formulation: 1) What ist the relation between an connection and a covariant derivative? Can you explain the exact difference? 2) One a lorentzian manifold, what ist the relation between the...
  12. Saracen Rue

    Derivative of a fractional function without quotient rule

    Homework Statement The displacement of a particle can be modeled by the function x(t)=\frac{2x-5}{4x^2+2x}, where t is in seconds, x is in meters, and t ∈ [1,10] a) Determine the derivative of the function without using the quotient rule. b) Hence, find exactly when the particle is...
  13. A

    Deriving a Function Without Quotient Rule

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  14. R

    MHB How Do You Find the Derivative of This Function?

    Hello, I am having trouble finding the derivative of this function. Any help would be appreciated! \frac{x\cos(x)+\sqrt{x}}{x+\sin(x)} What I tried was expanding it first to like (x+\sin(x)(x\cos(x)+\sqrt{x})'-(x\cos(x)+\sqrt{x} But I ended up with a long weird answer and doesn't seem to work...
  15. M

    Derivative of y = (1/x) + sqrt(cos x)

    Homework Statement Evaluate the derivative of the function y = (1/x) + sqrt(cos x) at the given point, (pi/2, 2/pi) Homework EquationsThe Attempt at a Solution I used power rule on (1/x) and chain rule on sqrt(cos x) but when I was simplifying, there is a sqrt(cos x) in one of the...
  16. G

    Unable to understand vector derivative

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  17. wololo

    How Can dv/dx Be Determined to Solve for dv/dt?

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  18. C

    Using chain rule to obtain the derivative dz/dt

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  19. Ravendark

    Functional derivative of effective action

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  20. T

    Metric variation of the covariant derivative

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  21. S

    Feynman rules for Lagrangian with derivative Interaction

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  22. C

    Directional derivative at a point

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  23. T

    Splitting Derivative: Rules & Effects in Physics

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  24. K

    MHB Partial Derivatives: Solving Difficult Problems

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  25. N

    Distance formula maximization problem

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  26. M

    I'm curious what anyone gets for part d

    Homework Statement After a wild ride on a windsurfer, you head to a lifeguard tower to fly a kite. You stand on the tower, which is 3 meters above the ground and release your kite. You let out 10 meters of string and the kite begins moving according to the following equation: y(t) = (0.9...
  27. G

    How do I find the equation of a derivative using a TI-84?

    I want to know how to do this so after I find the equation by hand, I can check to see if I am correct.
  28. K

    Multivariable Derivative Error

    Homework Statement f(x,y) = (xy) / (x2 + y4), when (x, y) ≠ (0,0) 0, when (x,y) = (0,0) Homework Equations Explicitly show that f(x,y) does not satisfy lim h -> 0 [ E(v,h) / ||h|| ] = 0 when v = 0 (h, v, and 0 are all vectors; I'm not sure how to put a hat on them) The Attempt at a Solution I...
  29. T

    Understanding transition from full derivative to partial

    I was looking over a derivation to find the laplacian from cartesian to cylindrical and spherical coordinates here: http://skisickness.com/2009/11/20/ Everything seems fine, but there is an instance (I have attached a screenshot) where implicit differentiation is done to find $$ \frac {\partial...
  30. Kingyou123

    Calculating the 46th Derivative of a Quotient Function

    Homework Statement If f(x) =(x^46 + x^45 + 2)/(x + 1) , calculate f(46)(3), the forty-sixth derivative of f(x) at x = 3. Express your answer using factorial notation: n! = n (n 1) (n 2) 3 2 1. Homework Equations Quotient rule The Attempt at a Solution I have tried trying to find a pattern...
  31. T

    MHB General question, which formula to use to find derivative?

    Hi this is my first post ever in MHB, and I'm in Calculus 1 wondering which formula to use to find derivatives. There are 2 as far as I know: (1) and the one one at the beginning of this: (2) Example Problem: How would i know which formula to use? Is there a particular reason the...
  32. T

    Approximation of second derivative of a smooth function

    Hi, I've attached an image of an equation I came across, and the text describes this as an approximation to the second derivative. Everything seems to be exact to me (i.e. not an approximation) if the limit of h was taken to 0. Is that the only reason why it's said to be an approximation or is...
  33. gracy

    Derivatives: Solving F'(x) = [2f(x)+g(x)]' Problem

    Homework Statement ##F'(x)##=##[2f(x)+g(x)]'## ##F'(0)=## Given g'(0)=2 and f'(0)=5 Homework Equations The Attempt at a Solution I can solve this if the questions is as follows ##F'(x)##=##[f(x)+g(x)]'## by applying sum rule but I don't know what to do about...
  34. P

    Derivative operator on both sides

    A basic question, not a homework problem. Say I have the expression: 5x = 10 Can I apply the derivative operator, d/dx, to both sides? d/dx(5x)=d/dx(10) would imply 5=0. I thought you can apply operators to both sides of an equation. Why can't you not do it in this case?
  35. N

    Why is the first derivative velocity?

    Just something that has been bugging me. Can someone bestow why the first derivative is velocity and the second derivative is acceleration. I want to conceptually understand this. Thank you
  36. P

    Double covariant derivative of function of scalar

    If R is Ricci scalar ∇i∇j F(R) = ? , where ∇i is covariant derivative.
  37. K

    Curvilinear coordinate, derivative

    https://www.particleincell.com/2012/curvilinear-coordinates/ http://www.jfoadi.me.uk/documents/lecture_mathphys2_05.pdf Hi, I have a question about the curvilineare coordinate system. I wonder why is normal to the isosurfaces?isnt ei a tangent vector to the surface ui since "With these...
  38. T

    Derivative of an integral and error functions

    Homework Statement differentiate ∫ e^(-x*t^4)dt from -x to x with respect to x.[/B]Homework Equations erf(x) = (2/sqrt(π)) ∫e^(-t^2)dt from 0 to x. Leibniz rule. I know that ∫t^2e^(-t^2)dt from 0 to x = (√π/4)*erf(x) - (1/2)*x*e^(-x^2)[/B]The Attempt at a Solution By using Leibniz rule...
  39. B

    MHB Need Help finding Derivative (full explanation)

    F(x)=5/(x^2)-9 Find F'(x)&F''(x) Find the Vertical Asymptote, Horizontal, and Slant
  40. X

    How Do You Calculate the Uncertainty in the Derivative of y = a / (x - b)^3?

    y = a / (x - b)3 a = 77.1 ± 15.2 b = -1.78 ± 1.18 x = 21 ± 1 --- y' = -3a / (x - b)4 How do I find the uncertainty of y'? Thanks.
  41. J

    Calc BC derivative problem with trig and double angle -- Help please

    Homework Statement Find f'(x) if f(x) = 8^(sin^2(3x)) Hint: you will need to use the double angle formula for trig functions and your answer should only have one trig function in it. Homework Equations if y=a^u then y' = ln a * a^u * du sin(2x) = 2sinxcosx The Attempt at a Solution We're...
  42. Hercuflea

    Curl and Convective Derivative

    Suppose u is a vector-valued function. Is it true that (∇×u)⋅( (u⋅∇)u ) = (u⋅∇)(∇×u)⋅u ? Please note the lack of a dot product on the first two terms of the RHS and the parenthesis around the second term of the LHS. I'm trying to understand whether these differential operators are associative.
  43. O

    Derivative of a trigonometric function

    Homework Statement \frac{d}{dx}7.5sin(\frac{pi}{10}x) The Attempt at a Solution 7.5(\frac{pi}{10})cos(\frac{pi}{10}x) Maximum: f'(x) = 0 7.5(\frac{pi}{10})cos(\frac{pi}{10}x) = 0 7.5(\frac{pi}{10})cos^{-1}(0)= \frac{pi}{10}x ** (\frac{pi}{10}\frac{10}{pi})7.5(90) = x (1)(7.5)(90) = x...
  44. S

    Derivative of Action Integral Equals Generalized Momentum?

    Homework Statement I need to find the partial derivative of the action S with respect to the generalized coordinate q(tf) and according to my textbook, it should equal the generalized momentum p(tf). How can I derive this? Homework Equations S = integral of L dt, with boundary ti to tf. (ti...
  45. Summer95

    Derivative of complex exponential differs by a sign

    I know this is probably the least of my worries at the moment but my quantum textbook solves ##\frac{\mathrm{d}\phi (t) }{\mathrm{d} t}=\frac{iC}{h}\phi (t) ## as ##\phi (t) = e^{-i(\frac{C}{h})t}##. Is this not off by a sign? Its really bugging me.
  46. T

    An equality about derivative of a polynomial?

    Why is $$ \left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n = 2nx\left(x^2-1\right)^{n-1} $$? This is in a textbook and says that its proof is left as an exercise. It seems to be a difficult equality. I believe this should just be $$ \left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n =...
  47. O

    Derivative of Force in terms of distance?

    Hi, Suppose I have a function on a graph with a vertical axis is Force and the horizontal axis is distance. Then the area under the curve is given by F*d = Work = Energy, correct? If so, then what would the slope of the curve represent? F/d = ? Thank you.
  48. Mr Davis 97

    What is meant by "take the derivative of a function"?

    Bourbaki defines a function as follows: We give the name of function to the operation which associates with every element x the element y which is in the given relation with x; y is said to be the value of the function at the element x, and the function is said to be determined by the given...
  49. A

    Quaternion Derivative: Product Rule Explained

    How does the quaternion derivative work in the presence of a quaternion product. More specifically, does the standard product rule apply for quaternion derivatives? Say, I have a function f(q) = q* x a x q [where q -> quaternion, a -> const vector x-> quat prod] what is the result of the...
  50. E

    Intuition for sign of third derivative

    [I asked this question over a year ago, but I thought I'd try again.] Let ##I\subseteq \mathbb R## be an interval and ##f:I\to\mathbb R## be a ##C^\infty## function. I have the following characterizations: 1) ##f'\geq 0## everywhere iff ##f## is increasing. 2) ##f''\geq 0## everywhere iff...
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