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. K

    I Derivative of a function is equal to zero

    Suppose: - that I have a function ##g(t)## such that ##g(t) = \frac{dy}{dt} ##; - that ##y = y(x)## and ##x = x(t)##; - that I take the derivative of ##g## with respect to ##y##. One one hand this is ##\frac{dg}{dy} = \frac{dg}{dx}\frac{dx}{dy} = \frac{d^2 y}{dxdt}\frac{dx}{dy}##. On the other...
  2. karush

    MHB 2.2.1 AP Calculus Exam .... derivative with ln

    If $f(x)=7x-3+\ln(x),$ then $f'(1)=$ $a.4\quad b. 5\quad c. 6\quad d. 7\quad e. 8$ see if you can solve this before see the proposed solution
  3. V

    I Derivative consisting Levi-Civita

    I've got here so far, but first of all I'm not sure if i did it right till the last line and second, if I've been right, i do not know what to do with the rest. should i consider each of levi-civita parentheses in the last line zero? and one additional question about the term in the first line...
  4. jk22

    I Integration : Are a function and it's derivative independent?

    The question is a bit confused, but it refers to if the following integration is correct : $$I=\int \frac{1}{1+f'(x)}f'(x)dx$$ $$df=f'(x)dx$$ $$\Rightarrow I=\int\frac{1}{1+f'}df=?\frac{f}{1+f'}+C$$ The last equality would come if I suppose $f,f'$ are independent variables.
  5. M

    A Solving Covariant Derivative Notation Confusion

    I've stumbled over this article and while reading it I saw the following statement (##\xi## a vectorfield and ##d/d\tau## presumably a covariant derivative***): $$\begin{align*}\frac{d \xi}{d \tau}&=\frac{d}{d \tau}\left(\xi^{\alpha} \mathbf{e}_{\alpha}\right)=\frac{d \xi^{\alpha}}{d \tau}...
  6. M

    Dyson's series and the time derivative

    I'm having a hard time understanding how exactly to evaluate the expression} $$\partial_t \mathcal{T}\exp\left(-i S(t)\right)\quad \text{where}\quad S(t)\equiv\int_{t_0}^tdu \,H(u) .$$ The confusing part for me is that if we can consider the following: $$\partial_t \mathcal{T}\exp\left(-i...
  7. karush

    MHB How to Find the Second Derivative with Given Equation at a Specific Point?

    If $(x+2y)\cdot \dfrac{dy}{dx}=2x-y$ what is the value of $\dfrac{d^2y}{dx^2}$ at the point (3,0)? ok not sure of the next step but $\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$
  8. Arman777

    What is the Derivative of the Scale Factor in Cosmology?

    In cosmology we have a scale factor that depends only on time ##a(t)##. Now how can I solve this thing $$\frac{d}{da}(\dot{a}(t)^{-2}) = ?$$ Is it 0 ? Or something else ?
  9. Vyrkk

    A Covariant derivative and connection of a covector field

    I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors. What people usually do is take the covariant derivative of the covector acting on a vector, the result being a scalar Invoke a product rule to...
  10. M

    Covariant derivative of a (co)vector field

    My attempt so far: $$\begin{align*} (\nabla_X Y)^i &= (\nabla_{X^l \partial_l}(Y^k\partial_k))^i=(X^l \nabla_{\partial_l}(Y^k\partial_k))^i\\ &\overset{2)}{=} (X^l (Y^k\nabla_{\partial_l}(\partial_k) + (\partial_l Y^k)\partial_k))^i = (X^lY^k\Gamma^n_{lk}\partial_n + X^lY^k{}_{,l}\partial_k)^i\\...
  11. V

    Finding the derivative and rate of change

    I tried solving this question a few ways and this one logically made the most sense however I got it wrong and I am unsure of why. I first plugged in t=2 into p(t). p(2)=0.3(2)1/2+6.3 to obtain 6.724264069. I then found the derivative of D(p) which is D'(p)=-60000/p3. I plugged in...
  12. Physyx

    How to model a rocket equation from the derivative of momentum?

    I am using the derivative of momentum (dp/dt) with Newton’s 3rd Law with the gravitational force of Earth. F - [Force of gravity on rocket] = dp/dt F - (G * m_e * m_r / r2 ) = v * dm/dt + ma F = Force created by fuel (at time t) G = Gravitational Constant m_e = Mass of Earth m_r = Mass of...
  13. NP04

    Derivative of a cylinder's volume

    Using product rule, we have: [d/dx] (πr^2)(h) = (πr^2)(1 ) + (2πr)(h) Why is the two there? V = 2 πrh+2πr^2 The derivative of h is 1, not 2. Please help!
  14. karush

    MHB 3.3.04 AP Calculus Exam 2nd derivative

    Ok not sure if I fully understand the steps but presume the first step would be divide both sides deriving$$\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$$offhand don't know the correct answer $\tiny{from College Board}$
  15. T

    I How Does the Dot Product of Vector Derivatives Relate to Their Original Vectors?

    Summary: The Dot Product of a Vector 'a' and a Derivative 'b' is the same like the negative of the Derivative 'a' and the Vector 'b'. Hello, I have the following Problem. The Dot Product of a Vector 'a' and a Derivative 'b' is the same like the negative of the Derivative 'a' and the Vector...
  16. sviego

    Why do we use"dx" in the derivative dy/dx?

    My theory is that dx = 1(x^0) = 1, which would mean d/dx(x^2) = 2(x^1)/1(x^0) = 2x/1 = 2x. I know that the derivative is literally the change in "y" over change in "x," but am confused as to what value the change in "x" has.
  17. DiracPool

    B Derivative and integral of the exponential e^t

    If I take the exponential function e^t and take the derivative, I think I get the same e^t. Even if I keep doing it over and over, second, third derivative, etc. My admittedly naive question, though, is this symmetric? Meaning...if I take the the integral of e^t, do I just get the reverse or...
  18. R

    What Does -∂V/∂x Represent in Newton's Law?

    In one of my textbooks about quantum mechanics, they mention a vehicle moving in a straight line along the x axis. With Newtons first law they take the second derivative from a which is d^2x/dt^2 and that should be equal to -∂V/∂x. What exactly does -∂V indicate? The complete equation...
  19. George Keeling

    I Metric compatibility and covariant derivative

    Sean Carroll says that if we have metric compatibility then we may lower the index on a vector in a covariant derivative. As far as I know, metric compatibility means ##\nabla_\rho g_{\mu\nu}=\nabla_\rho g^{\mu\nu}=0##, so in that case ##\nabla_\lambda p^\mu=\nabla_\lambda p_\mu##. I can't see...
  20. Another

    What is the Savitzky-Golay 2nd derivative method?

    How are Savitzky–Golay second derivative different from Savitzky–Golay smoothing?
  21. Matt & Hugh play with a Brick and derive Centripetal Acceleration

    Matt & Hugh play with a Brick and derive Centripetal Acceleration

    Matt and Hugh play with a tennis ball and a brick. Then they do some working out to derive the formula for the centripetal force (a = v^2/r) by differentiati...
    • scottdave
    • Media item
    • accelaration centrifugal centripetal derivative vector
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    • Category: Classical Mechanics
  22. barryj

    Having trouble finding the derivative of an inverse function

    Summary: Please see the attached problem and solution The answer is 1/5. I have tried various solutions and cannot get 1/5. What is my error? [Moderator's note: Moved from a technical forum and thus no template.]
  23. Math Amateur

    MHB Complex Derivative .... Remark in Apostol, Section 16.1 .... ....

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 16: Cauchy's Theorem and the Residue Calculus ... I need help in order to fully understand a remark of Apostol in Section 16.1 ... The particular remark reads as follows: Could someone...
  24. barryj

    How do I calculate the derivative of the inverse sin and inverse tan

    I calculated an expression for the derivative of the inverse tan but I did not use the identity as suggested. Why did I need to use this identity. Did I do the problem correctly? I got the correct answer. I tried to do the derivative of the inverse sin the same way. I used the same figure 1 on...
  25. V

    Find the derivative of the function V(P)= k/P

    I tried to find the derivative of the function V(P)= k/P which I found to be: V'(P) = kP-1 V'(P) = (1)(-1)(P)-1-1 = -1/(P2) And then I substituted in 1.30 into the derivative to obtain -0.5917 L/atm. And I am kind of confused how to actually find the derivative of this. I thought I was on...
  26. Kisok

    I Covariant Derivative: 2nd Diff - My Question

    My question is shown in Summary section. Please see the attached file.
  27. E

    B How to tell if a quantity is defined as a derivative or a ratio

    I appreciate that this is perhaps a strange question but it's been bugging me a little. For instance, velocity is defined as the time derivative of position, so will always appear as the gradient a graph of x vs t. However, something like resistance as R = V/I is defined in terms of a ratio...
  28. K

    Help with limits and a derivative

    I don't understand my textbook, so I simply don't understand how to solve this math problem. I really appreciate some help!
  29. V

    Time derivative jump of the electric/magnetic field

    So I just wanted to see if anyone could offer some suggestions. So in my mind this seems impossible, in the case of electric field a jump in time derivative of that field would indicated in my mind that electric charge was either introduced or removed from the system instantaneously which would...
  30. looseleaf

    A Derivative of a Lorentz-Transformed Field

    Please help me understand this line from P&S, or point me towards some resources: Why is there another Lorentz transformation acting on the derivative on the RHS? Thanks
  31. D

    I Lorentz transformation of derivative and vector field

    I'm currently watching lecture videos on QFT by David Tong. He is going over lorentz invariance and classical field theory. In his lecture notes he has, $$(\partial_\mu\phi)(x) \rightarrow (\Lambda^{-1})^\nu_\mu(\partial_\nu \phi)(y)$$, where ##y = \Lambda^{-1}x##. He mentions he uses active...
  32. Pencilvester

    I Lie derivative of hypersurface basis vectors along geodesic congruence

    Hello PF, here’s the setup: we have a geodesic congruence (not necessarily hypersurface orthogonal), and two sets of coordinates. One set, ##x^\alpha##, is just any arbitrary set of coordinates. The other set, ##(\tau,y^a)##, is defined such that ##\tau## labels each hypersurface (and...
  33. haushofer

    A Issue with the definition of a Lie derivative and its components (Carroll's GR)

    Dear all, I'm having a small issue with the notion of Lie-derivatives after rereading Carroll's notes https://arxiv.org/abs/gr-qc/9712019 page 135 onward. The Lie derivative of a tensor T w.r.t. a vector field V is defined in eqn.(5.18) via a diffeomorphism ##\phi##. In this definition, both...
  34. A

    A Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w))

    Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w)) Hello to my Math Fellows, Problem: I am looking for a way to calculate w-derivative of Fourier transform,d/dw (F{x(t)}), in terms of regular Fourier transform,X(w)=F{x(t)}. Definition Based Solution (not good enough): from...
  35. P

    I Christoffel symbols and covariant derivative intuition

    So I'm trying to get sort of an intuitive, geometrical grip on the covariant derivative, and am seeking any input that someone with more experience might have. When I see ##\frac {\partial v^{\alpha}}{\partial x^{\beta}} + v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}##, I pretty easily see a...
  36. maistral

    A Deriving the derivative boundary conditions from natural formulation

    PS: This is not an assignment, this is more of a brain exercise. I intend to apply a general derivative boundary condition f(x,y). While I know that the boxed formulation is correct, I have no idea how to acquire the same formulation if I come from the general natural boundary condition...
  37. P

    I Using the derivative of a tangent vector to define a geodesic

    I hope I'm asking this in the right place! I'm making my way through the tensors chapter of the Riley et al Math Methods book, and am being tripped up on their discussion of geodesics at the very end of the chapter. In deriving the equation for a geodesic, they basically look at the absolute...
  38. L

    I Understanding the definition of derivative

    As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in...
  39. karush

    MHB 2.2.206 AP Calculus Practice question derivative of a composite sine

    206 (day of year number) If $f(x)=\sin{(\ln{(2x)})}$, then $f'(x)=$ (A) $\dfrac{\sin{(\ln{(2x)}}}{2x}$ (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$ (C) $\dfrac{\cos{(\ln{(2x)}}}{2x}$ (D) $\cos{\left(\dfrac{1}{2x}\right)}$ Ok W|A returned (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$ but I didn't understand why...
  40. George Keeling

    I Question about a partial derivative

    I apologise for the length of this question. It is probably possible to answer it by reading the first few lines. I fear I have made a childish error: I am working on the geodesic equation for the surface of a sphere. While doing so I come across the partial derivative \begin{align}...
  41. F

    A The partial derivative of a function that includes step functions

    I have this function, and I want to take the derivative. It includes a unit step function where the input changes with time. I am having a hard time taking the derivative because the derivative of the unit step is infinity. Can anyone help me? ##S(t) = \sum_{j=1}^N I(R_j(t)) a_j\\ I(R_j) =...
  42. hilbert2

    I Idea about single-point differentiability and continuity

    Many have probably seen an example of a function that is continuous at only one point, for example ##f:\mathbb{R}\rightarrow\mathbb{R}\hspace{5pt}:\hspace{5pt}f(x)=\left\{\begin{array}{cc}x, & \hspace{6pt}when\hspace{3pt}x\in\mathbb{Q} \\ -x, &...
  43. E

    I Derivative of the Ad map on a Lie group

    Hi, let ##G## be a Lie group, ##\varrho## its Lie algebra, and consider the adjoint operatores, ##Ad : G \times \varrho \to \varrho##, ##ad: \varrho \times \varrho \to \varrho##. In a paper (https://aip.scitation.org/doi/full/10.1063/1.4893357) the following formula is used. Let ##g(t)## be a...
  44. snoopies622

    I The vanishing of the covariant derivative of the metric tensor

    I brought up this subject here about a decade ago so this time I'll try to be more specific to avoid redundancy. In chapter five of Bernard F. Schutz's A First Course In General Relativity, he arrives at the conclusion that in flat space the covariant derivative of the metric tensor is zero...
  45. Robin04

    I Mean of the derivative of a periodic function

    We have a periodic function ##f: \mathbb{R} \rightarrow \mathbb{R}## with period ##T, f(x+T)=f(x)## The statement is the following: $$\frac{1}{T}\int_0^T f(x)dx =0 \implies \frac{1}{T}\int_0^T\frac{d}{dx} f(x)dx =0$$ Can you give me a hint on how to prove/disprove it? The examples I tried all...
  46. berlinspeed

    A Semicolon notation in component of covariant derivative

    Can someone clarify the use of semicolon in I know that semicolon can mean covariant derivative, here is it being used in the same way (is expandable?) Or is a compact notation solely for the components of?
  47. Mcp

    B A function's derivative being not defined for some X but having a limit

    Let's say I have a function whose derivative is (tan(x)-sin(x))/x. It is not defined for X=0 but as X approaches 0 the derivative approaches 0, so should I conclude that my function is not differentiable at X=0 or should I conclude that the derivative at X=0 is 0.
  48. A

    I Partial Derivative: Correct Formulation?

    If given a function ##u(x,y) v(x,y)## then is it correct to write ##\frac{\partial }{\partial x}u(x,y)v(x,y)=\frac{u(x+dx,y)v(x+dx,y)-u(x,y)v(x,y)}{dx}##??
  49. Physics lover

    Derivative of a Series: How to Solve for x^n/((x-a1)(x-a2)...(x-an)) in Calculus

    I first solved the first two terms and then i solved the resulting term with the third term and so on.At last i was left with x^n/((x-a1)(x-a2)...(x-an)) .Thrn i took log on both sides and then differentiated both sides with respect to x.I got 1/y dy/dx=n/x -1/(x-a1)-1/(x-a2)...-1/(x-an).But now...
  50. R

    Commutativity of partial and total derivative

    Problem Statement: Use the definition of the total time derivative to a) show that ##(∂ /∂q)(d/dt)f(q,q˙,t) = (d /dt)(∂/∂q)f(q,q˙,t)## i.e. these derivatives commute for any function ##f = f(q, q˙,t)##. Relevant Equations: My approach is given below. Please tell if it is correct and if not ...
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