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. A

    Finding Directional Derivative

    The gradient is < (2x-y), (-x+2y-1) > at P(1,-1) the gradient is <3, -4> Since ∇f⋅u= Direction vector, it seems that we should set the equation equal to the desired directional derivative. < 3, -4 > ⋅ < a, b > = 4 which becomes 3a-4b=4 I thought of making a list of possible combinations...
  2. sergiokapone

    I Covariant derivative of the contracted energy-momentum tensor of a particle

    The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is \begin{equation} T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}. \end{equation} Let contract...
  3. S

    Velocity, acceleration, jerk, snap, crackle, pop, stop, drop, roll....

    Edit: I see this was discussed in the related thread sorry for a repost. If acceleration causes a change in velocity, and jerk causes a change in acceleration, snap causes a change in jerk, crackle causes a change in snap, pop causes a change in crackle, stop causes a change in pop, drop causes...
  4. D

    Drawing a derivative of a function

    Red line being the function and blue an approximation of the derivative. Does it look right?
  5. T

    I Derivative operators in Galilean transformations

    I'm studying how derivatives and partial derivatives transform under a Galilean transformation. On this page: http://www.physics.princeton.edu/~mcdonald/examples/wave_velocity.pdf Equation (16) relies on ##\frac{\partial t'}{\partial x}=0## but ##\frac{\partial x'}{\partial t}=-v## But this...
  6. A

    Finding the partial derivative from the given information

    It seems that the way to combine the information given is z = f ( g ( (3r^3 - s^2), (re^s) ) ) we know that the multi-variable chain rule is (dz/dr) = (dz/dx)* dx/dr + (dz/dy)*dy/dr and (dz/ds) = (dz/dx)* dx/ds + (dz/dy)*dy/ds ---(Parentheses indicate partial derivative) other perhaps...
  7. Celso

    I Partial derivative interpretation

    How do I interpret geometrically the partial derivative in respect to a constant of a function such as ##\frac{ \partial}{\partial c} (acos(x) + be^x + c)^2##?
  8. L

    I Lorenz gauge, derivative of field tensor

    Fμν = ∂μAν- ∂νAμ ∂μFμν = ∂2μAν - ∂ν(∂μAμ) = ∂2μAνWhy ∂ν(∂μAμ) and not ∂μ∂νAμ ? And why does ∂ν(∂μAμ) drop out? thank you
  9. maistral

    I Verification regarding Neumann conditions at time derivative

    Hi, just a question regarding neumann conditions, I seem to have forgotten these things already. I think this question is answerable by a yes or a no. So given the 2D heat equation, If I assign a neumann condition at say, x = 0; Does it still follow that at the derivative of t, the...
  10. cookiemnstr510510

    MATLAB MATLAB GUI derivative calculator

    Hello! Happy Sunday, I am trying to create a MATLAB GUI that can take an input ( i am starting out with just one variable) and take its derivative and display the result. I have attached pictures of what my GUI looks like, and also the code I wrote so far. Let me also describe my method: I have...
  11. A

    Trying to calculate the time derivative of a position differential

    here I am trying to find ##\frac{d}{dt}dx## where ##x(t)## is the position vector Now ##\frac{d}{dt}(v_x(x,y,z,t)dt)=\frac{dv_x}{dt}dt=\frac{\partial v_x}{\partial t}dt+\frac{\partial v_x}{\partial x}dx+\frac{\partial v_x}{\partial y}dy+\frac{\partial v_x}{\partial z}dz## Now dividing by ##dx##...
  12. E

    I The partial time derivative of Hamiltonian vs Lagrangian

    I have been reading a book on classical theoretical physics and it claims: -------------- If a Lagrange function depends on a continuous parameter ##\lambda##, then also the generalized momentum ##p_i = \frac{\partial L}{\partial\dot{q}_i}## depends on ##\lambda##, also the velocity...
  13. fazekasgergely

    Infinite series to calculate integrals

    For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
  14. A

    A Independent variables in higher derivative Lagrangian

    Consider the following Lagrangian density $$\mathscr{L}=\mathbf{E}\cdot\left(\nabla^{2}\mathbf{E}\right)$$ where $$E_{i}=\partial_{i}\phi\;(i=x,y,z )$$. In th is case the potential and its 3rd derivatives are the independent variables. Acording to Barut's classical theory of fields book, for...
  15. olgerm

    The derivative of velocity with respect to a coordinate

    Why ##\frac{\partial (0.5*m*\dot{x}^2-m*g*x)}{\partial x}=-mg##? why ##\frac{\partial \dot{x}}{\partial x}=0##? Why ##\frac{\partial (0.5*m*\dot{x}^2-m*g*x)}{\partial \dot{x}}=m*\dot{x}## ? why ##\frac{\partial x}{\partial \dot{x}}=0##? Does it assume that speed is same at every location? I...
  16. S

    Positive derivative implies growing function using Bolzano-Weierstrass

    I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement: $$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the...
  17. karush

    MHB 205.8.9 Find the derivative of the function

    205.8.9 Find the derivative of the function $y=\cos(\tan(5t-4))\\$ chain rule $u=\tan(5t-4)$ $\frac{d}{du}\cos{(u)} \frac{d}{dt}\tan{\left(5t-4\right)}\\$ then $-\sin{\left (u \right )}\cdot 5 \sec^{2}{\left (5 t - 4 \right )}\\$ replacing u with $\tan(5t-4)$ $-\sin{(\tan(5t-4))}\cdot 5...
  18. SamRoss

    B Justification for cancelling dx in an integral

    In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))... ## \int_0^\phi \frac {d} {dx} f(x) dx =...
  19. M

    How to Solve a Derivative Presented in a Non-Standard Format?

    I have attached a word document demonstrating the working out cos i was too lazy to learn how Latex primer works and writing it like I did above would've been too hard too read. I tried to make it as understandable as possible, presenting fractions as ' a ' instead of ' a / b ' . ------ b
  20. M

    I Uncovering the Derivative of a Tensor: Understanding its Equations and Origins

    How/why does the first equal sign hold? Where does each derivative come from:
  21. M

    Applying the Limit Method to Find Derivatives: A Simple Explanation

    First I am sorry if this question is futile, or even stupid but I find it confusing, and I would like to clarify in my mind. Taking the derivative ##y=x^2## with respect to x by using limit is very easy stuff. But would you please illustrate how to apply this to the equation ##y^2=x^2## It...
  22. H

    A Variational derivative of mean curvature

    Hello Physics Forums, I have a simple parametric surface in R3 <x,y,z(x,y)>. I've calculated the the usual mean curvature: H= ((1+hx^2)hyy-2hxhyhxy+(1+hy^2)hxx)/(1+hx^2+hy^2)^3/2 I needed to take the variational derivative of this expression. Since it has second order spatial derivatives the...
  23. A

    I Connecting Geodesic Curves and the Covariant Derivative

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  24. A

    I Covariant derivative of tangent vector for geodesic

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  25. E

    A Variational derivative and Euler-Poincare equations

    Hi, I'm trying to understand the Euler-Poincare equations, which reduce the Euler-Lagrange equations for certain Lagrangians on a Lie group. I'm reading Darryl Holm's "Geometric mechanics and symmetry", where he suddenly uses what seems to be a variational derivative, which I'm having a hard...
  26. CricK0es

    Derivative of a term within a sum

    Homework Statement [/B] From the Rodrigues’ formulae, I want to derive nature of the spherical Bessel and Neumann functions at small values of p. Homework Equations [/B] I'm going to post an image of the Bessel function where we're using a Taylor expansion, which I'm happy with and is as far...
  27. navneet9431

    What Is the Limit of (x - tan(x)) / x³ as x Approaches 0?

    <Moderator's note: Moved from a technical forum and thus no template.> $$\lim_{x\rightarrow 0} (x-tanx)/x^3$$ I solve it like this, $$\lim_{x\rightarrow 0}1/x^2 - tanx/x^3=\lim_{x\rightarrow 0}1/x^2 - tanx/x*1/x^2$$ Now using the property $$\lim_{x\rightarrow 0}tanx/x=1$$,we have ...
  28. JosephTraverso2

    Find the derivative of a balloon's circumference

    A balloon's volume is increasing at a rate of dV/dt. Express the rate of change of the circumference with respect to time (dc/dt) in terms of the volume and radius. Homework Equations Vsphere = (4/3)(π)(r^3) C = (2)(π)(r) The Attempt at a Solution [/B] My strategy was to come up with two...
  29. Math Amateur

    MHB Remarks by Fortney Following Theorems on Directional Derivative ....

    I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Chapter 2: An Introduction to Differential Forms ...I need help to understand some remarks by Fortney following Theorems 2.1 and 2.2 on the directional...
  30. filip97

    I The Lagrangian and the second derivative?

    Why Lagrangian not depend of higher derivatives of generalised coordinates ?
  31. U

    I Proving Alternating Derivatives with Induction in Mathematical Analysis I

    Hi forum. I'm trying to prove a claim from Mathematical Analysis I - Zorich since some days, but I succeeded only in part. The complete claim is: $$\left\{\begin{matrix} f\in\mathcal{C}^{(n)}(-1,1) \\ \sup_{x\in (-1,1)}|f(x)|\leq 1 \\ |f'(0)|>\alpha _n \end{matrix}\right. \Rightarrow \exists...
  32. Abhishek11235

    A Calculating Lie Derivative for Case (ii)

    I am relatively new to differential geometry. I am studying it from Fecko Textbook on differential geometry. As soon as he introduces the concept of lie derivative,he asks to do exercise 4.2.2 in picture. The question is,how do I apply ##\phi^*## to given function ##\psi## . I know that...
  33. Jozefina Gramatikova

    I How to differentiate with respect to a derivative

    Hi guys, I am reading my lecture notes for Mechanics and Variations and I am trying to understand the maths here. From what I can see there we differentiated with respect to a derivative. Could you tell me how do we do that? Thanks
  34. J

    A derivative identity (Zangwill)

    Homework Statement Without using vector identities, show that ##\nabla \cdot [\vec{A}(r) \times \vec{r}] = 0##. Homework Equations The definitions and elementary properties of the dot and cross products in terms of Levi-Civita symbols. The "standard" calculus III identities for the divergence...
  35. D

    I Derivative of f() as a function of a Laplacian

    I need a little help with understanding a differential relationship between functions. If g and f are vector fields and f(g(x,y),q(x,y))=∇2g(x,y) How could you, if possible, express ∂f/∂g explicitly? Please help a bit confused.
  36. K

    Meaning of subscript in partial derivative notation

    Homework Statement I'm given a gas equation, ##PV = -RT e^{x/VRT}##, where ##x## and ##R## are constants. I'm told to find ##\Big(\frac{\partial P}{\partial V}\Big)_T##. I'm not sure what that subscript ##T## means? Homework Equations ##PV = -RT e^{x/VRT}## Thanks a lot in advance.
  37. D

    I Component derivative of a tensor

    This is a simple and maybe stupid question. Can you take a derivative of a vector component with respect to a vector? Or even more generally,can you take the derivative of a component of a tensor with respect to the whole tensor? For instance in the cauchy tensor could you take the xx component...
  38. A

    I Contravariant derivative of tensor of rank 1

    If we have two sets of coordinates such that x1,x2...xn And y1,y2,...ym And if any yi=f(x1...,xn)(mutually dependent). Then dyi=(∂yi/∂xj)dxj Again dyi/dxk=(∂2yi/∂xk∂xj)dxj+∂yi/∂xk Is it the contravariant derivative of a vector?? Or in general dAi/dxk≠∂Ai/∂xk
  39. A different perspective to derivatives rather than slopes-3b1b

    A different perspective to derivatives rather than slopes-3b1b

    A different perspective to derivatives.
    • YoungPhysicist
    • Media item
    • calculas derivative
    • Comments: 0
    • Category: Calculus
  40. E

    I Second Derivative of Time Dilation Equation

    Hello all. I was playing around with the time dilation equation : √(1-v2/c2) Specifically, I decided to take the derivative(d/dv) of the equation. Following the rules of calculus, as little of them as I know, I got this: d/dv(√(1-v2/c2) = v / (c2√(1-v2/c2)). Now, this seems reasonable enough...
  41. Math Amateur

    I Computing the Directional Derivative ....

    I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Chapter 2: An Introduction to Differential Forms ... I need help with Question 2.4 (a) (i) concerned with computing a directional derivative ...
  42. YoungPhysicist

    Understanding the Derivative of a Polynomial with Exponent 10

    Homework Statement ##f(x) = (5x+6)^{10} , f'(x)=?## Homework Equations ##\frac{d}{dx}x^n = nx^{n-1}##? 3. The Attempt at a Solution [/B] I do know the solution ##f'(x) = 50(5x+6)^9##,but I don't know how this solution came to be.I downloaded this problem from the web and it only comes with...
  43. C

    I Mass counter term is a derivative at tree level?

    I've heard the statement that by computing just the leading-order (tree level) diagrams of a process and then computing the derivative of this result with respect to the mass should correspond to the evaluation of the mass counter term diagrams. Can someone explain why this statement is...
  44. Math Amateur

    MHB The Derivative in Several Variables .... Hubbard and Hubbard, Section 1.7 ....

    I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard. I am currently focused on Section 1.7: Derivatives in Several Variables as Linear Transformations ... I need some help in order to understand some...
  45. Specter

    Derivative of a sinusoidal function

    Homework Statement What is the derivative of ##f(x)=\frac {2x^2} {cos x}##? Homework EquationsThe Attempt at a Solution ##F(x)=\frac {2x^2} {cos x}## So... ##f(x)=2x^2## and ##f'(x)=4x## ##g(x)=cosx## and ##g'(x)=-sinx## If I plug these into the quotient rule I thought that I would get...
  46. 1

    I Derivative of the area is the circumference -- generalization

    I thought you guys might appreciate this. A lot of people notice that the derivative of area of a circle is the circle's circumference. This can be generalized to all regular polygons in a nice way.
  47. T

    Is this derivative in terms of tensors correct?

    Homework Statement Solve this, $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}$$ where q is a constant vector. Homework EquationsThe Attempt at a Solution $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial x^{\nu}}...
  48. SebastianRM

    I What is the 'formal' definition for Total Derivative?

    A total derivative dU = (dU/dx)dx + (dU/dy)dy + (dU/dz)dz. I am unsure of how to use latex in the text boxes; so the terms in parenthesis should describe partial differentiations. My question is, where does this equation comes from?
  49. J

    Determine the period of small oscillations

    Homework Statement Two balls of mass m are attached to ends of two, weigthless metal rods (lengths l1 and l2). They are connected by another metal bar. Determine period of small oscillations of the system Homework Equations Ek=mv2/2 v=dx/dt Conversation of energy 2πsqrt(M/k) The Attempt at a...
  50. YoungPhysicist

    B Is This a Valid Proof for the Number of Roots in a nth Degree Polynomial?

    Recently I came up with a proof of “ for a nth degree polynomial, there will be n roots” Since the derivative of a point will only be 0 on the vertex of that function,and a nth degree function, suppose ##f(x)##has n-1 vertexes, ##f’(x)## must have n-1 roots. Is the proof valid?
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