What is Derivative: Definition and 1000 Discussions

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. mopit_011

    B Derivative of Square Root of x at 0

    When you use the power rule to differentiate the square root, the result is 1/2(sqrt. x) which is undefined at 0. But, when you use the definition of the definition of the derivative to calculate it, the result is infinity. What causes this difference between these two methods?
  2. chwala

    Find the total derivative of ##u## with respect to ##x##

    see attached below; the textbook i have has many errors... clearly ##f_x## is wrong messing up the whole working to solution...we ought to have; ##\frac {du}{dx}=(9x^2+2y)+(2x+8y)3=9x^2+2y+6x+24y=9x^2+6x+26y##
  3. Purplepixie

    MHB Why is the derivative of |x| not defined at x=0?

    I would like to know how to differentiate |sin(t)| to obtain d(|sin(t)|)/d(t). Thank you!
  4. L

    I The time derivative of kinetic energy

    Lets consider T(\vec{p})=\frac{\vec{p}^2}{2m}=\frac{\vec{p}\cdot \vec{p}}{2m}. Then \frac{dT}{dt}=\vec{v}\cdot \vec{F}. And if we consider T=\frac{p^2}{2m} than \frac{dT}{dt}=\frac{1}{2m}2p\frac{dp}{dt} Could I see from that somehow that this is \vec{v}\cdot \vec{F}?
  5. rudransh verma

    B Speed/velocity as a derivative

    [Note: Link to the quote below has been pasted in by the Mentors -- please always provide attribution when quoting another source] https://www.feynmanlectures.caltech.edu/I_08.html Let s=16t^2 and we want to find speed at 5 sec. s = 16(5.001)2 = 16(25.010001) = 400.160016 ft. In the last...
  6. ergospherical

    I Exterior Covariant Derivative End(E)-Valued Forms

    i) Let ##\pi : E \rightarrow M## be a vector bundle with a connection ##D## and let ##D'## be the gauge transform of ##D## given by ##D_v's = gD_v(g^{-1}s)##. Show that the exterior covariant derivative of ##E##-valued forms ##\eta## transforms like ##d_{D'} \eta = gd_D(g^{-1}\eta)##. ii) Show...
  7. T

    A Covariant Derivative of Stress Energy Tensor of Scalar Field on Shell

    Hi all, I am currently trying to prove formula 21 from the attached paper. My work is as follows: If anyone can point out where I went wrong I would greatly appreciate it! Thanks.
  8. yucheng

    Contravariant derivative of a tensor field in terms of generalized coordinates?

    1. The laplacian is defined such that $$ \vec{\nabla} \cdot \vec{\nabla} V = \nabla_i \nabla^i V = \frac{1}{\sqrt{Z}} \frac{\partial}{\partial Z^{i}} \left(\sqrt{Z} Z^{ij} \frac{\partial V}{\partial Z^{j}}\right)$$ (##Z## is the determinant of the metric tensor, ##Z_i## is a generalized...
  9. yucheng

    Derivative of Determinant of Metric Tensor With Respect to Entries

    We know that the cofactor of determinant ##A##, is $$\frac{\partial A}{\partial a^{r}_{i}} = A^{i}_{r} = \frac{1}{2 !}\delta^{ijk}_{rst} a^{s}_{j} a^{t}_{k} = \frac{1}{2 !}e^{ijk} e_{rst} a^{s}_{j} a^{t}_{k}$$ By analogy, $$\frac{\partial Z}{\partial Z_{ij}} = \frac{1}{2 !}e^{ikl} e^{jmn}...
  10. Z

    Mathematica Mathematica bug? (Solving PDEs when the initial conditions contain a derivative)

    hello I own mathematica 10.02 it is virtually impossible to solve PDE's ,even with NDSolve,if the initial conditions contain a derivative I write Derivative[1,0] [0,x] == f[x] I mean the first t derivative of u[t,x] for x at t=0 is f[x] I own a book based on Mathematica 10.3 Even if a...
  11. msrultons

    Find the derivative of y= u^5/(1+u^3) from 8 to 8-7x

    Here is the problem Here is my work on it. I thought I did it correct, but again, was told it was wrong.
  12. S

    I Derivative of F = f(x)/f(x+dx)

    Hello, I'm struggling with this for some time. So I have the function: f(x) = sqrt(1 - 1/x) The derivative of this function can be easily calculated. Now we define the function: F(x) = f(x)/f(x + dx) = sqrt(1 - 1/x)/sqrt(1 - 1/(x+dx)) I have a hard time to find F'(x) due to the presence of...
  13. F

    Derivative of the deformation gradient w.r.t Cauchy green tensor

    What's the derivative of deformation gradient F w.r.t cauchy green tensor C, where C=F'F and ' denotes the transpose?
  14. G

    Numerical computation of the derivative

    I'm not sure if this is the correct forum to post this question, or should I post it in a math forum. But I was looking at some code when I found a 'strange' implementation to compute the derivative of a function, and I wanted to know if any of you has an idea of why such an implementation is...
  15. S

    Mathematica Derivative, i.e. D[ ] , of Re [ something ]

    Consider these two examples: D[ Re[ Exp[ I*t ] ], t ] D[Re[Exp[I*t]],t] /. t-> 0.5 Mathematica seems to get stuck differentiating the "Re[ ]" function after (rather naively) applying the chain rule. This is a trivial example, but we might have a more complicated function defined like...
  16. L

    I Second derivative of chained function

    Let's say we have a function ##M(f(x))## where ##M: \mathbb{R}^2 \to \mathbb{R}^2## is a multivariable linear function, and ##f: \mathbb{R} \to \mathbb{R}^2## is a single variable function. Now I'm getting confused with evaluating the following second derivative of the function: $$ [M(f(x))]''...
  17. Vividly

    I Question about Inverse Derivative Hyperbola function

    Im confused about a certain part of solving an equation. So I used the hyerbola formula to find the answer but I think I did the math wrong. X^2-y^2=c^2 X=1 Y= (2x^5-1)^2 I did the calculations as you can see in the picture but I know I messed up on the square root part. When you square one...
  18. M

    I Time derivative using the quotient rule

    Hi Guys Sorry for the rudimentary post. I am busy with a numerical solution to a mechanics problem, and the results are just not as expected. I am re-checking the mathematics to ensure that all is in order in doing so I am second guessing a few things Referring to the attached scan, is the...
  19. redtree

    B The norm of the derivative of a vector

    Is the following true? ##\left| \frac{d\vec{u}}{d t} \right| \overset{?}{=} \frac{d |\vec{u}|}{d |t|}##
  20. Poetria

    Directional derivative and hiking

    $$h_x=y$$ $$h_y=x$$ Substituting the coordinates of a given point: $$y'=-\frac {y} {x}$$ $$y'=-\frac {1} {2}$$ A unit vector: $$\frac {1} {\sqrt{5}, \frac {2} {\sqrt{5}}$$ $$D_\vec u h(2,1) = \frac {1} {\sqrt{5}, \frac {2} {\sqrt{5}} \cdot \vec (1,2)$$ $$D_\vec u h(2,1) = \frac {5} {\sqrt{5}}$$
  21. ArisMartinez

    Taking the derivative of a function of a function

    Summary:: According to Yale’s University PHYS: 200: v*(dv/dt) = d(v^2/2)/dt Could someone explain how has he reached that conclusion? He claims to be some standard derivation rules, but I can’t find anything about it. As much as I can tell: (dv/dt)* v = v’ * v = a* v thanks! [Moderator's...
  22. A

    I Time derivative of the angular momentum as a cross product

    I am trying to find the equations of motion of the angular momentum ##\boldsymbol L## for a system consisting of a particle of mass ##m## and magnetic moment ##\boldsymbol{\mu} \equiv \gamma \boldsymbol{L}## in a magnetic field ##\boldsymbol B##. The Hamiltonian of the system is therefore...
  23. chwala

    Finding the second derivative of a given parametric equation

    ok this is pretty straightforward to me, my question is on the order of differentiation, i know that: ##\frac {d^2y}{dx^2}=####\frac {d}{dt}.####\frac {dy}{dx}.####\frac {dt}{dx}## is it correct to have, ##\frac {d^2y}{dx^2}=####\frac {d}{dt}##.##\frac {dt}{dx}##.##\frac {dy}{dx}##? that is...
  24. M

    I Calculating Covariant Derivative of Riemann Tensor in Riemann Normal Coordinates

    Hello everyone, in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the...
  25. Arman777

    A Calculating Lie Derivatives for Tensors & Vectors

    I am writing a code to calculate the Lie Derivatives, and so far, I have defined the Covariant derivative 1) for scalar function; $$\nabla_a\phi \equiv \partial_a\phi~~(1)$$ 2) for vectors; $$\nabla_bV^a = \partial_bV^a + \Gamma^a_{bc}V^c~~(2)$$ $$\nabla_cV_a = \partial_cV_a -...
  26. I

    B Second derivative and inflection points

    Q: See f(t) in graph below. Does the graph of g have a point of inflection at x=4? There is a corner at x=4, so I don't think there is a point of inflection. Does a point of inflection exist where f''(x) does not exist? The solution says there is a point of inflection, could anyone explain why...
  27. E

    A How can I take the derivative?

    The action considered is \begin{align*} S[\Phi] = \int_M dt d^3 x \sqrt{-g} \left( -\frac{1}{2}g^{ab} \partial_a \Phi \partial_b \Phi - \frac{1}{2}m^2 \Phi^2\right) \end{align*}I can "see" unrigorously that variation with respect to ##\partial_t \Phi(x)## it is going to be\begin{align*}...
  28. J

    I Partial Derivative of Convolution

    Hello, I am trying to calculate the partial derivative of a convolution. This is the expression: ##\frac{\partial}{\partial r}(x(t) * y(t, r))## Only y in the convolution depends on r. I know this identity below for taking the derivative of a convolution with both of the functions only...
  29. Eclair_de_XII

    B Proving the Existence and Value of a Derivative at a Point

    This proof has three steps and is very similar to (if not the same as) that other proof I posted here.(1) Prove the existence of a ball centered around ##a## with the property that ##f'## evaluated at any point in the ball is positive. (2) Prove that the right end-point of this ball is bounded...
  30. J

    Does the derivative of a P(V) eqn give the eqn for change in Pressure?

    I know the integral of a P(V) eqn gives an eqn for work. I was wondering if taking the derivative of a P(V) eqn gives an eqn for change in pressure?
  31. L

    A Heisenberg equation of motion -- Partial derivative question

    Heisenberg equation of motion for operators are given by i\hbar\frac{d\hat{A}}{dt}=i\hbar\frac{\partial \hat{A}}{\partial t}+[\hat{A},\hat{H}]. Almost always ##\frac{\partial \hat{A}}{\partial t}=0##. When that is not the case?
  32. J

    Doubt regarding functional derivative for the Thomas Fermi kinetic energy

    I have some doubts with respect on how the functional derivative for the kinetic energy in density functional theory is obtained. I have been looking at this article in wikipedia: https://en.wikipedia.org/wiki/Functional_derivative In particular, I'm interested in how to get the...
  33. GrimGuy

    A Difficulties with derivative of a vector [Landau Textbook]

    Hi guys, I'm having trouble computing a pass 1 to 106.15. It's in the pictures. So, what a have to do is the derivative of ##f## with respect to time and coordinates. Then I need to rearrange the terms to find the equation 106.15. I am using the following conditions. ##r## vector varies in...
  34. J

    I Commutation between covariant derivative and metric

    First, we shall mention that it is known that the covariant derivative of the metric vanishes, i.e ##\nabla_i g_{mn} = 0##. Now I want tro prove the following: $$ \nabla_i A_k = g_{kn}\nabla_i A^n$$ The demonstration I encounter takes advantage of the Leibniz rule: $$ \nabla_i A_k = \nabla_i...
  35. stevendaryl

    I Covariant derivative notation

    [Moderator's note: Thread spun off from previous thread due to topic change.] This thread brings a pet peeve I have with the notation for covariant derivatives. When people write ##\nabla_\mu V^\nu## what it looks like is the result of operating on the component ##V^\nu##. But the components...
  36. S

    A Index notation and partial derivative

    Hi all, I am having some problems expanding an equation with index notation. The equation is the following: $$\frac {\partial{u_i}} {dx_j}\frac {\partial{u_i}} {dx_j} $$ I considering if summation index is done over i=1,2,3 and then over j=1,2,3 or ifit does not apply. Any hint on this would...
  37. docnet

    Help taking a partial derivative

    Hi all, I was wondering is if the following partial derivative can be computed without a specific ##u(t,x)## $$\partial_tu\big[(t,x-t\kappa V)\big]$$ I was thinking it can't be done, because we could have $$u_a(t,x)=tx \Rightarrow \partial_tu\big[(t,x-t\kappa...
  38. mcastillo356

    Thoughts on the derivative of a function

    (a) ##f(x)## is continuous only at ##x=3##: 1- If ##x\in\mathbb Q##, ##f(x)=9## at ##x=3##; around, there is ##\mathbb Q## 2- If ##x\in \mathbb R\setminus \mathbb Q##, this is the set of irrational numbers. Intuitively, if ##x## was in ##\mathbb R##, ##x^2## and ##6(x-3)+9## would meet at...
  39. S

    B Critical points of second derivative

    If the sign on the sign diagram of f" changes from positive to negative or from negative to positive, this means the critical points of f" is non-horizontal inflection of f But what about if the sign does not change? Let say f"(x) = 0 when ##x = a## and from sign diagram of f", the sign on the...
  40. SchroedingersLion

    From the limit of the derivative, infer the behavior of the antiderivative

    Greetings! In statistical mechanics, when studying diffusion processes, one often finds the following reasoning: Suppose there is a strictly positive differentiable function ##f: \mathbb{R} \rightarrow \mathbb{R}## with ## \lim_{x \rightarrow +\infty} {f'(x)} = a > 0##. Then for sufficiently...
  41. mcastillo356

    B Derivative of the product of a function by a constant (possible typo)

    Hi, PF, I think I've found a typo in my textbook. It says: "In the case of a multiplication by a constant, we've got $$(Cf)'(x)=\displaystyle\lim_{h \to{0}}{\dfrac{Cf(x+h)-Cf(x)}{h}}=\displaystyle\lim_{h \to{0}}{\dfrac{f(x+h)-f(x)}{h}}=Cf'(x)$$" My opinion: it should be...
  42. chwala

    Find the derivative of given function and hence find its integral

    ##y=x^2ln x-x## ##\frac {dy}{dx}=2x ln x+x-1## ##\int [2xln x+x-1]\,dx##=##x^2ln x-x##, since ##\int -1 dx= -x## it follows that, ##\int [2x ln x +x]\,dx##=##x^2 ln x## ##\int 2x ln x \,dx = x^2ln x##+##\int x\,dx## ##\int_1^2 xln x\,dx =\frac {x^2ln x}{2}##+##\frac{x^2}{4}##=##2ln2+1-0.25##
  43. Z

    Derivative of Cost Function with Respect to Output Layer Weight

    This is an issue I've been stuck on for about two weeks. No matter how many times I take this derivative, I keep getting the same answer. However, this answer is inevitably wrong. Please help me to understand why it incorrect. To start, I will define an input matrix ##X##, where ##n## is the...
  44. Mr Davis 97

    Help with picking out which point has the most negative derivative

  45. F

    I Why is the derivative defined as the limit of the quotient?

    Hello, The derivative ##\frac {dy} {dx}## "appears" at first glance to be just the ratio of two infinitesimal quantities ##dy## and ##dx##. However, infinitesimals are not really very very small numbers even if sometimes it is useful and practical to think about them as such. Infinitesimal are...
  46. Strand9202

    Derivative of the square root of the function f(x squared)

    I started out by rewriting the function as (f(x^2))^(1/2). I then did chain rule to get 1/2(f(x^2))^(-1/2) *(f'(x^2). - I think I need to go further because it is an x^2 in the function, but not sure.
  47. A

    I What is the logarithm of the derivative operator?

    I found this article which claims to have found the logarithm of derivative and even gives a formula. But I tried to verify the result by exponentiating it and failed. Additionally, folks on Stackexchange pointed out that the limit (6) in the article is found incorrectly (it does not exist)...
  48. abhinavabhatt

    A Second derivative of Heaviside step function

    In QFT by peskin scroeder page 30 the action of Klein Gordon Operator on propagator (∂2+m2)DR(x-y)=∂2θ(x0-y0)... how to compute this ∂2θ(x0-y0)?
  49. J

    Solving a Partial Derivative Problem Step-by-Step

    So I start by isolating v the speed here would be the square root of the partial t derivative divided by the sum of the partial x and y derivatives. the amplitude, phi and the cos portion of the partial derivatives would all cancel out. What I am left with is the sqrt(43.1 / ( 2.5 + 3.7 ) =...
  50. winnie_d_poop

    Prove that the derivative of the position vector equals the velocity vector

    1.)##\dot{\vec{r}}=\dot{x}\hat{i}+\dot{y}\hat{j}+\dot{z}\hat{k}=\dot{r}\hat{r}## since the unit vector is constant 2.) ##\dot{r}\hat{r}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{\sqrt{x^2+y^2+z^2}}\frac{\dot{x}x+\dot{y}y+\dot{z}z}{\sqrt{x^2+y^2+z^2}}##...
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