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. Safinaz

    I Partial derivative is terms of Kronecker delta and the Laplacian operator

    How can the following term: ## T_{ij} = \partial_i \partial_j \phi ## to be written in terms of Kronecker delta and the Laplacian operator ## \bigtriangleup = \nabla^2 ##? I mean is there a relation like: ## T_{ij} = \partial_i \partial_j \phi = ?? \delta_{ij} \bigtriangleup \phi.## But...
  2. G

    I How to apply tensor transformation rule

    Suppose I have a Cartesian Coordinate system (x,y) and a polar coordinate system (##r, \theta##). The position vector (3,4) and (5, ##\arctan \frac{4}{3}##) are the same except the representation. The position vector is a tensor, how does the position vector follow the tensor transformation...
  3. H

    I Intuition regarding Riemann curvature tensor

    The Riemann curvature tensor contains second derivatives of metric and squares of the first derivatives. The second derivatives have to be there because they are the ones that cannot be eliminated locally by a choice of coordinates. But other than being a mathematical artifact, is there a...
  4. S

    Proof related to derivative and Big O notation

    My attempt: Since ##\frac{f(a+h)-f(a-h)}{2h}-f'(a)=O(h^2)## as ##h \to 0##, then: $$\lim_{h \to 0} \frac{\frac{f(a+h)-f(a-h)}{2h}-f'(a)}{h^2} < \infty$$ So $$\lim_{h \to 0} \frac{\frac{f(a+h)-f(a-h)}{2h}-f'(a)}{h^2} = \lim_{h \to 0} \frac{f'(a)-f'(a)}{h^2}=0 < \infty$$ Because the value of...
  5. T

    Confusion on product rule for mass of differential volume element

    Good evening, I'm running into some trouble with this problem, and I have a hint as to why, but I'm not completely sure. Please see the steps below for context. I've been able to set up the proper equation representing the density as a function of distance from the center which looks like this...
  6. Hak

    Doubt about the derivative of a Taylor series

    My doubt arises over the definition of L'(v^2). If we are using ##x= v'^2##, shouldn't the derivative be made with respect to that very term? In essence, shouldn't it be: L'(v^2) = \frac{\partial L(v^2)}{\partial (v'^2)}? In the article I read, L'(v^2) = \frac{\partial L(v^2)}{\partial (v^2)} is...
  7. abobik

    Find the partial diameter error of the surface area of cylinder

    (ΔSA/ΔD) = 2πHΔD Something is wrong I guess as I get wrong value.
  8. L

    What is the derivative of ln(x)^e ?

    Can't figure it out, here's a screenshot with better typography.
  9. M

    I Finding the derivative of a characteristic function

    Problem summary I have the characteristic function of a probability distribution but I'm having difficulty obtaining its derivative. Background I am reading the following paper: Schwartz, Lowell M. (1980). On round-off error. Analytical Chemistry, 52(7), 1141-1147. DOI:10.1021/ac50057a033. The...
  10. chwala

    Find the derivative of the given function

    Let's see how messy it gets... ##\dfrac{dy}{dx}=\dfrac{(1-10x)(\sqrt{x^2+2})5x^4 -(x^5)(-10)(\sqrt{x^2+2})-x^5(1-10x)\frac{1}{2}(x^2+2)^{-\frac{1}{2}}2x}{[(1-10x)(\sqrt{x^2+2})]^2}##...
  11. mcastillo356

    Calculus Confusion over Calculus Book example footnote

    Hi,PF The book is "Calculus" 7th ed, by Robert A. Adams and Christopher Essex. It is about an explained example of the first conclusion of the Fundamental Theorem of Calculus, at Chapter 5.5. I will only quote the step I have doubt about: Example 7 Find the derivatives of the following...
  12. nomadreid

    I Want to understand how to express the derivative as a matrix

    In https://www.math.drexel.edu/~tolya/derivative, the author selects a domain P_2 = the set of all coefficients (a,b,c) (I'm writing horizontally instead off vertically) of second degree polynomials ax^2+bx+c, then defines the operator as matrix to correspond to the d/dx linear transformation...
  13. N

    Correct Usage of Partial Derivative Symbols in PDEs

    Some may say that ##\frac{ \partial g }{ \partial t }## is correct because it is a term in a partial differential equation, but since ##g## is a one variable function with ##t## only, I think ##\frac{ dg }{ dt }## is correct according to the original usage of the derivative and partial...
  14. Kumail Haider

    Find nth derivative of e^m(arcsinx) where x=0

    Here is the problem that I'm trying to solve. I've done the first part that is prove but I'm. facing difficulty in finding nth derivative I'm attaching pics of my attempt of solving this problem as I've no idea that how to type all these mathematical expressions. Can anyone please guide me...
  15. Baela

    A Infinitesimal Coordinate Transformation and Lie Derivative

    I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...
  16. Baela

    A Covariant derivative of Weyl spinor

    What is the expression for the covariant derivative of a Weyl spinor?
  17. snoopies622

    B Easy derivative but with a pesky singularity

    Setting: a plane with the standard Cartesian coordinate system. A particle is constrained to the x axis, with position x and moving at speed x dot. Another particle is constrained to the y axis, with position y and moving at speed y dot. The distance between them at any moment is s. It is...
  18. binbagsss

    I 4d integration/differentiation notation and the total derivative

    This is probably a stupid question but, ## \frac{d\partial_p}{d\partial_c}=\delta^p_c ## For the notation of a 4D integral it is ##d^4x=dx^{\nu}##, so if I consider a total derivative: ##\int\limits^{x_f}_{x_i} \partial_{\mu} (\phi) d^4 x = \phi \mid^{x_f}_{x_i} ## why is there no...
  19. homeworkhelpls

    I Solving an equation with a parameter and a derivative

    idk how to start after finding the second derivative
  20. C

    The derivative of uv wrt x using st function (homework problem)

    TL;DR Summary: I attempt to find the derivative of uv with respect to x using non standard analysis, hyperreals, and the standard part function st; I take u to be a function of x, and I also take v to be a function of x. Hello everyone! I've been learning about non standard analysis concepts...
  21. iceninja3

    Finding acceleration from Velocity vs Position graph

    The answer is E. I was initially very confused as to why the answer was not A but realized that the graph was velocity vs position (rather than velocity vs time) which means I can't simply take the derivative of the given graph. One thing I tried was writing out the equation first(c being a...
  22. J

    Chemistry Why is ethanol considered the parent chain in naming this benzene derivative?

    The name of this molecule is 1‐(3‐nitrophenyl)ethanol. I'm confused why ethanol is treated as the parent chain in this case, not the phenyl group. If the ring is composed of more atoms, should it be the parent chain? Thank you.
  23. AJSayad

    A Thermodynamics Derivative Reduction Problems

    Hi everyone, I'm in a graduate level mechanical engineering thermodynamics class. We're working on derivative reductions using the gibbs and maxwell relations. I was wondering if anyone has any good sources of practice problems that I could use. I've looked through my textbook and there are...
  24. N

    I Function min and max help

    What should I do when the f(x, y) function's second derivatives or Δ=AC-B² is zero? When the function is f(x) then we can differentiate it until it won't be a zero, but if z = some x and y then can I just continue this process to find what max and min (extremes) it has? What I've done is...
  25. S

    Understanding how time derivative = acceleration

    I'm having a hard time understanding some concepts and would really appreciate some help(not super smart so I need some things basically dumbed down). In my physics lab we're going over Newton's Second Law. There's a statement in the lab papers I don't understand. It states "As you should know...
  26. N

    How do I calculate physics formulas containing derivatives and real numbers?

    Hi, I'm trying to calculate my own physics problem but didn't get it something. When I'm trying to calculate the impulse of the object when it's hit by F=10N force in the smallest possible time, then should I write: dP/dt = Fnet => dP = Fnet*dt ? Another question: In general, if I calculate...
  27. Delerion24

    Help with the derivative of the Dirac delta

    My goal is to develop the equation 21. You should asume that \delta(r_2-r_1)^2 =0. These is named renormalization. Then my question is , do my computes are correct with previous condition ?
  28. Fady Megally

    I Second derivative, chain rules and order of operations

    So the chain rule for second derivatives is $$ \frac {d^2 y} {d t^2} = \frac{d}{dx}(\frac {dy} {dx}) \cdot \frac {dx} {dt} \cdot \frac {dx} {dt} + \frac {dy} {dx} \cdot \frac {d^2 x} {d t^2} = \frac{d^2 y}{d x^2} \cdot (\frac {dx} {dt})^2 + \frac {dy} {dx} \cdot \frac {d^2 x} {d t^2}$$ Today I...
  29. H

    Calculating total derivative of multivariable function

    This isn't a homework problem exactly but my attempt to derive a result given in a textbook for myself. Below is my attempt at a solution, typed up elsewhere with nice formatting so didn't want to redo it all. Direct image link here. Would greatly appreciate if anyone has any pointers.
  30. L

    B Question about the definition of a partial derivative

    I just started to study thermodynamics and very often I see formulas like this: $$ \left( \frac {\partial V} {\partial T} \right)_P $$ explanation of this formula is something similar to: partial derivative of ##V## with respect to ##T## while ##P## is constant. But as far as I remember...
  31. P

    I Definition of functional derivative

    In the book Quantum Field Theory for the Gifted Amateur, they define the functional derivative as: $$ \frac{\delta F}{\delta f(x))} = \lim_{\epsilon\to 0} \frac{F[f(x') + \delta(x'-x)) ] - F[ f(x') ]}{\epsilon} $$ Why do they use the delta function and not some other arbitrary function?
  32. Vladimir_Kitanov

    Derivative of ##\vec s = \vec \theta \times \vec r##

    My try: ##\vec s = \vec \theta \times \vec r## ##\frac {d}{dt} (\vec s) = \frac {d}{dt} (\vec \theta \times \vec r)## ##\vec v = \frac {d}{dt}(\vec \theta) \times \vec r + \vec \theta \times \frac {d}{dt}(\vec r)## ##\vec v = \vec \omega \times \vec r + \vec \theta \times \vec v## And I...
  33. G

    I Understanding Covariant and Partial Derivatives in General Relativity

    In the 128 pages of 《A First Course in General Relativity - 2nd Edition》:"The covariant derivative differs from the partial derivative with respect to the coordinates only because the basis vectors change."Could someone give me some examples?I don't quite understand it.Tanks!
  34. R

    I Inequality with integral and max of derivative

    Hi. I was reading Lighthill, Introduction to Fourier Analysis and Generalised Functions and in page 17 there is an example/proof where I can't make sense of the following step: $$ \left| \int_{-\infty}^{+\infty} f_n(x)(g(x)-g(0)) \, \mathrm{d}x \right| \le \max{ \left| g'(x) \right| }...
  35. chwala

    Find the second derivative of the relation; ##x^2+y^4=10##

    Find text (question and working to solution here ...this is very clear to me...on the use of implicit differentiation and quotient rule to solution). I am seeking an alternative approach. Now from my study we can also have; using partial derivatives...
  36. brotherbobby

    Spirit evaporating from a bowl

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  37. N

    The connection between potential energy and force

    Hi, if the force is the derivative of potential energy, does it mean that the force is equal to mg and with a constant gravity, it will be the same at any height? But in real life, F (or mg) would be different on the Earth's surface and 400 km above it (~8 m/s^2). So, this formula is used to...
  38. L_ucifer

    How do I compute the derivative of (x^x)^(x^x)^(x^x)....

    Summary: How do I solve the derivative of (((x^{x})^{x})^{x})... How would I solve this derivative?
  39. B

    I How do I compute the second derivative of a one-dimensional array?

    How do I compute the second derivative of an one dimensional array?
  40. LCSphysicist

    Covariant derivative in coordinate basis

    I need to evaluate ##\nabla_{\mu} A^{\mu}## at coordinate basis. Indeed, i should prove that ##\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt(|g|)}\partial_{\mu}(|g|^{1/2} A^{\mu})##. So, $$\nabla_{\mu} A^{\mu} = \partial_{\mu} A^{\mu} + A^{\beta} \Gamma^{\mu}_{\beta \mu}$$ The first and third terms...
  41. G

    B Derivative of a particle’s energy

    I’d appreciate some help with a mathematical block that I’m sure is trivial to most of you. Given the expression (1): Take the derivative of E with respect to a, set to zero and solve for a. Answer is shown at the bottom. This is not homework; I’m following an account of the development of...
  42. J

    Calculating the partial derivative in polar coordinates

    Hello, I am trying to solve the following problem: If ##z=f(x,y)##, where ##x=rcos\theta## and ##y=rsin\theta##, find ##\frac {\partial z} {\partial r}## and ##\frac {\partial z} {\partial \theta}## and show that ##\left( \frac {\partial z} {\partial x}\right){^2}+\left( \frac {\partial z}...
  43. mopit_011

    Classical Understanding Derivative of Position Function: Is Velocity Wrong?

    Hello! So, I was beginning to skim Kleppner and Kolenkow for an upcoming course I’m taking over the summer. I saw this on pg. 17 and was wondering if I’m making a silly mistake in understanding what the book is saying. When they take the derivative of the position function, isn’t the velocity...
  44. AL107

    Derivatives and the chain rule

    I originally thought you’d have to use the chain rule to get h’, as in: f’(g(x))*g’(x). Plugging in 1 for x, I got an answer of 10. An online solution, however, said that you only had to get f(g(1)), which was f(-1), then look up f’(-1) in the table. Both approaches seem logical to me, but they...
  45. George Keeling

    I Time derivative of the moment of inertia tensor

    I am completely stuck on problem 2.45 of Blennow's book Mathematical Models for Physics and Engineering. @Orodruin It says "We just stated that the moment of inertia tensor ##I_{ij}## satisfies the relation$${\dot{I}}_{ij}\omega_j=\varepsilon_{ijk}\omega_jI_{kl}\omega_l$$Show that this relation...
  46. Salmone

    I Derivative of the retarded vector potential

    In a problem of an oscillating electric dipole, under appropriate conditions, one can find, for the potential vector calculated at the point ##\vec{r}##, the expression ##\vec{A}=\hat{k}\frac{\mu_0I_0d}{4\pi}\frac{cos(\omega(t-r/c))}{r}## where: ##\hat{k}## is the direction of the ##z-axis##...
  47. N

    A Noether and the derivative of the Action

    I know that the Action has units Energy·time or Momentum·position. A second fact is that the derivative of the action with respect to time is Energy and similar with momentum-position, consistent with a units ie. dimensions check.Is it a coincidence that both are Noether conserved quantities...
  48. 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?
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