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.

View More On Wikipedia.org
Recent contents Top users
  1. STEMucator

    Derivative of Dirac Delta Distribution

    Hello, I am wondering about the derivative of the Dirac delta distribution ##\delta(t)##. I know: $$\frac{d}{dt} u(t) = \delta(t)$$ So what is ##\frac{d}{dt} \delta(t)##? How do we take the derivative of a distribution? I've heard about distributional derivatives, but I don't think any of...
  2. I

    Question regarding a derivative

    Homework Statement Write the derivative of y = (x2+4x+3)/(x1/2) I got the correct answer, but my question is, why can't I rewrite this as: y = (x^2+4x+3)*(1/x1/2) Then see my attempted solution for the result... Homework Equations y = (x2+4x+3)/(x1/2) The Attempt at a Solution y =...
  3. J

    Derivative of position question

    Homework Statement Q: The position of a dragonfly that is flying parallel to the ground is given as a function of time by r⃗ =[2.90m+(0.0900m/s2)t^2]i^− (0.0150m/s3)t^3j^ . At what value of t does the velocity vector of the insect make an angle of 30.0 ∘ clockwise from the x-axis? . Homework...
  4. J

    How to find the equation of this tangent?

    Mod note: Thread moved from Precalc section Homework Statement F(x)=sqrt(-2x^2 +2x+4) 1.discuss variation of f and draw (c) 2.find the equation of tangent line to (c) that passes through point A(-2,0) The Attempt at a Solution I solved first part I found the domain of definition and f'(x) and...
  5. TyroneTheDino

    Finding the distance from origin to a tangent line

    Homework Statement Find the distance between the origin and the line tangent to ##x^\frac{2}{3}+y^{\frac{2}{3}}=a^{\frac{2}{3}}## at the point P(x,y) Homework Equations [/B] Distance= ##\frac{\left |a_{0}+b_{0}+c \right |}{\sqrt{a^{2}+b^{2}}}## The Attempt at a Solution To begin I find...
  6. M

    Intuitive ways to think of integration and second derivative

    Hi, I feel sometimes when I'm doing calculus I lose the logic and intuition behind what I'm doing, especially when integrating. I have yet to find a way to think about it in a way it makes sense to me why the definite integral would tell us the area under a curve. Same with why the second...
  7. A

    Shouldn't Emf generated be Derivative of Flux?

    Hey, I don't understand why emf generated is not the first derivative. For example I have a graph of magnetic flux through a wire against time. I thought that emf generated was the rate of change of flux, however this doesn't work. Emf=-Dflux/Dtime . In this example I have a y=sinx curve for...
  8. N

    What version of the definition of derivative

    If my understanding is correct the definition of a derivative is lim h->0 (f(x+h)-f(x))/h However, I've also seen this used: lim x->c (f(x)-f(x))/(x-c) are these both considered valid definition for the derivative or does the derivative have to tend towards zero? I am a bit confused because I...
  9. G

    Time derivative of tensor expression

    I was trying to compute the time derivative of the following expression: \mathbf{p_k} = \sum_i e_{ki}\sum_{n=0}^{\infty} \frac{(-1)^n}{(n+1)!} \mathbf{r_{ki}}(\mathbf{r_{ki}\cdot \nabla})^n \delta(\mathbf{R_k}-\mathbf{R}) I am following deGroot in his Foundations of Electrodynamics. He says...
  10. J

    Difficulty computing second derivative value in SHM problem

    Homework Statement The displacement of a machine is given by the simple harmonic motion as x(t) = 5cos(30t)+4sin(30t). Find the amplitude of motion, and the amplitude of the velocity. Homework Equations x''(t) = -4500cos(30t)-3600sin(30t) The Attempt at a Solution [/B] I should note that...
  11. H

    Index Notation, taking derivative

    Can anyone explain how to take the derivative of (Aδij),j? I know that since there is a repeating subscript I have to do the summation then take the derivative, but I am not sure how to go about that process because there are two subscripts (i and j) and that it is the Kronecker's Delta (not...
  12. D

    Lie derivative of tensor field with respect to Lie bracket

    I'm trying to show that the lie derivative of a tensor field ##t## along a lie bracket ##[X,Y]## is given by \mathcal{L}_{[X,Y]}t=\mathcal{L}_{X}\mathcal{L}_{Y}t-\mathcal{L}_{Y}\mathcal{L}_{X}t but I'm not having much luck so far. I've tried expanding ##t## on a coordinate basis, such that...
  13. W

    Help understanding equation involving a partial derivative

    Mod note: Moved from a homework section 1. Homework Statement N/A Homework Equations f(x + Δx,y) = f(x,y) + ∂f(x,y)/∂x*Δx The Attempt at a Solution Sorry this isn't really homework. We were given this equation today in order to derive the Taylor expansion formula in two variables and I'm not...
  14. terryds

    Why isn't Un the derivative of Sn in sequence and derivative?

    I see that derivative of y with respect to x is just like the ratio of y over x. But, Why Un (the formula to find nth term) is not the derivative of Sn (the sum of sequence formula) ?? For example, 1 2 5 10 -> y = x2+1 +1...
  15. T

    Are a vector and its derivative perpendicular at all times?

    i'm dumb, sorry
  16. S

    Does derivative of wave function equal zero at infinity?

    I understand that ψ goes to zero as x goes to infinity. Is it also true that dψ/dx must go to zero as x goes to infinity?
  17. P

    Second derivative with parametric equations

    http://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx On this page the author makes it very clear that: $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$ provided ##\frac{dx}{dt} \neq 0##. In example 4, ##\frac{dx}{dt} = -2t##, which is zero when ##t## is zero. In simplifying...
  18. A

    Derivative of L-sqrt(L^2-x^2)How do I take the derivative of L-sqrt(L^2-x^2)?

    Homework Statement I did an experiment to test the conservation of mechanical energy in an oscillating pendulum. As part of the analysis I had to find the pendulum's vertical position with time using the formula: y = L-sqrt(L^2-x^2) where L was the pendulum's length (L=1 m). Then for the next...
  19. Mr Davis 97

    Derivative and function terminology

    In mathematical parlance, we say "take the derivative of a function f" to indicate that we are computing a new function, which maps slopes, that derives from f. However, in physics, we say "take the derivative of velocity". However, velocity is a quantity, not a function. What does it mean to...
  20. S

    Partial Derivative of x^2 on Manifold (M,g)

    How can I figure out ##\partial_\mu x^2## on the manifold ##(M,g)##? I thought that it should be ##2x_\mu##, but I think I'm wrong and the answer is ##2x_\mu+x^\nu x^\lambda \partial_\mu g_{\nu\lambda}##, right?! In particular, it seems to me, we can't write...
  21. loops496

    Directional Derivative of Ricci Scalar: Lev-Civita Connection?

    I have a question about the directional derivative of the Ricci scalar along a Killing Vector Field. What conditions are necessary on the connection such that K^\alpha \nabla_\alpha R=0. Is the Levi-Civita connection necessary? I'm not sure about it but I believe since the Lie derivative is...
  22. B

    Evaluate the partial derivative of a matrix element

    Homework Statement A determinant a is defined in the following manner ar * Ak = Σns=1 ars Aks = δkr a , where a=det(aij), ar , Ak , are rows of the coefficient matrix and cofactor matrix respectively. The second term in the equation is the expansion over the columns of both matrices, δkr is...
  23. M

    Prove existence of open interval with non-zero derivative

    I'm struggling to get started with the proof that an open interval D containing x0 exists such that f'(x) ≠ 0 for all x∈D, given f'(x0)≠0. It seems like it should be easy but I've been scratching around for an hour now and have gotten nowhere, could anyone give me some advice to help me along?
  24. Ravendark

    Is the Sign in the Covariant Derivative Important for Local Gauge Invariance?

    Homework Statement Consider the fermionic part of the QCD Lagrangian: $$\mathcal{L} = \bar\psi (\mathrm{i} {\not{\!\partial}} - m) \psi \; ,$$ where I used a matrix notation to supress all the colour indices (i.e., ##\psi## is understood to be a three-component vector in colour space whilst...
  25. Titan97

    Question on Mean Value Theorem & Intermediate Value Theorem

    Homework Statement for ##0<\alpha,\beta<2##, prove that ##\int_0^4f(t)dt=2[\alpha f(\alpha)+\beta f(\beta)]## Homework Equations Mean value theorem: ##f'(c)=\frac{f(b)-f(a)}{b-a}## The Attempt at a Solution I got the answer for the question but I have made an assumption but I don't know if...
  26. pellman

    What is a derivative in the distribution sense?

    Never mind. I got this one. Couldn't figure out how to delete the post though.
  27. N

    Prove that the derivative of tan(2x) - cot(2x) equals....

    Let f(x) = tan(2x) - cot(2x) defined on x∈]0,π/4[ Prove that derivative of f(x) is 16/1-cos(8x) What I did was: 2 * Sin^2(2x) + 2 * Cos^2(2x) / Cos^2(2x) + Sin^2(2x) If I factor the 2, I reach: 2 * (Sin^2(2x) + Cos^2(2x) / 1+cos(4x)/2 + 1-cos(4x)/22 * 1/ 1 = 2? What went wrong?
  28. J

    Exterior derivative of hodge dual

    Hello all, I'm having a minor annoyance in proving an identity. The identity is the following \star\text{d}\star A_p = \frac{(-)^{p(D-p+1)-1+t}}{(p-1)!}\nabla_\mu A^\mu_{\,\, \mu_1 \cdots \mu_{p-1}}\text{d}^\mu_1\wedge \cdots \wedge \text{d}^\mu_{p-1} I'm stuck at the first step of proving...
  29. Ravendark

    Second functional derivative of fermion action

    Homework Statement [/B] Consider the following action: $$\begin{align}S = \int \mathrm{d}^4 z \; \bar\psi_i(z) \, (\mathrm{i} {\not{\!\partial}} - m)_{ij} \, \psi_j(z)\end{align}$$ where ##\psi_i## is a Dirac spinor with Dirac index ##i## (summation convention for repeated indices). Now I would...
  30. N

    Confused on these two derivative problems

    Homework Statement a. k(t) = (sqrt(t+1))/(2t+1)b. y = (3^(x^2+1))(ln(2))The Attempt at a Solution For the first problem, I know I use the quotient rule for derivatives (L)(DH)-(H)(DL)/((L)^2) which would go to: ((2t+1)(1/(2sqrt(t+1)) - (sqrt(t+1))(2))/((2t+1)^2) I get stuck here, maybe it's...
  31. Y

    Find the derivative of the function(chain rule)

    Homework Statement Various values of the functions f(x) and g(x) and their derivatives are given in the table below. Find the derivative of f(x+g(x)) at x=0. at x=0 f(x)=5 f'(x)=2 g(x)=1 g'(x)=3 at x=1 f(x)=7 f'(x)=3 g(x)=-2 g'(x)=-5 2. Homework Equations Chain ruleThe Attempt at a Solution...
  32. U

    Total derivative involving rigid body motion of a surface

    This stems from considering rigid body transformations, but is a general question about total derivatives. Something is probably missing in my understanding here. I had posted this to math.stackexchange, but did not receive any answers and someone suggested this forum might be more suitable. A...
  33. ognik

    MHB 1st Derivative of Cauchy Integral formula

    Hi - I know the final result for the n'th derivative, I am looking though at getting an expression for the 1st derivative of f(z). From $ f({z}_{0}) = \frac{1}{2\pi i} \oint_{c} \frac{f(z)}{z - {z}_{0}}dz $ we get $ \frac{f({z}_{0} + \delta {z}_{0}) -{f({z}_{0}}) }{\delta {z}_{0}} =...
  34. N

    Why Does (\partial_{\mu}\phi)^2 Equal (\partial_{\mu}\phi)(\partial^{\mu}\phi)?

    I just came across this in a textbook: ## (\partial_{\mu}\phi)^2 = (\partial_{\mu}\phi)(\partial^{\mu}\phi) ## Can someone explain why this makes sense? Thanks.
  35. Tanny Nusrat

    Need Help Solving nth Derivative of e^ax*Sin(ax+b)

    i have solved the following one but not sure...anyone give me the solve..i want to be sure.. nth derivative of {e^ax * Sin(ax+b)}
  36. haael

    How does the Kalman filter calculate derivatives?

    Suppose we have a Kalman filter. We have a position sensor, for example GPS. We use the filter to estimate position. However in all examples I see higher derivatives in the state vector: speed, acceleration and sometimes jerk. There is no sensor that calculates these values directly, so they...
  37. dhalilsim

    D'Alembert operator is commute covariant derivative?

    For example: [itex] D_α D_β D^β F_ab= D_β D^β D_α F_ab is true or not? Are there any books sources?
  38. Z

    What does 'velocity is derivative of distance' mean?

    What does v equals dx/dt mean? I interpret v as: the limiting value as a vanishingly small value for time t (dt) goes to 0. Or lim as dt-->0 of dx/dt.
  39. R

    Help finding the derivative for Faraday's equation?

    I need to find the derivative with respect to time of the magnetic flux (dΦB/dt). I have a time of .0085 seconds, and a magnetic flux of .0008 Wb. I am a little hazy on my calc skills.
  40. hideelo

    Deriving Commutation of Variation & Derivative Operators in EL Equation

    I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality δ(dq/dt) = d(δq)/dt Where q is some coordinate, and δf is the first variation in...
  41. S

    Finding derivative of x^x using limit definition

    I am trying to find the derivative of x^x using the limit definition and am unable to follow what I have read. Can someone help me understand why lim [(x+h)^h -1]/h as h ---> 0 = ln(x). This part of the derivatio
  42. S

    What is the partial derivative of f with respect to w?

    Homework Statement Define f(x,y) = x+2y, w = x+y. What is ∂f / ∂w? Homework EquationsThe Attempt at a Solution f = w+y so: ∂f/∂w = ∂(w+y)/∂w = ∂w/∂w + ∂y/∂w = 1 + ∂y/∂w. But I'm really not sure if this is right and if it right so far, I can't figure out what ∂y/∂w should be...
  43. B

    Product rule for vector derivative

    Say I have a position vector p = e(t) p(t) Where, in 2D, e(t) = (e1(t), e2(t)) and p(t) = (p1(t), p2(t))T And if I conveniently point the FIRST base vector of the frame at the particle, I can use: p(t) = (r1(t), 0)T I want the velocity, so I take v = d(e(t))/dt p(t) + e(t) d(p(t))/dt...
  44. K

    What Value of x Maximizes the Function y=axln(b/x)?

    Homework Statement $$y=a\cdot x\cdot ln\left(\frac{b}{x}\right)$$ The derivative should be 0 (to maximize), what's x? Homework Equations $$(ln\:x)'=\frac{1}{x}$$ $$(x^a)'=ax^{(a-1)}$$ $$(uv)'=u'v+v'u$$ The Attempt at a Solution $$\dot y=a \left[ ln \left( \frac{b}{x} \right)-x\frac{x}{b}x^{-2}...
  45. C

    Definition of derivative - infinitesimal approach, help :)

    Hi I'm reading Elementary calculus - an infinitesimal approach and just wan't to make sure my understanding of what dy, f'(x) and dx means is correct. I do understand the basic idea: You make the secant between 2 points on a graph approach one of the points and at this point you get the...
  46. F

    Hamiltonian defined as 1st derivative

    Why is Hamiltonian defined as 1st derivative with respect to time ? From the units of energy (kgm2s-2) I would expect it to be defined as 2nd derivative with respect to time. (I'm reading http://feynmanlectures.caltech.edu/III_11.html#Ch11-S2)
  47. B

    Derivative of the mixed metric tensor

    So i am studying GR at the moment, and I've been trying to figure out what the derivative (not covarient) of the mixed metric tensor $$\delta^\mu_\nu$$ would be, since this tensor is just the identity matrix surely its derivative should be zero. Yet at the same time $$\delta^\mu_\nu =...
  48. MidgetDwarf

    Can relative maximum and minimum exist when the derivative....

    Can relative maximum and minimum points exist when a function is defined at say x=c, however the derivative does not exist or tends to infinity? Ie the graph of. F (x)= |x|, for x=c=o. If I am correct the relative minimum is at o, can it also be the abs minimum? I recalled the theorem by...
  49. E

    Derivative of best approximation

    Say that we have a continuous, differentiable function f(x) and we have found the best approximation (in the sense of the infinity norm) of f from some set of functions forming a finite dimensional vector space (say, polynomials of degree less than n or trigonometric polynomials of degree less...
  50. Joseph Nechleba

    Second Derivative of Circle Not a Constant?

    Using the standard equation of a circle x^2 + y^2 = r^2, I took the first and second derivatives and obtained -x/y and -r^2/y^3 , respectively. I understand that the slope is going to be different at each point along the circle, but what does not make sense to me is that the rate of change of...
Back
Top