What is Second derivative: Definition and 178 Discussions

In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation:














{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}{\boldsymbol {x}}}{dt^{2}}},}
where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change. The last expression







{\displaystyle {\tfrac {d^{2}{\boldsymbol {x}}}{dt^{2}}}}
is the second derivative of position (x) with respect to time.
On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.

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  1. Hill

    Inflection Point Calculation: Reduction of Cubic with Second Derivative Method

    (i) I take the second derivative of Y: Y'' = 6X + 2A. Y'' = 0 when X = -A/3. Moreover, as Y'' is linear it changes sign at this X. Thus, it is the point of inflection. (iii) After the substitution, the term x^2 appears twice: one, from X^3 as -3(x^2)(A/3), and another from AX^2 as Ax^2. They...
  2. 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...
  3. 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...
  4. 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?
  5. 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))]''...
  6. 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...
  7. 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...
  8. 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...
  9. 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)?
  10. greg_rack

    B Find Local Max/Min: 2nd Derivative=0

    How do I distinguish between a point of local maxima or minima, when the second derivative in that point is equal to zero?
  11. T

    A Accelerated Hubble expansion -- Is the second derivative positive?

    Since distances increase, their first derivative which is velocity (Hubble constant) should be positive if not increasing too. Accelerated expansion needs the velocity to increase. What about the third derivative which is acceleration? An accelerated universe could have third derivative (called...
  12. J

    MHB Understanding Derivatives: Exploring the First and Second Derivative

    How would someone answer derivative question
  13. N

    B Second Derivative Question -- Help Understanding the Importance Please

    Hello. My understanding of the importance of second derivatives is that they help us to know whether the graph of a function is concave upward or concave downward. In the equation ## f(x) = x^2 + 2x ## we already know from the first derivative, ## f\prime (x) = 2x + 2 ##, that the graph is...
  14. 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 $\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$
  15. R

    Second derivative of 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...
  16. filip97

    I The Lagrangian and the second derivative?

    Why Lagrangian not depend of higher derivatives of generalised coordinates ?
  17. 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...
  18. A

    Uniform convergence of a sequence of functions

    Homework Statement This is a translation so sorry in advance if there are funky words in here[/B] f: ℝ→ℝ a function 2 time differentiable on ℝ. The second derivative f'' is bounded on ℝ. Show that the sequence on functions $$ n[f(x + 1/n) - f(x)] $$ converges uniformly on f'(x) on ℝ...
  19. R

    I "Undo" Second Derivative With Square Root?

    In my classical mechanics course, the professor did a bit of algebraic wizardry in a derivation for one of Kepler's Laws where a second derivative was simplified to a first derivative by taking the square root of both sides of the relation. It basically went something like this: \frac{d^2...
  20. K

    Determine the second derivative of a function

    Homework Statement Let ##f: \mathbb{R} \rightarrow \mathbb{R}## a function two times differentiable and ##g: \mathbb{R} \rightarrow \mathbb{R}## given by ##g(x) = f(x + 2 \cos(3x))##. (a) Determine g''(x). (b) If f'(2) = 1 and f''(2) = 8, compute g''(0). Homework Equations I'm not aware of...
  21. I

    A Second derivative of a complex matrix

    Hi all I am trying to reproduce some results from a paper, but I'm not sure how to proceed. I have the following: ##\phi## is a complex matrix and can be decomposed into real and imaginary parts: $$\phi=\frac{\phi_R +i\phi_I}{\sqrt{2}}$$ so that $$\phi^\dagger\phi=\frac{\phi_R^2 +\phi_I^2}{2}$$...
  22. K

    Second derivative in parametric equations

    Homework Statement Only the second part Homework Equations Second derivative: $$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}$$ The Attempt at a Solution $$dx=(1-2t)\,dt,~~dy=(1-3t^2)\,dt$$ Do i differentiate the differential dt? $$d^2x=(-2)\,dt^2,~~d^2y=(-6)t\,dt^2$$...
  23. EEristavi

    An Error Formula for Linearization (involving second Derivative)

    Homework Statement In textbook i was given formula to calculate error. I know that: E(t) = f(t)- L(x) = f(t) - f(a)- f'(a)(t- a) [L(x) is linear approximation]; [Lets call this Formula 1] I understand that, but that I have formula: E(x) = f''(s)/2 * (x-a)^2 [lets call this Formula 2] Here...
  24. Saracen Rue

    B Second derivative differential equations in terms of y?

    Firstly I know how to do this with first derivatives in differential equations - for example say we had ##\frac{dy}{dx}=4y^2-y##, and we're also told that ##y=1## when ##x=0##. ##\frac{dy}{dx}=4y^2-y## ##\frac{dx}{dy}=\frac{1}{4y^2-y}=\frac{1}{y\left(4y-1\right)}=\frac{4}{4y-1}-\frac{1}{y}##...
  25. F

    Second derivative of friction force question

    I'm studying boundary layers. I am confused by what I am reading in this book. The book says the friction force (F) per unit volume = $$\frac{dF}{dy}=\mu\frac{d^2U}{dy^2}$$ They say $$\frac{dU}{dy}=\frac{U_\infty}{\delta}$$ This makes sense to me, delta is the thickness in the y direction...
  26. M

    MHB Calculating the First and Second Derivative of a Twice Differentiable Function

    Hey! :o I want to find the first and second derivative of the function $$\psi (\lambda )=f(\lambda x_1, \lambda x_2)$$ where $f(y_1, y_2)$ is twice differentiable and $(x_1, x_2)$ is arbitrary for fix. I have done the following: $$f(g(\lambda), h(\lambda)) : \\...
  27. mastermechanic

    Finding the Second Derivative Using the Chain Rule

    Homework Statement Question has been attached to topic. Homework Equations Chain rule. The Attempt at a Solution $$\frac {dy}{dt} . \frac{dt}{dx} = \sqrt{t^2+1}.cos(π.t)$$ $$\frac{d^2y}{dt^2}.(\frac{dt}{dx})^2 = 2 $$ $$\frac{d^2y}{dt^2}.(t^2+1).cos^2(π.t)= 2 $$ and for the t=3/4...
  28. Mr Davis 97

    I Second derivative of a curve defined by parametric equations

    Quick question. I know that if we have a curve defined by ##x=f(t)## and ##y=g(t)##, then the slope of the tangent line is ##\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}##. I am trying to find the second derivative, which would be ##\displaystyle \frac{d}{dx}\frac{dy}{dx} =...
  29. Kanashii

    Finding the second derivative using central difference formula

    Homework Statement Develop aprogram that will determine the second derivative of pi(16 x^2 - y^4) at y=2 with step sizes of 0.1, 0.01, 0.001…. until the absolute error (numerical-analytical) converges to 0.00001. Use the 2nd order Central Difference Formula. User Input: y, tolerance Output: h...
  30. T

    A Computing first derivative based on second derivative

    I am trying to numerically solve a PDE, and just had a question as to the validity of a certain approach. For example, given the PDE: $$ \frac {\partial ^2 E}{\partial t^2} = - k\frac {\partial E}{\partial t} + c^2 \frac {\partial ^2 E}{\partial z^2} - c\frac {\partial E}{\partial z}$$ If I...
  31. Q

    Second Derivative (Implicit Differentiation)

    Homework Statement Find y'' Homework Equations 9x^2 +y^2 = 9 The Attempt at a Solution y' 18x+2y(y')=0 y'=-18x/2y y'=9x/y For the second derivative, I get the correct answer (same as the book) up until the very last step. Here's where I'm left at: -9( (-9x^2 - y^2) / y^3 ) The book then...
  32. A

    MATLAB How to Calculate the Second Derivative of a Curve in Matlab?

    I have a set of data as follows, How can I calculate the second derivative of the curve obtained from these data. x=[0.1;0.07;0.05;0.03;0]; r=[-98.9407;-105.7183;-111.2423;-116.0320;-120.0462];
  33. S

    I Why does the kinetic operator depend on a second derivative?

    The formula T = -(ħ/2m)∇2 implies that T is proportional to the second spatial derivative of a wavefunction. What is the origin of this dependence? In classical mechanics, T = p2/2m. Is it also the case in classical mechanics that p2/2m is proportional to a second spatial derivative? I...
  34. StanEvans

    I Magnitude of the Second Derivative

    So to find the x values of the stationary points on the curve: f(x)=x3+3x2 you make f '(x)=0 so: 3x2+6x=0 x=0 or x=-2 Then to find which of these points are maximum or minimum you do f ''(0) and f ''(-2) so: 6(0)+6=6 6(-2)+6=-6 so the maximum has an x value of -2 and the minimum has an x value...
  35. C

    Concave/convex -- second derivative

    Hello. I have a question regarding curvature and second derivatives. I have always been confused regarding what is concave/convex and what corresponds to negative/positive curvature, negative/positive second derivative. If we consider the profile shown in the following picture...
  36. T

    Approximation of second derivative of a smooth function

    Hi, I've attached an image of an equation I came across, and the text describes this as an approximation to the second derivative. Everything seems to be exact to me (i.e. not an approximation) if the limit of h was taken to 0. Is that the only reason why it's said to be an approximation or is...
  37. kostoglotov

    Very basic Q about solns to y" = y

    Wolfram and the Linear Algebra text I'm currently working on, give the two possible solutions of \frac{d^2y}{dx^2}=y as being e^{x} and e^{-x}, or rather, constant multiples of them. Here wolfram agrees: http://www.wolframalpha.com/input/?i=d^2y/dx^2=y My question is, why isn't y = e^{x} + x...
  38. 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...
  39. 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...
  40. 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...
  41. 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...
  42. P

    Is continuity of the second derivative necessary for the second derivative test?

    According to this link: http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx The second derivative test can only be applied if ##f''## is continuous in a region around ##c##. But according to this link...
  43. H

    Hermite Interpolation extended to second derivative

    SOLVED 1. Homework Statement Find polynomial of least degree satisfying: p(1)=-1, p'(1)=2, p''(1)=0, p(2)=1, p'(2)=-2 Homework Equations In general, a Hermite Polynomial is defined by the following: ∑[f(xi)*hi(x)+f'(xi)*h2i(x)] where: hi(xj)=1 if i=j and 0 otherwise. Similarly with h'2...
  44. A

    Parity operator commutes with second derivative?

    How do I prove that the parity operator Af(x) = f(-x) commutes with the second derivative operator. I am tempted to write: A∂^2f(x)/∂x^2 = ∂^2f(-x)/∂(-x)^2 = ∂^2f(-x)/∂x^2 = ∂^2Af(x)/∂x^2 But that looks to be abuse of notation..
  45. H

    Eigenstuff of Second Derivative

    Hi, I'm trying to find the eigenvalues and eigenvectors of the operator ##\hat{O}=\frac{d^2}{d\phi^2}## Where ##\phi## is the angular coordinate in polar coordinates. Since we are dealing with polar coordinates, we also have the condition (on the eigenfunctions) that ##f(\phi)=f(\phi+2\pi)##...
  46. C

    Very very short question on second derivative

    What does it mean when I have to find the second derivative of a circle at a given point? (Implicit diffing) In specifics, the equation is 9x2 +y2 =9 At the point (0,3) You don't really need the rest at all, but it was just my process. This seems to make no sense. first D'v 18x+2yy'=0 Second...
  47. M

    Second derivative of an autonomous function

    For the derivative: dy/dt = ry ln(K/y) I am trying to solve the second derivative. It seems like an easy solution, and I did: d^2y/dt^2 = rln(K/y)y' + ry(y/K) which simplifies to: d^2y/dt^2 = (ry')[ln(K/y) + 1/Kln(K/y) Unfortunately, the answer is d^2y/dt^2 (ry')[ln(K/y) - 1] and I don't...
  48. G

    If Integral with Sine Limits What is Second Derivative?

    Homework Statement If f(x) = ∫sin x0 √(1+t2)dt and g(y) = ∫3y f(x)dx, find g''(pi/6)? Homework Equations FTC: F(x) = ∫f(x)dx ∫ab f(t)dt = F(b) - F(a) Chain Rule: f(x) = g(h(x)) f'(x) = g'(h(x))h'(x)The Attempt at a Solution I tried u-substition setting u = tan(x) for the first dirivative...
  49. ZetaOfThree

    Second derivative of a unit vector from The Feynman Lectures

    In the Feynman Lectures on Physics chapter 28, Feynman explains the radiation equation $$\vec{E}=\frac{-q}{4\pi\epsilon_0 c^2}\, \frac{d^2\hat{e}_{r'}}{dt^2}$$ The fact that the transverse component varies as ##\frac{1}{r}## seems fairly obvious to me since what matters is just the angle...
  50. J

    System of equations (multivariable second derivative test)

    I am doing critical points and using the second derivative test (multivariable version) Homework Statement f(x,y) = (x^2+y^2)e^{x^2-y^2} Issue I am having is with the system of equations to get the critical points from partial wrt x, wrt y The Attempt at a Solution f_{x} =...