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:

a

=

d

v

d
t

=

d

2

x

d

t

2

,

{\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

d

2

x

d

t

2

{\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. 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. 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. 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. 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?

27. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. If Integral with Sine Limits What is Second Derivative?

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49. 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. 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} =...