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.
Good day,
I don't understand the following:
\frac{d^{2}}{dt^{2}}\int_{0}^{t}(t-\epsilon )\phi (\epsilon)d\epsilon=\phi''(t)
All I know is:
\frac{d^{2}}{dt^{2}}\int_{0}^{t}(t-\epsilon )\phi (\epsilon)d\epsilon=\frac{d^{2}}{dt^{2}}\int_{0}^{t}t \cdot \phi...
Hi. I have a question about steady-state stability and the second derivative test. I have been reading about it in a book on mathematical modeling, and the section concerns differential equations. I believe this forum is more appropriate than "General Math," but let me know if it is not...
Homework Statement
http://imageshack.us/photo/my-images/717/unleddym.png/
Homework Equations
The Attempt at a Solution
I was wondering why the second derivative at t=1 does not exist but exists at the first derivative. What I did was draw the graph of the function, then the...
Homework Statement
If xy + 9e^y = 9e, find the value of y'' at the point where x = 0.
Homework Equations
product rule
The Attempt at a Solution
Okay so first I found the first derivative using implicit differentiation and I got:
y'=\frac{-y}{x+9e^{y}}
then, I found the second...
Homework Statement
Let y = s(t) represent the number of students who have contracted measles at time t (days). Give an interpretation for each condition:
e) s' = 0, s" > 0
The Attempt at a Solution
This seems counterintuitive to me, to think that the second derivative is also not...
Homework Statement
Find the second derivative of x^3+y^3=1 by implicit differentiation.
The Attempt at a Solution
I found the first derivative to be x^2/y^2. Do I then use the first derivative and take the derivative of that? I tried to do this, but got stuck on what to do.
Hello!
I am wondering if someone could let me know if my understanding is right or wrong. The Taylor series gives the function in the form of a sum of an infinite series. From this an approximation of the change in the function can be derived:
f_{a} and f_{a,a} are the first and second...
urgent! second derivative test for functions of 2 variables
Homework Statement
f(x,y)=x^4 - y^2 - 2x^2 + 2y - 7
Homework Equations
classify points (0,1) and (-1,1) as local maximum, local minimum or inclusive
The Attempt at a Solution
f(x,0)=4x^3 - 0 - 4x + 0 - 0 = 4x^3-4x...
Homework Statement
Find the second derivative of 4x^2 + 3x - 9y^2. Answer in terms of y.
Homework Equations
All derivative formulas.
The Attempt at a Solution
[PLAIN]http://http://i52.tinypic.com/2w2ptex.jpg
I can't get much further than this; the thing that gets me is how to put...
Hi everybody.
I have a question regarding an example problem at about 22min on this lecture http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-2-eulers-numerical-method-for-y-f-x-y/
The equation in question is y'=x^{2}-y^{2}.
In an...
Second Derivative
Homework Statement
f(x) = sin (2x)^1/2 find the f"(x).
Homework Equations
The Attempt at a Solution
I don't know homework to do it...
[b]1. Homework Statement [/
Using the approximation, explain why the second derivative test works
approximation=f(x0+delta x, y0+delta y)
delta x and delta y are small...
Homework Equations
f(x0+delta x,y0+delta y)
The Attempt at a Solution
ok so i know the first derivative...
Homework Statement
I am doing the various ins and outs of curve sketching and the mean value theorem and all that jazz with this function:
f(x)=sec(x)+tan(x)
Homework Equations
The Attempt at a Solution
I took the first derivative to be:
f'(x)=sec(x)tan(x)+sec^{2}(x)
I am having...
Homework Statement
Knowing that:
\frac{dz}{dx}=\sqrt{\frac{(-T)^2}{(k*(z-a)*(z-2*b+a)/2-T*cos(phi))^2}-1}
What is:
\frac{d^2z}{dx^2} ?
Homework Equations
I'm trying to solve using this general equation I found on Wikipedia...
Alright. So I have dy/dx = -1-y2. I want to take the second derivative to get some information about the concavity of the solution, but I can't wrap my head around what's really going on.
What I think I know: I have an ODE that is dependent on the dependent variable, so my solution will only...
Hey there,
I have a problem to do, in which I need to determine the acceleration of an object and the uncertainty of the acceleration. The position vs. time equation is given by:
s(t) = 0.205t2 + 0.3001t
Therefore, after differentiation, I can state the the velocity vs. time equation...
Hi, I am having a little trouble understanding something my lecturer said about using the table of signs to check whether there exists a point of inflection when y'' = 0. I understand that in order for there to be a point of inflection at x0 say, I require to check the value of y'' at either...
Homework Statement
r = 2/(2 - cos (\pi*t))
Homework Equations
N/A
The Attempt at a Solution
Hello everyone, first I would just like to say (which is obvious since I'm asking :tongue: ) That it's been a long long time since I've had to do derivatives, hence my total cluelessness...
Homework Statement
if d/dx(f(x)) = g(x) and d/dx(g(x)) = f(x^2), then d^2/dx^2 ( f(x^3) ) = ?
Homework Equations
The Attempt at a Solution
from 1 and 2 we get d.dx(g(x)) = d2/dx2(f(x)) = f(x^2)
but then what? That doesn't tell me anything about f(x^3)
Please help.
Homework Statement
Use Maclaurin's theorem to estimate d^{2}y/dx^{2} at x=0
It's the deflection of a 2m beam, where x is the distance along the beam, y is the deflection in mm.
Homework Equations
The Attempt at a Solution
I didn't know where to start so I tried to solve it a different...
Second derivative...
Homework Statement
Okay, this is a rough one for me. It was a question I got on my test, and (obviouslly) didnt get right. I am studying all my old exams for my final in 2 days and this is the last of the problems that I can't wrap my head around...any help would be...
I have this problem for homework dealing with second derivatives and graphs. I have no problem finding derivatives usually, but this one is giving me trouble. I cannot figure out how to get the second derivative. I have an idea of what to do, but need some extra guidance.
Find f '(x) where...
Homework Statement
Find the second derivative (y") of y=xtanx.
The attempt at a solution
I got the first derivative (y')
y=xtanx
y'=x(secx)+tanx
I started the second derivative and got stuck
y"=xsec^2x+tanx
Hello,
I am facing a diffusion equation..
\frac{du(x,t)}{dt} = D \frac{d^2u}{dx^2}
.. with slightly exotic boundary conditions:
u(0,t) = 0
\frac{d^2u(a,t)}{dx^2}+ \alpha \frac{du(a,t)}{dx} = 0
I expected the solution to be relatively easy to find, since separation of variables quickly...
Homework Statement
x = t - e^{t}
y = t + e^{-t}
Find dy/dx and d^{2}y/dx.Homework Equations
Derivative equations.The Attempt at a Solution
dy/dt = 1 - e^{-t}
dx/dt = 1 - e^{t}
The dy/dx I came up with is:
dy/dx = (1 - e^{-t}) / (1 - e^{t})
Second derivative I came up with is:
d^{2}y/dx = -...
Homework Statement
I trying to find the second derivative of xe^x
Homework Equations
chain rule
The Attempt at a Solution
Two find the first derivative I use the chain rule.
f'(y)g(y)+f(y)g'(y)
so I get
e^x+xe^x
is the second derivative
e^x+f'(y)g(y)+f(y)g'(y)...
f(x)= 1/125(e5x)(5x-2)I think this is the first derivative but I ain't good at math and gives me some headaches I used the product rule.. but still I have doubts =(
f '(x)= 1/125(5)(e5x)(5x-2) + (5)(1/125(e5x)
Please help me! :cry:
I need to understand how to do this
Hello--
I'm in the process of implementing a PML for FDTD modeling.
I would like to take the derivative of the partial derivative shown below, but I am uncertain with respect to how I might proceed.
\[
\frac{\partial }{{\partial x}} \to \frac{1}{{1 + \frac{{i\sigma \left( x \right)}}{\omega...
Can someone tell me what this actually is.
So, in the case when the Hessian is positive (or negative) semidefinite, the second derivative test is inconclusive.
However, I think I've read that even in the case where the Hessian is positive semidefinite at a stationary point x, we can still...
Homework Statement
For example with f(x,y) = x2y + xy2
Homework Equations
The Attempt at a Solution
Well I know there is a critical point at (0,0). So I calculated the second derivatives but they are all 0 here so that doesn't help.
I also tried using the Taylor expansion to...
Homework Statement
Find y''(x) of the parametric equation 9x^2+y^2=9 using implicit differentiation.
Homework Equations
I already came up with y'(x) = -9x/y
The Attempt at a Solution
Here is what I have for y''(x) so far
y''(x) = d/dx (-9xy^-1)
=-9(d/dx)(xy^-1)...
Homework Statement
Find the second derivative of:
e^{ax}
and
e^{-ax}
Homework Equations
The Attempt at a Solution
The book that I am using seems to have been very vague on how to take the derivatives of exponential functions. I am aware that:
\frac {d(e^{x})}...
Homework Statement
f(x)= x^(3)e^(x)
Find f'(x) in simplest form.
Find f"(x) in simplest form.
Homework Equations
The Attempt at a Solution
I found the first derivative to be: (using the product rule)
f'(x) = (e^x)[3x^2] + (x^3)[e^x]
f'(x) = 3x^(2)e^(x) + x^(3)e^(x)
f'(x) = x^[2]e^[x](3 + x)...
Homework Statement
Find the exact value of f''(2) if f(x)=\sqrt{3x-4}
Homework Equations
See above
The Attempt at a Solution
I've tried to use the product rule to differentiate.
f= x(3x -4)^{\frac{1}{2}}
f'= (3x -4)^{\frac{1}{2}} + \frac{3}{2}^{\frac{-1}{2}}
f''=...
Homework Statement
The function Sh(t) = 30[cos(16.04*)]t models the horizantal position of a pellet with respect to time.
Find the first & second derivatives of Sh(t).
Homework Equations
The Attempt at a Solution I attached a word document because I lack the ability to put...
Homework Statement
A curve has equation y=e^2x-x^2+x-3 , find value of x for which d^2y/dx^2=0.
Homework Equations
The Attempt at a Solution
well. i started by finding out the 1st and 2nd derivative:
y=e^2x-x^2+x-3
dy/dx= 2e2^x-2x+1 and d2y/dx2=4e^2x-2 = 0
dy/dx =>2e^2x=2x-1...
Homework Statement
Assuming sufficient differentiability, find second derivative of F(x) = integ[a,x] (t-x)2 f(t) d(t)
Homework Equations
Probably Fund.Thm of Calculus and some properties
The Attempt at a Solution
I really have no idea..I tried evaluating but with t=x but I get...
binary mixture.
Na=moles of a
Nb=moles of b
(using Peng Robinson Equation of state)
(second order partial derivative below)
d^2P/(dNa^2) holding T, molar volume, Nb constant
I can't figure out how to do this?
I know that Peng Robinson is a function of concentration of Na and...
Homework Statement
Suppose f(3)=2 , f'(3)=5 , and f''(3)= -2 . Then d²/dx² (f²(x)) at x=3 is equal to ____?
A. -20
B. 20
C. 38
D. 42
E. 10
The Attempt at a Solution
I am confused about how to find the function to get the derivative from that function. Any Ideas? Thanks.
If x and y are defined in terms of a third vatiable say t , then to find d2y/dx2 , we cannot find d2y/dt2 and d2x/dt2 and divide them to get d2y/dx2 , i am unable to fingure out the reason for this !
Does the second derivative test fail for x3 at x=0:
f'(x)=3x2 f''(x)=6x ,
for x=0,
f'(0)=0 & f''(0)=+ve ,
so it should be a point of local maxima , but it is not!
How would you find the second derivative of an implicit function?
y^2-x^2=16
Heres my attempt:
2y(dy/dx)-2x=0
2y(dy/dx)=2x
2y(dy/dx)/2y=2x/2y
dy/dx= x/y
This is only the first derivative. I think I'm suppose to plug in dy/dx back into the original equation. Am I on the right track?
The Attempt at a Solution
I have calculated a Lagrangian for a particular system (I can post the problem upon request). The system has two degrees of freedom, but I have applied a constraint to remove one of the degrees of freedom. In doing so, I have introduced a second time-derivative of the...
I often see the second derivative written down like this:
\frac{d^2y}{dx^2}
Although it seems more logical to me to write
\frac{d^2y}{d^2x^2}
Or
\frac{d^2y}{(dx)^2}
Since it represents
\frac{d}{dx} \frac{dy}{dx}
Is there any logic behind this or is it just a shortcut notation to omit...
Let f(x)=x^2/(1-x^2 )
a) Find f'(x)
b) Find f"(x)
For the answer to a) they give f'(x)=2x/〖(1-x^2)〗^2
and for b) f"(x)=2 (1+〖3x〗^2)/〖(1-x^2)〗^3
Now after many rounds of trying i have not been able to get an answer remotely close to what they have given. i don;t know if it is due to me...
Homework Statement
find f''(x) if f(x) = sqrt(x) * e^(-x) and then find the roots of f''(x)
// I am trying to do the 2nd derivative test (need f''x) and then find inflection points//
Homework Equations
my methodology| d/dx sqrt(x) = 1/(2*sqrt(x)) and d/dx e^(-x) = -e^(-x)...