In finance, a derivative is a contract that derives its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be used for a number of purposes, including insuring against price movements (hedging), increasing exposure to price movements for speculation, or getting access to otherwise hard-to-trade assets or markets.
Some of the more common derivatives include forwards, futures, options, swaps, and variations of these such as synthetic collateralized debt obligations and credit default swaps. Most derivatives are traded over-the-counter (off-exchange) or on an exchange such as the Chicago Mercantile Exchange, while most insurance contracts have developed into a separate industry. In the United States, after the financial crisis of 2007–2009, there has been increased pressure to move derivatives to trade on exchanges.
Derivatives are one of the three main categories of financial instruments, the other two being equity (i.e., stocks or shares) and debt (i.e., bonds and mortgages). The oldest example of a derivative in history, attested to by Aristotle, is thought to be a contract transaction of olives, entered into by ancient Greek philosopher Thales, who made a profit in the exchange. Bucket shops, outlawed in 1936, are a more recent historical example.
Homework Statement
If f(x) can be differentiated, find expressions for the derivatives of the following functions.
a) g(x) = f(x6)
b) h(x) = [ f(x)]6
c) f(x) = x2/ f(x)
The Attempt at a Solution
a)
b) Use the product rule first then multiply that expression by the expression for...
Lee: Introduction to Smooth Manifolds, definition A.18:
He then shows, by the chain rule, that
D_vf(a_0)= \sum_{i=1}^n v^i \frac{\partial }{\partial x^i}f(a) \bigg|_{a_0}
It seems to me, though, that this number depends not only on the direction of v but also on its length. For example...
I know that Mathcad only takes partial derivatives. I set up my equations using this general format:
L:= x(t)
Then, I take the derivative of L with respect to t and get the following:
dL/dt -> d/dt*x(t)
However, when I take the derivative of L with respect to x, I should get 1, but...
I want to interpolate a function between the points A, B, C. At A and C I only know the value of the function, but at B (lying between them) I also know the function's first and second derivatives. How would you interpolate between these points?
Dear PF,
I am having very heavy supersymmetric derivatives to do (and in general heavy calculations) could you advise me any software/package or something that will help me to do my calculations on computer?
Thank you very much
Rgds
GT
Homework Statement
Use Clairaut's Theorem to show that is the third order partial derivatives are continuous, then fxxy=fyxy=fyyz
Clairaut's Theorem being: fxy(a,b)=fyx(a.b)
Homework Equations
fxyy=d/dy(d2f/dydx)=d^3f/dy^2dx
The Attempt at a Solution
Tried to differentiate...
Homework Statement
See attachment.
2. Homework Equations /solution attempt
Part (a)
Well, the gradient evaluated at (1,2-1) will give the rate of change. If we want the maximum rate of change then we need the directional direction such that the unit vector \mathbf{u} is in the same...
hi does anyone know why the 2nd derivative chain rule is as such?
i roughly know that
if u = f(x,y) and x=rcos(T) , y = rsin(T)
then
du/dr = df/dx * dx/dr + df/dy * dy/dr
but if i am going to have a second d/dr, then how does it work out?
find the directions in which the directional derivative of f(x,y)=x^2 + sin(xy) has a value of 1 at the point (1,0)
Fx=2x+ycos(xy)=2
Fy=xcos(xy)=1
So we have <2,1> and we need to find vectors that dotted with <2,1> =1
<2,1>.<x1,x2>=1
2x1+x2=1
So whn x1 is 0 we have x2 is 1
so...
Hello! My problem is that I want to find (\frac{\partial}{{\partial}x}, \frac{\partial}{{\partial}y}, \frac{\partial}{{\partial}z}) in spherical coordinates. The way I am thinking to do this is...
I am working on an advanced fundamental engineering theory. For that I need to solve a system of differential equations in R2 by expanding my variables as Legendre series expansion.
Thus: u(x,y)=\sum\sumAmnPm(x)Pn(y)
The equations contain of each variable derivatives up to the fourth...
Homework Statement
Everyline tanjent to the function y=sin x has a y-intercept. Among all these tanjent lines, somewere between 0<x<2pi, find the equation of the line with the highest y-intercept.
Homework Equations
derivative of sinx=cosx
Second derivative is -sinx
The Attempt at a...
Homework Statement
function f is differentiable when x=0,
f'(0) is not equal to zero for all real Numbers
f(a+b) = f(a)f(b)
show f'(x) = f'(0)f(x)Homework Equations
The Attempt at a Solution
f(x+0) = f(x) = f(x)f(0)
this shows f(0) = 1
then i get stuck..
Homework Statement
function f is differential when x=0,
f'(0) is not equal to zero for all a,b(real Numbers)
f(a+b) = f(a)f(b)
show f'(x) = f'(x)f(x)
Homework Equations
The Attempt at a Solution
f(a+b) = f(a)f(b) for all a,b(real numbers)
f(0), a+b=0
then f(0) =...
Suppose u(x) is periodic with period 2\pi. Also m\le u(x)\le M.
Then is it possible for some derivatives of u(x) to be outside [m, M]? In other words, can any derivative be 2pi-periodic and have a different amplitude?
When u(x) is a sine curve, then it is not true because, the frequency of...
Hello all,
This is a question on a problem set for my Calculus 1 class. Please help if you can.
Homework Statement
A function, f defined on the set of real numbers is said to have even symmetry if f(-x) = f(x) for all x, and is said to have odd symmetry if f(-x) = -f(x) for all x. Use...
Homework Statement
http://www.math.wvu.edu/~hjlai/Teaching/Tip-Pdf/Tip3-27.pdf
Example 7.
Not this question in particular, but it shows what I'm talking about.
I understand how they get the first partial derivative, but I'm completely lost as how to take a second one.
I have tried...
I recently read a paper on fractional derivatives. That is how to take derivatives of fractional order rather than the usual integral order. The paper made perfect sense to me, however I wondered:
1) Are there geometric interpretations of fractional derivatives? Kind of like how first...
I started to answer this question, and I have quite a bit an answer, but still not complete, let's say that we write a Lagrangian in QFT, which an unknown function of the scalar field \phi and its derivative \partial \phi. We can always Taylor-expand it and get: L(\phi,\partial\phi) = a + b \phi...
This is a small part in converting between rectangular to polar coords for laplace equation with a problem of circular geometry:
what I have in my notes:
\tan\theta=\frac{y}{x}\implies\sec^{2}\theta\frac{\partial\theta}{\partial x}=\frac{-y}{x^{2}}
I can't figure out how he went from...
Homework Statement
Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f '_n converges to zero uniformly in D = {z : |z| < 1/2}.Homework Equations
Cauchy inequalities (estimates from the Cauchy integral formula)The...
Homework Statement
Use L'Hopital's rule to calculate the following derivatives.
Homework Equations
1. lim x-> pie/2 tan3x/tan5x
2. lim x->0 e^x - 1/sin x
3. lim x->1 e^x - e/In x
The Attempt at a Solution
i have attempted to solve the...
I understand that, having a function f(x), it's derivative function is the rate of change of f.
That df/dx means how much f changes, given an infinitesimal change in x, denoted as dx.
In second derivatives,how is d2f / dx2explained ?
Help me on the intuition please.
Find the first and second derivative--simplify your answer.
y=x tanx
I solved the first derivative.
y'=(x)(sec^2(x)) +(tanx)(1)
y'=xsec^2(x) +tanx
I don't know about the second derivative though.
Homework Statement
x(t) = -0.01t^3 + t^2 - 20t + 4
Homework Equations
Min is when t = 12.3
Max is when t = 54.4
The Attempt at a Solution
I got -0.03t^2 + 2t - 20 as the derivative.
I substituted in t = 12.3 and 54.4 and got 0.02 and 0.19 which don't seem right at all.
Because...
Homework Statement
Let z = z (x,y) be a function with x = x(s), y = y(t) satisfying the partial differential equation
(Ill write ddz/ddt for the partial derivative of z wrt t and
dz/dt for the total derivative of z wrt t, as I have no idea how to use Latex.)
ddz/ddt +...
Please help me with the geometric interpretation...
I am wondering why they can define u(directional derivative) as dr/ds instead of r(s) ?
Below are my interpretations, i don't not know whether I'm right... :
The r in dr is <dx,dy,dz> and it refers to the change in position of the graph...
Ok my last post was trivial, but it led to this question
Assume f is unbounded and analytic in some domain D, and f' is bounded in D
does there exist a function for which the above holds and f'',f''',... are all unbounded in D?
Can someone explain to me what are differential and derivatives used for (intergrals ?) in some well known stuff from dynamics or thermodynamics:
dA=Fdr or in thermodynamics PdV
For example what is that dV ... why not just V.
Why do I sometimes write a=d^2r/dt^2 instead of a=r(:)/t(.)...
Homework Statement
I just started in calculus ii ,and I remember that most I didn't use specific rules when dealing with derivatives and I sometimes managed to get away with it(especially in physics. now I have been playing with relations and I got unacceptable results so i will post what I...
Homework Statement
(1+4x)^5(3+x-x^2)^8
Homework Equations
The Attempt at a Solution
I get to this point, but I don't know how to break it down.
5(1+4x)^4(4x)(3+x-x^2)^8+8(3+x-x^2)(1-2x)
Help! Solving Derivatives using "Tables" and Equation on Tangent Line
1. I got a worksheet that was given to me on Derivatives to finish for homework but hardly understand it since my teacher assigned it for the same night and she had taught it
I was wondering if an explanation on how to do...
According to wiki
"The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. "
Is this to say that that derivatives are not 100% accurate? They're linear approximation? Which is confusing to me, with math being such an...
Homework Statement
Given the following graph of h(x), identify:
1. The intervals where h(x) is increasing and decreasing.
2. The local maximum and minimum points of h(x)
3. The intervals where h(x) is concave up and concave down
4. The inflection point
5. Sketch the graph of h'(x) and...
Homework Statement
Find the derivative of f(x) = x^2+2x+1Homework Equations
f(x + h) - f(x) / h
lim(h->0) f (x+h) - f(x) / hThe Attempt at a Solution
Hi everyone. I keep calculating the derivative for this function incorrectly. I haven't learned the rules of derivatives yet, I am only using...
Homework Statement
A 1.8 m tall student is trying to escape from the minimum security prison in Tono.
She runs in a straight line towards the prison wall at a speed of 4.0 m/s. The guards
shine a spotlight on the prisoner as she begins to run. The spotlight is located on
the ground 30 m...
Homework Statement
Let f be a differentiable function, defined for all real numbers x, with the following properties:
1. f'(x) = ax^2 + bx
2. f'(1) = 6 and f"(1) = 18
3. \int_{1}^{2} f(x)dx = 18
Find f(x).
Homework Equations
The Attempt at a Solution
Using the first two properties, I...
Let g(x) =x^asin(1/x) if x is not 0
g(x)=0 if x=0
Find a particular value for a such that
a) g is differentiable on R but such that g' is unbounded on [0,1].
b) g is differentiable on R with g' continuous but not differentiable at zero
c) g is differentiable on R but and g' is...
Homework Statement
Sorry I tried to use Latex but it didn't work out :/
Make the change of variables r = x + vt and s = x vt in the wave equation
partial^2y/partialx^2-(1/v^2)(partial^2y/partialt^2)=0
Homework Equations...
Sorry it's not the best Latex, I hope that you can still help me grasp this.
y=2xy1+y(y1)2; y2=C1(x+1/4C1)
So, the solution says to implicitly differentiate and gives y1=C1/2y
So, how did they get the derivative to be this? This is the first chapter in my DE class and I'm rusty with my...
Homework Statement
Derivatives and elasticity:
The demand equation for a product is q = \left(\frac{20-p}{2}\right)^{2} for 0 \leq p \leq 20.
a) find all values of p for which demand is elastic.
Homework Equations
Elasticity: \eta = \frac{p}{q} x \frac{dq}{dp}
The Attempt at a...
I couldn't decide whether to place this in the Physics or the Math section of the forums, deep down it is really a Math question for Physics problems. So mods please move if you feel it would be more appropriate in the Physics section.
So when doing calculations, I always like to make sure my...
Hello;
You can have positive integer derivatives, such as this:
\frac{d^{2}}{dx^{2}}(x^{2}) = 2
You can have fractional derivatives too;
\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}(x) = \frac{2\sqrt{x}}{\sqrt{\pi}}
But what about negative derivatives?
\frac{d^{-2}}{dx^{-2}}(x^{2})
Or even...
Domain of f(x,g(x)), and partial derivatives
Watching http://www.khanacademy.org/video/exact-equations-intuition-1--proofy?playlist=Differential%20Equations Khan Academy video on exact equations, I got to wondering: if x is a real number, what is the domain of a function defined by f(x,g(x))...
Homework Statement
Find an equation of the tangent line to y = sin x at the point x = 0. Graph both functions on the same set of axes on the interval [-pie/4, pie/4]. What does this illustrate?
Homework Equations
y = mx + b
The Attempt at a Solution
y = sin x ---> y' = cos x...
This is a very basic question, but I just started wondering about it. First which is the original quantity and which is the derived one. I mean is the displacement by definition is integral from a to b v(t) dt or is it the distance from the starting point to the finishing point or change in...