What is Derivatives: Definition and 1000 Discussions
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
It says:
\displaystyle f:{{\mathbb{R}}^{2}}\to \mathbb{R}
\displaystyle f\left( x,y \right)=\left\{ \begin{align}
& 1\text{ if 0<y<}{{\text{x}}^{2}} \\
& 0\text{ in other cases} \\
\end{align} \right.
Show that all the directional derivatives about (0,0) exist but f...
I'm just experimenting with graphs of derivatives and have noticed something
that our teacher has never adverted to but strikes me as very interesting:
the third derivative when set to zero will give the inflection points of the first, which makes me reflect that the relationship which...
I am trying to understand the notion of a covariant derivative and Christoffel symbols. The proof I am looking at starts out with defining a tensor, Tmn = ∂Vm/∂xn where V is a covariant vector. I am having a mental block with regard to the indeces. How is it that the derivative of a...
A 3-dimensional graph has infinite number of derivatives (in different directions) at a single point. I've learned how to find the partial derivative with respect to x and y, simply taking y and x to be constant respectively. But what do I do if I want to take the partial derivative with respect...
Hi all,
I have the following partial derivatives
∂f/∂x = cos(x)sin(x)-xy2
∂f/∂y = y - yx2
I need to find the original function, f(x,y).
I know that df = (∂f/∂x)dx + (∂f/∂y)dy
and hence
f(x,y) = ∫∂f/∂x dx + g(y) = -1/2(x2y2+cos2(x)) + g(y)
Then to find g(y) I took the...
What I have learned in school is that differentiation and integration are opposites.
By integrating a function we find the area under the graph. So, integration gives us the area. Differntiation gives slope of the function.
If I am right by saying differentiation and integration are...
Homework Statement
The problem is attached in the picture. The top part shows what is written in the book, but I am not sure how they got to (∂I/∂v)...The Attempt at a Solution
It's pretty obvious in the final term that the integral is with respect to 't' while the differential is with...
Homework Statement
What does it mean when the derivative of a function f(x) is in the form:
d ln f(x) / d ln x
?
Is it the logarithmic scale derivative, or something?
Homework Equations
d ln f(x) / d ln x
The Attempt at a Solution
Googling.
Hi. I am reviewing for an exam, and this is a topic that I did not go through very thoroughly.
I understand how to calculate the derivative of an inverse function when I am given a point, as I simply use the equation (1/f((f^-1))'. So if I am given the equation y=x^3, for example, and am asked...
Homework Statement
The equations xu^2 + yv = 2, 2yv^2 + xu = 3 define u(x,y) and v(x,y) in terms of x and y near the point (x,y) = (1,1) and (u,v) = (1,1).
Compute the following partial derivatives:
(A) ∂u/∂x(1,1)
(B) ∂u/∂y(1,1)
(C) ∂v/∂x(1,1)
(D) ∂v/∂y(1,1)
The answers are:
(A)...
Homework Statement
1. Is (∂P/∂x)(∂x/∂P) = 1?
I realized that's not true, but I'm not sure why.2. Say we have an equation PV = T*exp(VT)
The question wanted to find (∂P/∂V), (∂V/∂T) and (∂T/∂P) and show that product of all 3 = -1.The Attempt at a SolutionI tried moving the variables about...
Hello, I was pondering acceleration's relation to velocity, when I came across a problem that is more of an issue of differentiation.
Given our variables
v=velocity
a=acceleration
α=slope of 'a'
t=time
(at1→t2)=Total acceleration from t1 to t2
(αt1→t2)=Total slope of 'a' from t1 to t2...
Homework Statement
Hi
I have a technical question regarding the following paper: http://arxiv.org/pdf/quant-ph/0602170v1.pdf
In it they derive an equation for \left\langle a \right\rangle (equation #5) from the master equation (equation #2). My question is how they do this. Here is what...
I just got to a point in multivariable calculus where I realize I can solve problems in assignments and tests but have no actual idea of what I'm doing. So I started thinking about stuff and came up with a few questions:
1. Is picturing the derivative as the slope of the tangent line to a...
Homework Statement
From step 1 to step 2, what do they mean by "Taking the weighted sum of the two squares " ?
I tried and expanded everything in step 2 and it ends up as the same as step 1 (as expected),
The Attempt at a Solution
I tried looking up "weighted sum" and "...
Is there any clear explanation as to why exactly derivatives and integrals cancel each other [other than the integral is the anti-derivative]?
To my understanding the derivative gives the slope of a curve at given point whereas the the integral finds the area under the curve. How are these...
Hi!
I'm currently re-reading Griffiths introductory QM book and plan to do most of the exercises. I got stuck on one problem and had to look for some hints and found two solutions that both claim that:
\int_{-\infty} ^{\infty} \frac{\partial }{\partial x}\left[ \frac{\partial \Psi...
Here's the problem:
Find the directional derivative of the function $V(x,y) = x^3 -3xy +4y^2$ at the point $P(2,1)$ in the direction of the unit vector $\vec{u}$ given by the angle $\theta = \pi /6$.
So I found the gradient of the function $$\nabla V(x,y) = (3x^2 -3y, 8y -3x),$$ and then at...
Homework Statement
http://gyazo.com/d875970963124b7d4a64acc887f168fa
The Attempt at a Solution
I don't understand the part where they take the derivative of a diff eq to find the higher derivatives can somebody explain it?
Mainly the answer where they get
y''=(y'')'=(1-y^2)'=-2yy'(this...
I'll just make one thread for all the help I'll need with derivatives so I don't clutter up this forum.
Homework Statement
Find the derivative of y = sqrt(x)(x - 1).Homework Equations
Wolfram Alpha gets this:
http://www.wolframalpha.com/input/?i=derivative+y+%3D+sqrt%28x%29%28x+-+1%29
I got...
Hi all! I'm sorry if this question has been already asked in another post...
I'm studying the path integrals formalism in QED. I'm dealing with the functional generator for fermionic fields. My question is:
The generating functional is:
$$Z_0=e^{-i\int{d^4xd^4y \bar{J}(x)S(x-y)J(y)}}$$...
Homework Statement
Consider the surface and point given below:-
Surface: f(x,y)= 4-x2-2y2
Point: P(1,1,1)
a) Find the gradient of f.
b) Let C' be the path of steepest descent on the surface beginning at P and let C be the projection of C' on the xy-plane. Find an equation of C in the...
Hi
I thought of something today coming home from school: If I have two arbitrary real functions f(x) and g(x) and I know that
\frac{df(x)}{x} = \frac{dg(x)}{dx}
Does this imply that f(x)=g(x)?
Niles.
In Spivak's Calculus, there is a theorem relating the derivative of the limit of the sequence {fn} with the limit of the sequence {fn'}.
What I don't like about the theorem is the huge amount of assumptions required:
" Suppose that {fn} is a sequence of functions which are differentiable on...
Homework Statement
Find a differential of second order of a function u=f(x,y) with continuous partial derivatives up to third order at least.Hint: Take a look at du as a function of the variables x, y, dx, dy:
du= F(x,y,dx,dy)=u_xdx +u_ydy.
Homework Equations
The Attempt at a Solution
I'll be...
Homework Statement
I have got a question concerning the following function:
f(x,y)=\log\left(x^2+y^2\right)
Partial derivatives are:
\frac{\partial^2f}{\partial x^2}=\frac{y^2-x^2}{\left(x^2+y^2\right)^2}
and
\frac{\partial^2f}{\partial y^2}=\frac{x^2-y^2}{\left(x^2+y^2\right)^2}
The...
Homework Statement
Let f be a polynomial function of degree n such that f(x) ≥ 0 for all x (note that n must be even). Prove that f + f' +f'' + f''' + ... + f^(n) ≥ 0.
Homework Equations
I believe that is all - the derivative of some term in the polynomial ax^n is anx^(n-1).
The...
Hello Everyone!
This has been confusing me a lot: consider a function $f(x) = x^2 + 2x + 3$. Now, $\frac{\partial f}{\partial x} = 2x + 2$. Now, someone tells me that $y = x^2$. What is $\frac{\partial f}{\partial x}$ now?
Why is partial derivative with respect to time used in the continuity equation,
\frac{\partial \rho}{\partial t} = - \nabla \vec{j}
If this equation is really derived from the equation,
\frac{dq}{dt} = - \int\int \vec{j} \cdot d\vec{a}
Then should it be a total derivative with...
I luckily went on to Newton's laws which I already know. I was stuck with a fantastic question ( I have been seeing the Newton's laws from my childhood, but I didn't notice these sort of observation ) . I was seeing the Force and Momentum. Force is based upon acceleration and Momentum on...
In the ordinary least squares procedure I have obtained an expression for the sum of squared residuals, S, and then took the partial derivatives of it wrt β0 and β1. Help me to condense it into the matrix, -2X'y + 2X'Xb.
∂S/∂β0 = -2y1x11 + 2x11(β0x11 + β1x12) + ... + -2ynxn1 + 2xn1(β0xn1 +...
Help Derivatives ASAP -- maximizing sunlight through a window
3. The amount of daylight a particular location on Earth receives on a given day of the year can be modeled by a sinusoidal function. The amount of daylight that Windsor, Ontario will experience in 2007 can be modeled by the function...
Derivatives Question -- temperature as a funtion of time
1. When a certain object is placed in an oven at 540°C, its temperature T(t) rises according to the equation T(t) = 540(1 – e–0.1t), where tis the elapsed time (in minutes). What is the temperature after 10 minutes and how quickly is it...
Dear all,
I have a confusion about partial derivatives.
Say I have a function as
y=f(x,t)
and we know that
x=g(t)
1. Does it make sense to talk about partial derivatives like \frac{\partial y}{\partial x} and \frac{\partial y}{\partial t} ?
I doubt, because the definition of...
Homework Statement
Hi
I've been giving the following problem:
We have a differentiable function f: [a,b] \rightarrow \mathbb{R} with f'(a) < 0 en f'(b) > 0. Let c \in \mathbb{R} such that f'(a) < c. Show that there exists a \delta >0 such that for every x \in ]a, a + \delta[ the following...
Why is it that the constant e is defined as being the unique case where the limit as h goes to 0 of (e^h - 1)/h = 1? I mean every exponential function like a^x equals 1 when x equals 0, right? So would it be fair to say that (a^h) approaches 1 as h approaches zero? And that (a^h - 1) approaches...
The total derivative of the function z=f(x,y) with respect to x is:
dz/dx = ∂z/∂x + (∂z/∂y)(dy/dx)
The way i see this is that the total derivative, dz/dx, gives the rate of change of z with x, allowing y to vary with x at the rate dy/dx. I don't know if this is right.
The directional...
Homework Statement
Here is the problem:
http://dl.dropbox.com/u/64325990/MATH%20253/help.PNG
The Attempt at a Solution
http://dl.dropbox.com/u/64325990/Photobook/Photo%202012-05-24%209%2037%2028%20PM.jpg
This seems to be wrong... Since I have fx and fy which I cannot cancel out. Why...
Homework Statement
Here is the problem with the solution:
http://dl.dropbox.com/u/64325990/MATH%20253/Capture.PNG
What I don't understand is circled in red, how did they combine dxdy with dydx? Is it with Clairaut's theorem? If it is can someone explain how it works in this case because...
Homework Statement
Ok, I'm trying to understand the proof for derivatives. I understand most of it, but there is one step that I cannot understand.
lim x-> 0 [xm - am]/[xn - an = (m/n)am-n
The Attempt at a Solution
I don't see how those are equal. The best I can do is...
Hi, I was hoping someone could help me out with my homework set.
I have done a lot of the questions, and it would help if someone could tell me if I have done them correctly. Thanks! :)
Q1: Find first derivatives for the following functions
(a)g(s,t)=sin(st^3)...
Directional and partial derivatives help please!
I have read that the partial derivative of a function z=f(x,y) :∂z/∂x, ∂z/∂y at the point (xo,yo,zo)are just the tangent lines at (xo,yo,zo) along the planes y=yo and x=xo. Directional derivatives were explained to be derivatives at a particular...
1. A 1000 L tank is draining such that the volume V of water remaining in the tank after t minutes is v=1000(1-t/60)2. Find the rate at which the water is flowing out of the tank after 10 min.
Calculate dV/dt using chain rule
u = 1 - t/60:
u = 1 - t/60 ==> du/dt = -1/60
V = 1000u^2 ==> dV/du =...
f=9-x^2-y^2 and u=i-j
The directional derivative comes out to be Du f(x,y)=-sqrt(2)+sqrt(2)
I'm going to find the directional derivative and could someone kindly point out the mistake because I am getting a different answer and it's important I understand how to do this question:
Du...
I was searching for the proof of \frac{d}{dx} e^x = e^x.
and I found one in yahoo knowledge saying that
\frac{d}{dx} e^x = \lim_{Δx\to 0} \frac {e^x(e^{Δx}-1)} {Δx}
= \lim_{Δx\to 0} \frac {e^x [\lim_{n\to\infty} (1+ \frac{1}{n})^{n(Δx)}-1]} {Δx}
Let h= \frac {1}{n} , So that n = \frac...
Homework Statement
Demonstrate that C_{Y,N}=\left ( \frac{ \partial H}{\partial T } \right ) _{Y,N} where H is the enthalpy and Y is an intensive variable.
Homework Equations
(1) C_{Y,N}=\frac{T}{N} \left ( \frac{ \partial S}{\partial T } \right ) _{Y,N}
(2) T= \left ( \frac{ \partial...
Homework Statement
Find the derivative of f(x):
f(x)= 1/ ((ln(x)^2))
Homework Equations
f(x)= ln(x)
f'(x)= 1/((ln(x))
The Attempt at a Solution
Dx(1/ln(x)^2) =
Dx((ln(x))^-2)=
-2*(ln(x)^-3) * Dx(ln(x)) =
-2*(ln(x)^-3) * 1/x =
-2/(x*ln(x)^3)
Are these the correct...
I'm not sure if it's OK to post this question here or not, the Calculus and Beyond section doesn't really look very heavily proof oriented.
I'm trying to prove that if continuous complex valued function f(z) is such that the directional derivatives(using numbers with unit length) preserve...