Derivatives Definition and 1000 Threads

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.

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  1. M

    Finding Derivatives with a Constant Radius

    Homework Statement the total surface area of a right circular cylinder is given by the formula: (A = 2Pir(r + h) ). where r is the radius and h is the height. sub part a) find the rate of change of A with respect to h is r remains constant i know how to take derivatives. the only...
  2. P

    What Is the Key Step Missing in Deriving Wirtinger Derivatives?

    Let \bar{z} = x+iy. We are given that x = \frac{z+\bar{z}}{2} & y = \frac{z-\bar{z}}{2i}. We are trying to derive \partial F/\partial\bar{z} = 1/2(\partial F/ \partial x + i \partial F/ \partial y), where F(x,y) is some function of two real variables. Using the chain rule I get \partial...
  3. 3

    Absolute and Relatiave Uncertainty (partial derivatives)

    Homework Statement Calculate the following, expressing all results with uncertainties both in absolute and relative (percentage) form: a) A + B b) A x B c) Asin(theta) d) A^2 / Bcos(theta) The relevant formula for the absolute uncertainty is below, but i have no idea how to...
  4. T

    Derivatives and Partial Derivatives

    ey guys Generally i just do these without thinking, however i was checking some work today with a friend and he is adament i did my derivative wrong... If i can double check with you Well firstly 'c' is simply a constant q1 and q2 are generalised coordinates IZG1 is simply the...
  5. M

    Learning Calculus: Chain Rule and Derivatives

    I am currently learning calculus and just had my lecture on the chain rule. I noticed that we haven't learned how to take the derivative of a function like 2^2+x or 3^4+x. Any example works.. Is this something I will learn later as I progress through calculus or what?
  6. P

    Is u(t) a unit vector of f(x,y)?

    Homework Statement f(x,y)=2Sin x Cos y g(x,y) = 2Cos x Sin y verify that d(fg)/dx = g(x,y) df/dx + f(x,y) dg/dx The Attempt at a Solution first of all I worked out the partials derivatives in respective to x and y, for both functions df/dx = 2Cos x (but I've a gut feeling that it...
  7. A

    Partial Derivatives of Discontinuous Fcn?

    f(x,y)= xy2/(x2+y2) if (x,y)\neq(0,0) =0 if (x,y)=(0,0) Show that the partial derivatives of x and y exist at (0,0). This may be a really stupid question, but would the partial derivatives of x and y at (0,0) just be 0? I tried taking that partial derivatives of xy2/(x2+y2) and...
  8. A

    Partial derivatives (Maxwell relations) in thermodynamics

    My professor did this in lecture, and I can't figure out his logic. Can someone fill in the gaps? He went from: dS = \left( \frac{\partial S}{\partial P} \right)_T dP + \left( \frac{\partial S}{\partial T} \right)_P dT (which I totally understand; it just follows from the fact that...
  9. N

    Problem with solving 'basic formula' derivatives

    Homework Statement http://webwork2.asu.edu/webwork2_files/tmp/equations/9f/ab106661843d52ded597f9bcb68ace1.png find f'(x) Homework Equations The Attempt at a Solution I tried 8x^.5(-4), 0, and a few others once my original try came up with nothing. I realize sqrtx should come...
  10. P

    Directional derivatives, SIMPLE

    f(x, y, z) = xe^y + ye^z + ze^x, at (0, 0, 0), directional vector v = <-2, 0, 5> i solved for gradient f = (e^y + ze^x, xe^y + e^z, ye^z + e^x), at f(0,0,0) to be... gradient f = (1,1,1) this would make the answer just be -2 + 0 + 5 = 3 but this isn't right. can someone show me...
  11. A

    Spherical coordinate derivatives

    1. Find the derivatives of the spherical coordinates in terms of df/dx, df/dy, and df/dz. 2. f(x,y,z) x=rcos\thetasin\varphi y=rsin\varthetacos\varphi z=rcos\varphi 3. The Attempt at a Solution [/b] I took the derivatives of the three equations and I got...
  12. S

    Do derivatives always exist in a neighborhood?

    Suppose that f:\mathbb{R} \to \mathbb{R} is continuous and f'(x_0) exists for some x_0 . Does it follow that f' exists for all x such that |x-x_0|<r for some r>0 ?
  13. N

    Partial derivatives of contour maps/level curves

    Homework Statement Basically I have two problems that are asking for the partial derivative with respect to x and y at a certain point on a level curve graph, and a contour map. How do you go about doing these? There is no function given, so I don't really know what they expect you to do...
  14. D

    Matlab- expressing derivatives in an equation with ode45?

    Homework Statement (-1)^4*xdx + (8y-y^2-13)dy=0; y(0)=4; 1. Use dsolve to obtain a solution. 2. As dsolve was not much help fi nd an implicit solution of the form f(x, y) = 4 Homework Equations --- The Attempt at a Solution the dsolve part was easy, i just did: syms x y t...
  15. Telemachus

    Partial derivatives for the sign function

    Homework Statement Hi there. Well, I've got some doubts on the partial derivatives for the next function: f(x,y)=sg\{(y-x^2)(y-2x^2)\} Where sg is the sign function. So, what I got is: f(x,y)=f(x)=\begin{Bmatrix}{ 1}&\mbox{ if }& (y-x^2)(y-2x^2)>0\\0 & \mbox{if}& (y-x^2)(y-2x^2)=0\\-1 &...
  16. Telemachus

    Solving Partial Derivatives: Is This Right?

    Homework Statement Hi there. Well, I got the next function, and I'm trying to work with it. I wanted to know if this is right, I think it isn't, so I wanted your opinion on this which is always helpful. f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x+y}) & \mbox{ si }&...
  17. P

    Polynomials, Kernels and Derivatives

    Is there a simple way to show that when we differentiate the following expression (call this equation 1): Y(x) = \frac{1}{n!} \int_0^x (x-t)^n f(t)dt that we will get the following expression (call this equation 2): Y'(x) = \frac{1}{(n-1)!}\int_0^x (x-t)^{n-1}f(t)dt It's simple...
  18. P

    Find f ' (2) for Simple Derivatives: g(2)=3, g ' (2)=-2, h(2)=-1, h ' (2)=4"

    Find f ' (2) given the following. g(2) = 3 , g ' (2) = -2 h(2) = -1 , h ' (2) = 4 a. f(x) = 2g(x) + h(x) b. f(x) = g(x) / h(x) c. f(x) = 4 - h(x) d. f(x) = g(x)*h(x)
  19. I

    Calc I derivatives, I know the answers, but how did I get there?

    Homework Statement Question #2 http://img830.imageshack.us/img830/8185/0000825.jpg Homework Equations The Attempt at a Solution I can of course find the derivatives for both functions. When I set them equal to each other I get x=1/2 that's where the functions have parallel tangent lines for...
  20. L

    Orthogonality of time dependent vector derivatives of constant magnitude

    I'm having trouble understanding why a derivative of a time dependent vector function is orthogonal to the original function. Can anybody give me some enlightenment? I searched around for some previous talk about this, and I can't find anything. Thanks.
  21. Rasalhague

    Covariant and exterior derivatives

    Reading Roger Penrose's The Road to Reality, I wondered what is the relationship/difference between these? Can one be expressed simply in terms of the other? The exterior derivative seems to be only defined for form fields. He says the covariant derivative of a scalar field (0-form field) is the...
  22. W

    Can Lie Derivatives be Defined Without Dragging on Manifolds?

    suppose there is a vector field V on a manifold M V generates a flow on M suppose \gamma(t) is an integral curce in this flow now there is another vector field W on M why not define the lie derivative of W with respect to V as the limit of the divide (W(\gamma+\delta...
  23. V

    Simple act of taking derivatives, I suppose

    I am having bit of a problem proving Eq. (0.2): (0.1) \text{ } G_{\omega}(t-t^\prime)=\theta(t-t^\prime) \frac{e^{-i\omega(t-t^\prime)}}{2\omega}+\theta(t^\prime-t) \frac{e^{i\omega(t-t^\prime)}}{2\omega} (0.2) \text { }\left (-\partial^2_{t}-\omega^2 \right ) G_\omega...
  24. J

    Derivatives in polar coordinates

    I appologise in the lack of distinction between curly d's and infinitesimals! All derivatives are partial and anything outside of brackets is an infinitesimal. also, I sincerely apologise for any dodgy terminology, but I am for the most part self taught (regarding calculus) :/ (also, 0 is my...
  25. T

    Question about domain of derivatives

    I was thinking how do I differentiate the domain of functions... Suppose I have a function: f(x) = \left\{\begin{matrix} x^2 -1, \;\; |x| \leq 1\\ 1 - x^2, \;\; |x| > 1 \end{matrix}\right. And I need to derive it: f'(x) = \left\{\begin{matrix} 2x, \;\; |x| \leq 1\\ -2x...
  26. T

    What are the implications of having different lateral derivatives at a point?

    I'm reading about lateral derivatives... I know that a function is said derivable on a point if the lateral derivative coming from left and the lateral derivative coming from right are both equal at that point. f'(a) exists only if both f'+(a) and f'-(a) exists and are equal... right? Ok...
  27. Char. Limit

    Can Derivatives be Taken with Respect to Functions?

    Is it possible to take a derivative with respect to a function, rather than just a variable? I'll give a simple example of how I imagine such a thing would work to try to explain... \frac{d}{d(sin(x))}\left(sin^2(x)\right) = 2 sin(x) Can you take a derivative this way? Also, can you...
  28. L

    Higher order derivatives in field theories

    It is common lore to write lagrangians in field theories in the form L(t)=\int d^{3}x\mathcal{L}(\phi_{a},\partial_{\mu}\phi_{a}). Nonetheless, is there any particular reason for doing that? Why do we neglect higher order derivatives? Does it mess around with Lorentz invariance or something...
  29. X

    Confusion with understanding derivatives in respect to

    Confusion with understanding derivatives "in respect to..." I have been teaching myself Calculus for the past 2 weeks or so and I've just barely started learning Implicit Differentiation. There's a few things I have trouble understanding, which this is probably a simple concept that I am...
  30. A

    Derivatives and Increments -help, again

    Homework Statement Find f'(c) and the error estimate for: f(x)= \sqrt{x^{2}+1} Homework Equations The error is given by: E(\Delta x) = \frac {1}{2}M \Delta x and f''(c) \leq M The Attempt at a Solution So the first derivative is: f'(x) = \frac...
  31. jegues

    What is the Directional Derivative at a Given Point?

    Homework Statement See first figure. Homework Equations N/A The Attempt at a Solution See second figure. I defined direction of the line by which the two planes intersect as, \vec{d} and found that the point they are asking about is when, t=1 and I'm stuck here. This...
  32. A

    Derivatives and Increments -help

    Homework Statement Find f'(c) and the error estimate for the limit: f'(c) = \lim_{x \to 0^{+}} \frac {f(c+\Delta x) - f(c)}{\Delta x} I just included that to show that we are working with one (right) sided limit the actual problem is: f(x) = \frac {1}{x} \;\; with\;\; c = 3...
  33. S

    Laplacian, partial derivatives

    Homework Statement Find the Laplacian of F = sin(k_x x)sin(k_y y)sin(k_z z) Homework Equations \nabla^2 f = \left( \frac{\partial}{\partial x} +\frac{\partial}{\partial y} + \frac{\partial}{\partial z} \right) \cdot \left( \frac{\partial}{\partial x} + \frac{\partial}{\partial y} +...
  34. Y

    What is the meaning by a function with continuous 1st and 2nd derivatives?

    The definitions of a harmonic function u are: It has continuous 1st and 2nd derivatives and it satisfies \nabla^2 u = 0. Is the second derivative equal zero consider continuous? Example: u=x^2+y^2 ,\; \hbox{ 1st derivative }=u_x + u_y = 2x+2y,\; \hbox{ 2nd derivative }=u_{xx} + u_{yy} +...
  35. J

    How Do You Prove This Limit Equals the Derivative?

    Homework Statement If f is differentiable at a, prove the following: \lim_{h,k \to 0^+} \frac{f(a+h)-f(a-k)}{h+k} = f'(a) Homework Equations N/A The Attempt at a Solution At the moment, I don't have a complete proof worked out, but I was wondering if someone could comment on the...
  36. G

    Question about notation on derivatives

    Homework Statement If I'm asked to find the acceleration at t=2 s I can just put the X with two dots on top of it parentheses(2 s)? X(2) = what ever I calculate it being equal to that's all I have to put right the X with two dots indicated the derivative of x( distance) the two dots...
  37. K

    How does distributional derivatives work in the context of linear mappings?

    When do derivatives in the sense of distributions and classical derivative coincide? Of course f needs to be differentiable. What else? Any reference?
  38. B

    Partial Derivatives for Functions f(z) of a Complex Variable.

    Hi, Everyone: I was never clear n this point: given that z is a single complex variable, how/why does it make sense to talk about z having partial derivatives.? I mean, if we are given, say, f(x,y); R<sup>2</sup> -->R<sup>n</sup> then it makes sense to talk about...
  39. M

    Proving an equation related to order of derivatives

    Homework Statement Prove that for all y(x)=ax^2+bx+c where a is a constant !=0 and x is a real number that \frac{y'(x2)^2- y'(x1)^2}{(x2-x1)} = 2y''(x)Homework Equations I don't know what to put here in mathematics but here... y(x)=ax^2+bx+c y'(x)=2ax+b y''(x)=2aThe Attempt at a Solution I...
  40. S

    Symmetry of higher order partial derivatives

    Hi, As per Clariut's theorem, if the derivatives of a function up to the high order are continuous at (a,b), then we can apply mixed derivatives. I am looking at http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives and I cannot understand in the example for non-symmetry, why the...
  41. T

    How Do You Handle Double Functional Derivatives?

    Hi all, Long time stalker, first time poster. I've finally got stumped by something not already answered (as far as I can tell) around here. I'm trying to make sense of double functional derivatives: specifically, I would like to understand expressions like \int dx \frac{\delta^2}{\delta...
  42. N

    Second derivatives to find max and min values then sketch graph

    Homework Statement Sketch the graph of each function. List the coordinates of where extrema or points of inflection occur. State where the function is increasing or decreasing, as well as where it is concave up or concave down. 9. f(x)=x3-12x Homework Equations The Attempt at a...
  43. pellman

    When derivates w/ resp to complex variables differ from real derivatives?

    If z is complex, the following rules are true, right? \frac{d}{dz}z^n = nz^{n-1} \frac{d}{dz}\ln{g(z)} = \frac{1}{g(z)} \frac{d}{dz}g(z) \frac{d}{dz}e^{g(z)}=e^{g(z)}\frac{d}{dz}g(z) These are of course the same rules as for real variables. When do I need to be careful about...
  44. P

    Relating 2nd order partial derivatives in a coordinate transformation.

    Homework Statement Could some mathematically minded person please check my calculation as I am a bit suspicious of it (I'm a physicist myself). This isn't homework so feel free to reveal anything you have in mind. Suppose I have two functions \phi(t) and \chi(t) and the potential V which...
  45. M

    Understanding Objective Derivatives in Continuum Mechanics: A Simplified Guide

    Hi there, I am trying to refine my continuum mechanics, which I learned as an enginner. I need to get a better undertstanding of the differences between upper and lowerc onvected derivative, and Jaumann derivative, as well as Lie derivative. I am not far, hopefully, from having gained a...
  46. R

    Derivatives of Cauchy Distribution

    Hi guys, I would like to ask you where you spot the mistake in the derivatives of the loglikelihood function of the cauchy distribution, as I am breaking my head :( I apply this to a Newton optimization procedure and got correct m, but wrong scale parameter s. Thanks! LLF =...
  47. R

    Ywhere \phi^X_t is the flow of X.

    Given a smooth manifold with no other structure (like a metric), one can define a derivative for a vector field called the Lie derivative. One can also define a Lie derivative for any tensor, including covectors. Incidentally, with antisymmetric covectors (differential forms) one can define...
  48. T

    Prove f'(x) = a(n) x^(n-1): Math Steps & Examples

    1. f(x)= ax^2 = ae^TR, nez Prove f '(x) = a(n) x^(n-1)2. n does not equal 03. I don't even understand it
  49. R

    Proving the Roots of Higher Derivatives of a Polynomial Function

    Let f (x) = (x^2 − 1)^n . Prove (by induction on r) that for r = 0, 1, 2, · · · , n, f^ (r) (x)(the r-th derivative of f(x)) is a polynomial whose value is 0 at no fewer than r distinct points of (−1, 1). I'm thinking about expanding f(x) as the sum of the (n+1) terms, then it's easier to...
  50. M

    Calculating second derivatives implicitly

    Hi I've just been learning about how to get first derivatives implicitly and I think I'm getting it. Then the book comes onto calculating second derivatives implicitly and I don't know how to handle the dy/dx terms you might have in your equation from the first implicit differentiation...
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