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.

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  1. T

    Partial derivatives, equation help

    Homework Statement Heat is being conducted radially through a cylindrical pipe. The temperature at a radius r is T(r). In Cartesian co-ordinates, r = \sqrt{(x^{2}+ y^{2}}) show that \frac{\partial T}{\partial x} = \frac{x}{r} \frac{dT}{dr}
  2. Char. Limit

    A quick check-up on directional derivatives

    Just a quick question... To calculate a directional derivative of f(x,y) at the point \vec{u} in the direction \hat{v}, can I just use the formula... \nabla f(\vec{u}) . \hat{v}? It would be so easy.
  3. J

    Laplace transformation of derivatives

    What are the laplace transformations of y"' and y"". Any table I can find only goes up to y". Thanks.
  4. D

    Two covariant derivatives of a vector field

    V_{a;b} = V_{a,b} - \Gamma^d_{ad}V_d Now take the second derivative... V_{a;b;c} = (V_{a;b})_{,c} - \Gamma^f_{ac}V_{f;b} - \Gamma^f_{bc}V_{a;f} But I have no idea how to get the parts with the Christoffel symbols. V_{a;b;c} = (V_{a;b})_{,c} - \Gamma^f_{(a;b)c}V_{af} = (V_{a;b})_{,c} -...
  5. C

    Solving Problem with Derivatives as Initial Conditions

    Homework Statement I've been given equations that have derivatives as initial conditions, rather than things like u(0,t)=u(L,t)=0 Things like this: http://img444.imageshack.us/img444/5082/mathu.th.jpg Uploaded with ImageShack.us Homework Equations The Attempt at a...
  6. M

    Integration & Derivatives ,Newtons

    Hi! Please can anyone help me to understand what exactly Integration & derivatives are. Please don't tell in form of limits & continuity. But tell in details of , what we exactly do when we use these functions. Please explain with a practicle example. I will appreciate your efforts...
  7. L

    Determining a function given points and the values of derivatives at points

    Homework Statement Using Maple, I'm asked to create a quintic function, in the form of ax^5+bx^4+cx^3+dx^2+ex+f given the following data: It will pass through the points (-5,15), (-5/2, 100), and (10, -5) -f'(5)=(-1) -f''(5)=1 Homework Equations How would I go about doing this? I'm allowed to...
  8. P

    Question about Leibniz's notation for derivatives

    Hi, I'm a new member to the forum, and I'm currently studying Calculus. Basically, derivatives can be written as (dy/dx) in Leibniz's notation, but I remember my teacher saying that it's just a symbol and shouldn't be used like two variables (dy and dx)... However, when there's some integral...
  9. C

    Partial derivatives with dependent variables (fixed) question.

    In statistical mechanics we express partial derivatives of functions, keeping some variables fixed. But these variables are functions of the other variables (which are not fixed). I'm just confused by this, what is the convention for taking these derivatives? For example, if we have S as a...
  10. L

    Derivatives of functions with matrices

    I try to understand how to calculate derivatives of functions, which contain matrices. For a start I am looking at derivatives by a single variable. I have x=f(t) and I want to calculate \frac{dx}{dt}. The caveat is that f contains matrices, that depend on t. Can I use the ordinary chain rule...
  11. R

    How to Calculate Implicit Derivatives for Understanding

    In short, I need to know how do you do them. I missed class, and our textbook is so bad that it might as well be written in a foreign language. I understand how to do dy/dx of an equation not in the form y =... ex. y^2 = x^3 + 2x + 5, (y'=(3x^2 +2)/2y) for example, but how would you take the...
  12. J

    How to take the Partial Derivatives of a Function that is Defined Implicitly?

    How does one take the partial derivatives of a function that is defined implicitly? For example, the function, x^2 / 4 + y^2 + z^2 = 3.
  13. S

    Upper and lower derivatives

    Homework Statement What are the upper derivative and lower derivative of the characteristic function of rationals? Homework Equations The Attempt at a Solution I think they are : upper derivative = 0 lower derivative = negative infinity
  14. K

    Understanding Derivatives: Function Relationships and Graph Interpretation

    Homework Statement What is the relationship with a function's rising, falling, high point or low point to it's derivative? The Attempt at a Solution I have plotted my graphs, I can see that they intersect at the high and low points. But what is the relationship Also on another...
  15. L

    Using Partial Derivatives To Prove Solution To Wave Equation

    I need to use partial derivatives to prove that u(x,t)=f(x+at)+g(x-at) is a solution to: u_{tt}=a^{2}u_{xx} I'm stuck on how I'm supposed to approach the problem. I'm lost as to what order I should do the derivations in. I tried making a tree diagram, and I came out like this. The arrow...
  16. B

    Acceleration and distance using derivatives.

    Now I understand the basic concept that if one derivative's velocity you get acceleration and if you integrate velocity you will get the distance. But what about in this case?Homework Statement Homework Equations The Attempt at a Solution
  17. H

    Derivatives using Logarithmic Differentiation

    Homework Statement Using logarithmic differentiation calculate the derivative of y=e^(x^x) The Attempt at a Solution y=e^x^x LNy=LNe^x^x LNy=x^xLNe ... Stuck! This seems to be the only way you can do it, but once I get to that part I'm not sure what else there is to do. I...
  18. T

    How do I find the derivative of the square root of (2x+1) using the chain rule?

    I need to find the derivative of the square root of (2x+1) (not sure how to do square root symbol here, sorry) I understand that the square root of (2x+1)= (2x+1)^(1/2), but I am getting a little confused on how to continue from there.
  19. V

    Application of partial derivatives

    sorry folks i don't even have an idea to this question`s solution so i hope u people may like to help me. i`m stuck to it since last week nd i hope its from partial derivative... please suggest me a book or a hint or the solution. Let a long circular cylinder of unit radius be placed in a large...
  20. A

    Do derivatives introduce loss of solutions?

    For example, if I want to show that there is no real # solution to x2 + 24x2 = -1 is it correct to show that d2/dx2( x4 + 24x2 ) = d2/dx2(-1) ---> 12x2+48 = 0 And since x^2 is >0 or =0, 12x2+48 ---> 0 + 48 \neq 0 Therefore, there is no real number solution to x2 + 24x2 = -1...
  21. P

    Derivatives of Trigonometric Function

    Homework Statement find the dy/dx of y = Sin4 x2 - Cos4 x2 Homework Equations derivatives and identities factoring dy/dx (Sinx) = Cosx dy/dx (Cosx) = -Sinx The Attempt at a Solution y = (Sin2 x2 - Cos2 x2) (Sin2 x2 + Cos2 x2) im stuck at this part i don't know how to...
  22. P

    Derivatives of Trigonometric Function

    Homework Statement y=(sin2x)(cos2) Homework Equations Product Rule for Derivatives identities: derivatives of Sinx = Cosx Cox = -Sinx The Attempt at a Solution i used the product and chain rule for derivatives then do the identities y = sin2x*cos2x dy/dx = (Cos2x)(2)...
  23. M

    Derivatives and shortest length

    Homework Statement A straight line is drawn from the point (0,a) to horizontal axis, and then back to (1,b). Prove that the total length is shortest when the angles \alpha and \beta are the same. 2. Homework Equations /graphs [PLAIN]http://dl.dropbox.com/u/23215/Graph.jpg The...
  24. R

    Manipulate partial derivatives to obtain desired physical expression

    Homework Statement Show that the expression A, T(dP/dT)|V - P is equal to expression B, T^2 * [d(P/T)/dT]|V Also, show that expression C, -[d(P/T)/d(1/T)]|V is also equal to expression B Homework Equations A: temperature * (dPresure/dTemperature at constant volume) -...
  25. J

    Is the Derivative the Same as the Slope of a Function?

    Homework Statement is the derivativethe same thing as the slope of the function for which we're finding the derivative? Homework Equations The Attempt at a Solution
  26. J

    Derivatives of a Constant in a Trigonometric Function

    Homework Statement Find y'' if y=1/3(1+cos^2(√x)) Homework Equations The Attempt at a Solution Now I believe I got the first derivative right since the teacher marked ir right, but my real question here is what do I do with the 1/3? Is it ok to throw away the constant when I...
  27. L

    Partial derivatives boundery point problems

    Homework Statement find the largest distance and shortest distance from the origin to the conic whose equation is 6x2 + 4xy +3y2 - 28=0 and hence determine the lengths of the semi axes of this conic Homework Equations Lagrange identity F= f + λφ = 0 distance = d2 =x2+ y2+...
  28. V

    Partial Derivatives Maximum and Minimum Values

    Homework Statement Find the absolute maximum and minimum values of f on the set D. f(x,y) = 1+4x-5y D is the closed triangular region with vertices (0,0) (2,0) (0,3) Homework Equations To find the absolute maximum and minimum values of a continuous function on a closed, bounded set : 1. Find...
  29. M

    Derivatives and Polynomial Functions

    Homework Statement Show that there is a polynomial function f of degree n such that: 1. f('x) = 0 for precisely n-1 numbers x 2. f'(x) = 0 for no x, if n is odd 3. f'(x) = 0 for exactly one x, if n is even 4. f'(x) = 0 for exactly k numbers, if n-k is odd Homework Equations The...
  30. F

    Coupled ODE with missing connecting derivatives

    Hi, I have a coupled system of ODE like: w1'' = A w2'' + B w1 + C w2 w2'' = D w1'' + E w1 + F w2 I need to solve it analytically but it seems it cannot be solved using eigenvalue method. My concern is first that if this system have sufficient equations and if so how it can be solved...
  31. O

    How to prove the derivatives of powers?

    Ok, I just had a lecture recently the attached picture is what it was about. I understand it all, however at the end it says we havnt proven it yet - just wondering how DO you prove it then? Thanks, Owen
  32. V

    I have a few questions about partial derivatives and potential functions.

    Homework Statement I have no homework problem to ask, but rather a general question. Ill give and example of a potential function V = 3x^2 + 2y^2 i know to find Fx i have to differentiate 3x^2 with respect to x and 2y^2 with respect to y. But i have seen cases where someone takes the...
  33. B

    Squeeze Theorem for derivatives

    Homework Statement Show, with appropriate examples, that the conditions g(x) < f(x) < h(x) and derivative(g(x0))=derivative(h(x0)) = m does not imply derivative(f(x0)) = m or even exists. And with some additional condition. Homework Equations derivative g(x) = lim(h tends to zero)...
  34. S

    Continuity of partial derivatives

    What exactly does it mean for a function to have continuous partial derivatives? How do we see this?
  35. J

    Derivatives, Sin and Cos, Rate of Change, Tangent Lines

    Hi, I am in calculus and am having major struggles. If someone could provide a walk through on how to answer these questions, that would be fantastic. Cheers! Let f(x)=−3x+6 if x<-3 = 15 if x > -3 Find the average rate of change of f(x) on the interval −5<x<5 . The average rate of...
  36. P

    Calculate Derivatives of f(x,y,z,t), g(x,y) & h(x,y,z)

    Homework Statement calculate the derivative of the following functions? f(x,y,z,t) = (x-1)(2-y)z + (t^3 - 1)xyz g(x,y) = 1/(1 + exp(-(ax + by + c)) h(x,y,z) = (x-1)^2 exp(x) + (y-2)^3 * z^3 The Attempt at a Solution the way i was thinking was may be split the problem into...
  37. Telemachus

    Second derivatives using implicit differentiation

    Hi there. Well, I wanted to know how to find the second derivatives of a function using implicit differentiation. Is it possible? I think it is. I think I must use the chain rule somehow, but I don't know how... I'm in multivariable calculus since the function I'm going to use could be seen as a...
  38. L

    Partial Derivatives of ln(x+y)/(xy)

    I need help with this one: Find fxy in: ln(x+y)/(xy) .. the ln applies to the whole problem.
  39. r-soy

    Questions in derivatives I want to check my answer

    Questions in derivatives I want to check my answer http://store1.up-00.com/Oct10/KpR43940.jpg
  40. r-soy

    I want check my answer in derivatives (part 2)

    I want check my answer in derivatives (part 2) http://store1.up-00.com/Oct10/7uV34128.jpg
  41. r-soy

    I want check my answer in derivatives (part 1)

    I want check my answer in derivatives (part 1) http://store1.up-00.com/Oct10/p0O34128.jpg
  42. M

    Why is the nth derivative of x to the n power equal to n factorial?

    Homework Statement proving the nth derivative of x to the n power is n factorial Homework Equations The Attempt at a Solution proving it for n=1 d^(1)x^1/dx = 1!=1 (a) d/dx x^1 =1 (b) a=b therefore at n=1 it is true supposing it is true for n=k then d^(k)x^k/dx = k...
  43. S

    I applying the difference/power rule (derivatives)

    Homework Statement The problem is : take the derivative of (x - a) Homework Equations Power Rule : f '(x) = r x^(r-1) Difference Rule : f '(x) = g '(x) - h '(x) The Attempt at a Solution This is such a simple problem but I don't understand how my solutions manual and Wolfram...
  44. B

    Derivatives of partial fractions

    I'm having issue with one problem. We're asked to break down the problem into partial fractions to solve for the integral. Well, I'm stuck on one. I'm being asked for the values of A, B, and C for the following problem. ∫((9x^2+13x-83)/((x-3)(x^2 + 16)))dx I can get it worked down...
  45. J

    Using Derivatives and Integrals to Find Velocity: Am I doing this right?

    Homework Statement The function given to me is F(x) = A + Bx. x is the displacement, F(x) is the force as a function of that displacement, and A and B are constants. From the function, I'm supposed to find the velocity of the function as a function of x. We also know that the items...
  46. C

    Derivatives of exponent x with a product

    Homework Statement f(x) = 10(sin(x))^x ----> find f '(1) The Attempt at a Solution I have tried several different approaches, but still get stuck with a wrong answer every time f(x) = 10(sin(x))^x let f(x) = y so y=10(sin(x))^x then ln y = ln10(sin(x))^x...
  47. B

    Directional Derivatives vs. Partial Derivatives

    I have a question about these two. I have a direction derivative at a in the direction of u defined as: f'(a;u) = lim [t -> 0] (1/t)[f(a + tu) - f(a)] And the partial derivative to be defined as the directional derivative in the direction of u = e_i. My text, Analysis on Manifolds by...
  48. L

    Tangent plane, directional derivatives

    Homework Statement find the equation on the tangent plane of yz=ln(x+z) at point (0, 0, 1 ) Homework Equations Tangent plane equation... The Attempt at a Solution I wasn't sure how to determine the partials on this equation. My attempt was to rearange as ln(x+z)-yz=0 so Fx =...
  49. 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...
  50. 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...
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