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. W

    Derivatives, continuity

    Homework Statement 2 problems. 1) Find an example of a function f such that : the line y=2 is a horizontal asymptote of the curve y=f(x) the curve intersects the line y=2 at the infinitive number of points 2) The position of an object moving along x-axis is given at time t by: s(t)= 4t-4 if...
  2. L

    What is the Impact of Derivatives on US Banks?

    The Office of the US Currency Comptroller has an interesting chart 4 titled '5 Banks Dominate in Derivatives, Insured U.S. Commercial Banks, Second Quarter 2011'. http://www.occ.gov/topics/capital-markets/financial-markets/trading/derivatives/dq211.pdf I'm speechless!
  3. O

    Triangle tangent to circle problem using derivatives

    Homework Statement A metal bar of length l in the figure below has one end attached at a point P to a circle ofradius a < l. Point Q at the other end can slide back and forth along the x–axis. (a) Find x as a function of θ (θ=angle POQ). (b) Assume the lengths are in centimeters and the...
  4. Dembadon

    Multi-Variable Calculus: Partial Derivatives Using Level Curves

    Homework Statement This is a bonus problem on our homework, and I'm having trouble figuring out how to setup what I need. Homework Equations Here are my best guesses: f_x=\frac{\partial f}{\partial x} f_y=\frac{\partial f}{\partial y} f_{xx}=\frac{\partial}{\partial...
  5. D

    Directional Derivatives: the maximum rate of change of f.

    The maximum rate of change of f at the given point and the direction in which it occurs. f(x,y)=(y^2)/x, (2,4) Answer: 4√2, <-1,1> _________________________ For this problem, since I couldn't find the upside down triangle, I am going to use delta to represent the gradient of a function...
  6. N

    Differentiating Trigonometric Functions: Find the Derivatives

    find the derivatives of differentiation of trigonometric functions 1. y=cos(3x^2+8x-2) 2. y=tan^3 2x 3. y=sin5x sin^5 x 4. y=Square root of 4sin^2x+9cos^2x help here please.. i can't understand trigonometric functions sorry admin or moderator, i just search the net on how...
  7. J

    Derivatives of the Natural Exponential and logarithmic Functions

    1. Hello there, I have a few questions on the derivative homework that I did but not sure if it's right or not. These are difficult and my teacher doesn't go over all of them so I don't know if it is right or not. Please help me so I can understand the concepts before the upcoming test. 1...
  8. S

    Can any one explain me about Integrals and derivatives in breif

    Homework Statement can anyone explain me about Integrals and derivatives in brief Homework Equations The Attempt at a Solution
  9. M

    Question about limits and derivatives.

    Hi to all of Physics forums community :), I'm not sure whether or not I am in the right section. If not, I apologize for my mistake. Also, I am new to Calculus (I am in Cegep --> Quebec schooling system...) and I am taking Calculus 1 single variable. My teacher just started our class by...
  10. T

    Finding derivatives for second order differential equation

    Homework Statement Hello, My first post here I have a numerical problem for Matlab but I get stuck with the basic math... For a circuit I have three equations: 1.Inductance: L=Lo/(1+I^2) 2.Voltage over the inductance: V=L*dI/dt 3.Current over a condensator: I=-C*dV/dt...
  11. P

    Partial derivatives of Gas Law

    In James Stewart's Calculus exercise 82 page 891 asks you to show that: \frac{\partial P}{\partial V}\frac{\partial V}{\partial T}\frac{\partial T}{\partial P} = -1 I can do this by noting that V = \frac{nRT}{P} so that: \frac{\partial V}{\partial T} = \frac{\partial}{\partial...
  12. S

    1st order PDE, quadratic in derivatives, two variables analytic solution?

    I have the PDE: (v_r)^2+(v_z)^2=p^2 where v=v(r,z), p=p(r,z). I have some boundary conditions, of sorts: p=c*r*exp(r/a)exp(z/b) for some constants a,b,c, at r=infinity and z=infinity p=0 at f=r, where (f_r)^2=p*r/v-v*v_r (f_z)^2=p*r/v+v*v_r Is it possible that one could obtain an...
  13. X

    Chain rule of partial derivatives

    Homework Statement Suppose f(x,y) = 2x^5 + 4xy + 2y^3 g1(u,v) = u^2 - v^2 g2(u,v) = uv h(u,v) = f(g1(u,v), g2(u,v)) Use chain rule to calculate: dh/du (1,-1) and dh/dv (1,-1) Homework Equations The Attempt at a Solution i let h (u,v) = 2(u^2 - v^2)^5 + 4(u^2-v^2)(uv) +...
  14. S

    How Do You Calculate f(x+Δx) for the Function x²+3x+2?

    it's just something thechnical let's say i have the function x2+3x+2 when they say f(x+\Deltax) is it (x+\Deltax)2+2(x+\Deltax)+2 or another way?
  15. B

    I am having trouble finding trig derivatives using chain rule

    Homework Statement cot^2(Cos\theta)Homework Equations chain rule f prime (x) = f prime(g(x) * g prime (x) The Attempt at a Solution I am not sure if I am just inputting the wrong numbers into webassign or I am just missing and important trig derivative and just completely off of the boat...
  16. R

    If Partial derivatives exist and are continuos then function is differentiable

    Homework Statement Hi I'm just looking for a link to the proof of this theorem: if the partial derivatives of function f exist and are continuous at a point then the function is differentiable there Or even the name would be helpful Its not a homework assignment per say, just something...
  17. H

    A fundamental solution and its derivatives

    Hello, if I have a fundamental solution, ,f, to a partial differential equation L(f)=0, where L is the differential operator, is that true that the derivatives of the fundamental solution, like D(f), will also be solution to the partial differential equation? Intuitively, is it because things...
  18. K

    Implicit Differentiation of Multivariable Functions

    Homework Statement Suppose that the equation F(x,y,z) = 0 implicitly defines each of the three variables x, y and z as functions of the other two: z = f(x,y), y = g(x,z), z = h(y,z). If F is differentiable and Fx, Fy and Fz are all nonzero, show that \frac{∂z}{∂x} \frac{∂x}{∂y} \frac{∂y}{∂z} =...
  19. T

    Exercise on Kinetics using derivatives.

    Homework Statement The acceleration of a body is defined as a=-K*u^2 , K is const. When t=o sec V=Vo. Find : a) V(t) b) X(t) c) V(x) . Homework Equations The Attempt at a Solution
  20. N

    Solve partial derivatives from a table

    Let a represent the area, p the perimeter, d the diagonal, b the breadth, and L the length of a rectangle. One can easily write down from analytical geometry all the various relationships between the above variables, and from these obtain directly a variety of partial differential quantities...
  21. QuarkCharmer

    Derivatives of exponentials (calc II)

    Homework Statement \frac{d}{dx}e^{ax^{3}} I'm simply trying to determine whether or not I am doing these correctly and applying the chain rule properly. Homework Equations Chain rule et al. The Attempt at a Solution \frac{d}{dx}e^{ax^{3}} e^{ax^{3}}\frac{d}{dx}ax^{3} e^{ax^{3}}a(3)x^{2}...
  22. V

    Solving Derivatives: A Puzzling Experience

    Homework Statement I was messing around online when I found this: \frac{dy}{2} = 2x. This was derived from the function y = x2. I had never really seen anything like this before. When I solved for "dy," I got 4x. However, for example, when x changes from 0 to 2, the y changes from 0 to 4...
  23. C

    Question about second-order partial derivatives

    Homework Statement If V=xf(u) and u=y/x, show that x^2.d2V/dx2 + 2xy.d2V/dxdy + y^2.d2V/dy2= 0 (This a partial differentiation problem so all the d's are curly d's) The Attempt at a Solution I have tried to work out d2V/dx2 and the other derivatives, then multiply them by x^2 or 2xy or...
  24. A

    A Singularity: Finite Function, Infinite Derivatives

    Hi, Do you know the name of this kind of singularity at A ? The function is finite but the left derivative is +\infty and the right derivative is -\infty. http://shareimage.org/viewer.php?file=mt79897bbpxxse1v8pzb.jpg Thanks
  25. P

    Applications of partial derivatives

    Dear Everybody! I'm searching for some real life applications of partial derivatives. I would be very thankful, if you sent me some example. Thanks from Hungary.
  26. AlexChandler

    Why Does Rewriting a Function Change the Partial Derivative Outcome?

    I have come to a bit of a misunderstanding with partial derivatives. I will try to illustrate my problem. Say we have a function f(x, y(x), y'(x)) where y'(x)=dy/dx. Now suppose that f does not explicitly depend on x. My physics book says at this point that ∂f/∂x=0, even though y(x) and y'(x)...
  27. T

    Understanding Partial Derivatives and the Wave Equation

    Homework Statement Let f = f(u,v) where u = x+y , v = x-y Find f_{xx} and f_{yy} in terms of f_u, f_v, f_{uu}, f_{vv}, f_{uv} Then express the wave equation \frac{\partial^2f}{\partial x^2} - \frac{\partial^2f}{\partial y^2} = 0 Homework Equations Chain rule, product rule...
  28. T

    Solving integration by parts using derivatives vs differentials?

    What is the difference? I was pretty bored last night so I got onto Yahoo Answers and answered a few calculus questions. It was a simple integration by parts question: \intxsin(x) dx I solved as: u = x du = dx dv = sin(x) dx v = -cos(x) uv - \intvdu -xcos(x) + \intcos(x)dx =...
  29. I

    Non-linear 2nd ODE involving squares of derivatives

    Homework Statement y''+(1/y)*(y')2=0 Homework Equations The Attempt at a Solution This is another problem I am having trouble with. I have done searches around the internet, but seen that all "non linear" ODE of second order involves a non linear form in a non differential term...
  30. D

    Year 11 Double variable derivatives

    URGENT !Year 11 Double variable derivatives I am having trouble with this question it is derivatives. Previously I have been able to complete these with no trouble but am a little confused with how start this one: y= a^2(3x+5)^3. I don't know whether to use the product rule and just leave...
  31. J

    Proving a set of derivatives to be a subset of real functions

    let C0 be the set of continuous functions f : R -> R. For n >= 1, let Cn denote theset of functions f : R -> R such that f is differentiable and such that f' is contained in C(n-1). (Therefore Cn is the set of functions whose derivatives f',f'',f''',...,f^(n) up to the nth order exist and are...
  32. T

    Lie derivatives of Quantum Fields

    Hello, this question will essentially concern quantum field theory in curved spacetime, and it has two parts to it. I have recently acquired DeWitt's treatment of the formalism, which immediately discusses the role of killing vectors in the theory. Specifically, given a killing vector field...
  33. E

    Is There a Discrepancy in Matrix Trace Derivative Rules?

    Hope this is the right section. I'm having trouble ironing out an apparent inconsistency in matrix trace derivative rules. Two particular rules for matrix trace derivatives are \frac{\partial}{\partial\mathbf{X}} Tr(\mathbf{X}^2\mathbf{A})=(\mathbf{X} \mathbf{A}+\mathbf{A} \mathbf{X})^T...
  34. N

    Derivatives of arctan((x+y)/(1-xy))

    Homework Statement Find all second partial derivatives of z=arctan((x+y)/(1-xy))Homework Equations d/dx of arctan(x) is 1/(1+x^2)The Attempt at a Solution Not sure how to proceed... I don't want the answer, just an idea as to how to move forward. My attempt at finding the first...
  35. J

    Derivatives of trigonometric functions - Question

    Homework Statement Find the Derivative of: (ln(cos4x)) / 12x^2 Homework Equations y' ln(x) = 1/x The Attempt at a Solution I have determined the correct answer, but I am still confused as to how I came to the solution. Starting with the numerator, the derivative of cos...
  36. E

    Related Rates (Derivatives)

    Homework Statement A man 6 feet tall walks away from a streetlight that is 18 feet tall. If the length of his shadow is changing at a rate of 3 feet per second when he is 25 feet away from the base of the light, how fast is he walking away from the light at this moment? Homework Equations...
  37. Y

    What is the definition of Lie derivatives?

    Let \varphi be a one-parameter group on a manifold M, and let f be a differentiable function on M, the derivative of f with respect to \varphi is the defined as the limit: \lim_{t\to 0} \frac{\varphi^*_t[f]-f}{t}(x)=\lim_{t\to 0}\frac{f\circ \varphi_x(t)-f\circ...
  38. W

    Understanding derivatives graphically.

    Hi. Doing a bit of self study. I would like to know how to understand the derivative. I understand the algebra and procedural stuff that you need to do to get the derivative of a function. Is there a way I can understand it graphically? Say I draw y=x^2 on a graph. Then I draw y=2x on...
  39. S

    Derivatives: Composites, normal lines, n-th derivatives and more.

    Homework Statement 1. The line perpendicular to the curve y = 2x^3 - x^2 + x - 3 at the point (1, -1) will intersect the x-axis at what point? 2. f(x) = |x^2 - 5| - x, for all x. Let g = f(f(f(x))), find g'(2). I tried just subbing in 2x - 1, the first derivative, to f(2x - 1) and...
  40. S

    Some Derivatives Questions

    Homework Statement Let f(x) = axe^((bx)^2). Find the value for a times b if it's known that there's a max value of 2 at x = 3. Second, There is one line which is tangent to the curve y = 1/x, at some point A and at the same time tangent to the curve x^2 at some point B. What is the...
  41. F

    ODE now made me think about derivatives and partial derivatives

    Homework Statement Let's say I have a function for a circle x^2 + y^2 = C where C is a constant. Then this is a cylinder with the z-axis. Now in my ODE book, we would normally define it as F(x,y) = C = x^2 + y^2 as a level surface. Now my question is about what the partial...
  42. S

    Prove: All Derivatives of f at 0 = 0 if Lim f(x)/x^n = 0 as x --> 0

    Homework Statement if f is infinitely continuously differentiable and f(0) = 0 then prove that all derivatives of f at 0 are 0 iff lim f(x)/x^n = 0 as x --> 0 Homework Equations The Attempt at a Solution I didnt know whether to use induction on this, I tried a base case so...
  43. R

    Integrating Partial Derivatives

    Homework Statement Find the general function f(x,y) that satisifes the following first-order partial differential equations \frac{df}{dx}=4x^3 - 4xy^2 + cos(x) \frac{df}{dy}=-4yx^2 + 4y^3 The Attempt at a Solution I integrated both to get: x^4 - 2x^2y^2 + sin(x) + c(y) and -2y^2x^2 + y^4...
  44. K

    Multivariable Calculus unit derivatives question

    Homework Statement The position on the ground in the xy plane that is hit by the sun given by (x,y)=(3t+tan(phi), -2t+tan(theta)), where t, phi, and theta, are controlled input variables. What is the velocity of the hit point if the input variables are at values (5, pi/4, pi/3) and changing...
  45. T

    Derivatives on boundary points

    Something has been bugging as of late: usually, derivatives (ordinary and partial) are defined for interior points. However, I often come across statements in which they seem to also be defined for boundary points. For example, Leibniz' rule of integration, as usually stated, assumes some...
  46. D

    Expressions for Derivatives

    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...
  47. Rasalhague

    Directional derivatives and non-unit vectors

    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...
  48. C

    Help with Derivatives in Mathcad

    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...
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