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. D

    How Do You Differentiate Complex Functions Involving Exponents and Operations?

    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...
  2. 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...
  3. 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...
  4. B

    Interpolation with knowledge of derivatives

    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?
  5. N

    What software can assist with heavy supersymmetric derivative calculations?

    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
  6. P

    How to use clairaut's theorem with 3rd order partial derivatives

    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...
  7. B

    Directional Derivatives and max rate of change

    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...
  8. Q

    Chain rule for 2nd derivatives

    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?
  9. I

    Solving Partial Derivatives with f(x-z)=x+y+z

    hi i have a problem for this if f(x-z)=x+y+z solve can i say u=x-z and write F(x,y,z)=x+y+z-f(u) and then or this isn't true ? thanks if u help me.
  10. P

    Finding the Directional Derivatives of f(x,y)=x^2 + sin(xy) at (1,0)

    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...
  11. B

    Spherical coordinates and partial derivatives

    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...
  12. V

    Derivatives of Legendre Series Expansions

    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...
  13. A

    Which Tangent Line to y=sin(x) Has the Highest Y-Intercept Between 0 and 2π?

    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...
  14. C

    Derivatives Adv Calc: Show f'(x) = f'(0)f(x)

    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..
  15. C

    How Does f'(x) Relate to f(x) in Functional Equations?

    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) =...
  16. C

    Derivatives of 2pi-periodic functions

    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...
  17. P

    Can the Chain Rule Prove Even and Odd Symmetry in Derivatives?

    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...
  18. C

    Second order partial derivatives and the chain rule

    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...
  19. S

    Exploring the Applications and Interpretations of Fractional Derivatives

    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...
  20. I

    Why is the lagrangian polynomial in fields and derivatives

    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...
  21. S

    Parametric equations and derivatives

    Just a quick question... if we have f(x,y,z) and x(t), y(t), z(t), without substituting in what x y and z are in f, how do we calculate df/dt?
  22. A

    Manipulating partial derivitaves/ general derivatives

    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...
  23. S

    Basic Complex Analysis: Uniform convergence of derivatives to 0

    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...
  24. R

    Use L'Hopital's rule to calculate derivatives.

    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...
  25. A

    Higher derivatives : d/dx notation and meaning

    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.
  26. I

    Partial Derivatives: Help & Thanks!

    please help about this if f(u,v)=f(y/x,z/x)=0 and z=g(x,y) and show thanks alot
  27. J

    Calculus - derivatives of xtan(x)

    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.
  28. T

    Acceleration from min/max derivatives

    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...
  29. P

    Second Order Partial Derivatives + Chain Rule

    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 +...
  30. icystrike

    Directional Derivatives definition

    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...
  31. H

    Bounding Analytic Functions by derivatives

    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?
  32. X

    Differential and derivatives [HELP]

    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(.)...
  33. M

    What kind of operations are allowed on derivatives?

    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...
  34. R

    How to Simplify the Derivative of \( (1+4x)^5(3+x-x^2)^8 \)?

    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)
  35. S

    Help Solving Derivatives using Tables and Equation on Tangent Line

    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...
  36. Femme_physics

    Are Derivatives Merely Approximations?

    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...
  37. D

    Graphing of second derivatives.

    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...
  38. D

    What Did I Miss in My Derivative Calculation Using First Principles?

    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...
  39. N

    Rate Of Change (derivatives) Word Problem

    Homework Statement A 1.8 m tall student is trying to escape from the minimum security prison in To no. 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...
  40. G

    Solution for Expressing Derivatives in Terms of u

    Homework Statement w(r,\theta)= u(rcos \theta ),rsin( \theta)) for some u(x,y) express \frac{ \partial w}{\partial r} and \frac{ \partial w}{ \partial \theta} in terms of \frac{ \partial u}{ \partial x} and \frac{ \partial u}{ \partial y} Homework Equations rewrite the...
  41. S

    Find f(x) given properties of the derivatives.

    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...
  42. K

    Derivatives - differentiabilty, continuity, unbounded

    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...
  43. M

    Partial derivatives and change of variables

    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...
  44. A

    LaTeX Help Me Understand Latex Derivatives: Struggling With Calculus After 2 Years

    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...
  45. P

    Finding Elasticity in Demand: Solving for p in the Demand Equation

    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...
  46. D

    Derivatives and Integrals of units

    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...
  47. FeDeX_LaTeX

    Imaginary/Complex and Negative Derivatives?

    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...
  48. Rasalhague

    Domain of f(x,g(g)), and partial derivatives

    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))...
  49. P

    What Does the Tangent Line at x=0 Reveal About y=sin(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...
  50. M

    Integrals and derivatives , velocity and displacement

    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...
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