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. D

    How do I find derivatives involving natural logarithms and multiple variables?

    So I have an exam tomorrow, and the teacher provided a review. f(x) = ln(x + y) I remember that d/dx ln[f(x)] = f'(x)/f(x) so would that not equal 2/(x + y) ? The answer she gave is 1/(x + y - 1) ... where that neg. one came from I have no idea. Come to think of it, there were no...
  2. D

    Find the equation of tangent line for derivatives of functions

    Homework Statement Find an equation of the tangent line at the point indicated f(x) = 5x2-2x+9 , x = 1 Homework Equations (d/dx) bx = ln(b)bx General Power Rule which states: (d/dx) g(x)n = n(g(x))n-1 * g'(x) The Attempt at a Solution So looking at a previous problem...
  3. D

    Derivatives of General Exponential and Logarithmic Functions

    So in my math class we're studying derivatives involving ln(), tanh, coth, etc.. I need to say this first. I skipped precalc and trig and went straight to calculus, so whenever I see a trig problem, I can only go off of what I've learned "along the way." This problem has baffled me, please...
  4. Petrus

    MHB Calculating Derivatives and Finding Roots in Math

    Hello MHB, I got one question, I was looking at a Swedish math video for draw graph and for some reason he did take derivate and did equal to zero and did calculate the roots and then he did take limit of the derivate function to the roots and it's there I did not understand, what does that...
  5. F

    Finding Derivatives of f(x): A & B

    Homework Statement f(x) is given by the forumula y=\sqrt{3x^2 + 2x + 1} Find A: The first derivative B: The second derivative Homework Equations chain rule quotient and product rule?The Attempt at a Solution I think I have made a good logical attempt at part A but only have an inclin when it...
  6. DiracPool

    Ordinary vs. partial derivatives

    I'm thinking in particular about Lenny Susskind's lectures, but I've seen other lecturers do it too. They'll be writing equation after equation using the partial derivative symbol: \frac{\partial f}{\partial a} And then at some point they'll realize that some problem they're currently...
  7. S

    Multivariable Chain rule for higher order derivatives

    Hello, Given is the function f = f(a,b,t), where a=a(b) and b = b(t). Need to express first and second order derivatives. \frac{\partial f}{\partial a} and \frac{\partial f}{\partial b} should be just that, nothing more to it here, correct? But \frac{df}{dt} = \frac{\partial...
  8. P

    Differentiate the function (derivatives, difference of sums rule)

    Homework Statement Differentiate f(x) = x^{1/2} - x^{1/3} Homework Equations f(x) = f'(x)- g'(x) The Attempt at a Solution I am a little stuck about what to do after the first couple steps. Here is my attempt. f(x) = x^{1/2} - x^{1/3} f'(x) = (x^{1/2})' -(x^{1/3})' =...
  9. P

    Differentiate the function (derivatives, chain rule for powers)

    Homework Statement Differentiate f(x) = (x^2 - 3x)^2 Homework Equations f'(x) = nf'(x)f(x)^(n-1) The Attempt at a Solution f’(x) = 2(x2-3x)’(x2-3x)2-1 = 2(2x-3)(x2-3x)1 = 2(2x3 – 6x2 – 3x2 + 9x) = 2(2x3 – 9x2 + 9x) = 4x3 – 18x2+ 18x Is this correct?
  10. P

    Taylor Expansion where the derivatives are undefined?

    Homework Statement Expand x/(x-1) at a=1 The book already gives the expansion but it doesn't explain the process. The expansion it gives is: x/(x-1) = (1+x-1)/(x-1) = (x-1)^(-1) + 1 Homework Equations The Attempt at a Solution I've already solved for the Mclaurin expansion for the same...
  11. A

    Derivatives of inverse functions-how two formulas relate?

    Derivatives of inverse functions--how two formulas relate? Homework Statement I know two formulas for calculating the derivative of an inverse function, both of which I know how to derive, but I don't know how to relate them to one another. Homework Equations...
  12. topsquark

    MHB Coordinate transformation derivatives

    I've had to hit my books to help someone else. Ugh. Say we have the coordinate transformation \bf{x}' = \bf{x} + \epsilon \bf{q}, where \epsilon is constant. (And small if you like.) Then obviously d \bf{x}' = d \bf{x} + \epsilon d \bf{q}. How do we find \frac{d}{d \bf{x}'}? I'm missing...
  13. D

    Derivatives: Finding Slopes and Tangent Lines

    Homework Statement Let f(x) = 2x2 -3x -5. Show that the slope of the secant line through (2, f(2)) and (2+h, f(2+h)) is 2h + 5. Then use this formula to compute the slope of : (a) The secant line through (2, f(2)), and (3, f(3)) (b) The tangent line at x = 2 (by taking a limit)...
  14. R

    Temp profiles through partial derivatives

    Homework Statement The separation of layers is considered to occur at the thermocline, which is defined as the location of the steepest slope in the temperature gradient. Mathematically, this occurs at the inflection point – so the position of the thermocline can be found from the following...
  15. M

    Finding derivatives of inverse trig functions using logarithms

    For some polynomial functions it is useful to logarithmize both sides of the eq. First. How can this be applied for inverse trig functions? Is it even possible?
  16. B

    Questions about Derivatives and Continuity.

    1. Is this the only example of a function ##f(x) \in C^1([0,1])## with discontinuous derivative $$f(x) = \begin{cases} x^2 sin(\frac{1}{x}) & \textrm{ if }x ≠ 0 \\ 0 & \textrm{ if }x = 0 \\ \end{cases}$$ It seems this example is over-used. Do we have other examples besides this one in...
  17. O

    Question about partial derivatives with three unknown

    Homework Statement If z=f(x,y) with u= x^2 -y^2 and v=xy , find the expression for (∂x/∂u). the (∂x/∂u) will be used to calsulate ∂z/∂u. my question is how to find (∂x/∂u). I don't know what to keep constant. Maybe the question has some problem. The answer is (∂x/∂u)=(x/2)/(x^2+y^2)...
  18. P

    Optimization problem using derivatives

    Homework Statement We want to make a conical drinking cup out of paper. It should hold exactly 100 cubic inches of water. Find the dimensions of a cup of this type that minimizes the surface area. Homework Equations SA = pi*r^2 + pi*r*l where l is the slant height of the cone. V =...
  19. B

    Relationship between Derivatives and Integrals

    Hi, I've recently taken a Calculus 1 (Differential Calculus) course and I've been looking ahead to see what sort of material is covered in the Calculus 2 (Integral Calculus) course. I am wondering about the relationship between derivatives and integrals. From what I understand, an integral...
  20. Z

    Partial Derivatives of the Position and Velocity Vectors of a Particle

    Hello guys! Lately I've been studying some topics in Physics which require an extensive use vector calculus identities and, therefore, the manipulation of partial redivatives of vectors - in particular of the position and velocity vectors. However, I am not sure if my understanding of partial...
  21. M

    How Does T Satisfy the Heat Equation?

    Homework Statement Where T(x,t)=T_{0}+T_{1}e^{-\lambda x}\sin(\omega t-\lambda x) \omega = \frac{\Pi}{365} and \lambda is a positive constant. Show that T satisfies T_{t}=kT_{xx} and determine \lambda in terms of \omega and k. I'm not to sure what is meant by the latter part of "determine...
  22. M

    Solution of wave equation, 2nd partial derivatives of time/position

    f(z,t)=\frac{A}{b(z-vt)^{2}+1}... \frac{\partial^{2} f(z,t)v^{2} }{\partial z^2}=\frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}}=\frac{\partial^2 f}{\partial t^2} \frac{-2Abv^{2}}{[b(z-vt)^{2}+1]^{2}}+\frac{8Ab^{2}v^{2}(z-vt)^{2}}{[b(z-vt)+1]^{3}} this...
  23. NATURE.M

    Position/acceleration derivatives problem

    Homework Statement An object is traveling along a linear path according to the equation s(t) = 4t^3 - 3t^2 + 5 where t is measured in seconds and s(t) measured in meters. How far has the object traveled when its acceleration is zero? Homework Equations The Attempt at a Solution...
  24. V

    Higher order time derivatives of position

    Newton's laws says ## F=ma ##. Which, as far as I can see, states that all physical interactions concern the second time derivative of position. And because there is no other way for two bodies to interact in the physical world, the "worst" I can do to a system is change its acceleration, right...
  25. Q

    How Do You Solve This Partial Derivatives Problem?

    I have z=(e^y)φ*[y*e^(x^2/2y^2)].I have to prove that y*(dz/dx) -x*(dz/dy)=0.First of all what does φ mean there?
  26. B

    MHB Trig derivatives general question

    Hi, here I come again now with a problem, this time is more because the boo didn't really explain much deeper than the minimum. This time it's not homework related ... Yet. Lol. I just wanted to know how I would separate an equation like y=sin^2x cos3x To find the derivativ; I guess how...
  27. V

    Double derivatives of F as a function of F and harmonicity

    Homework Statement Homework Equations I can't think of much. u(x,y) is harmonic, so its double derivatives with respect to x and y add up to zero. I'm not 100% sure, but does being harmonic also imply that u satisfies the cauchy riemann equations? That might come in handy in the...
  28. L

    How to do functional derivatives

    Here's an example from my homework. I already turned it in, though. I basically just copied what I could from my notes, but I have no idea how this is done. Could someone explain this to me? I can't find anything intelligible (at least to me) of this stuff on any website. My notes contain parts...
  29. L

    Request for tutorial on funtional derivatives

    Would anyone here know where I could find some kind of tutorial that goes step by step of how to evaluate functional derivatives? I'm looking over my notes, trying to make sense of this functional derivatives stuff the teacher did in class, and it makes no sense at all. I'm not even in the realm...
  30. 5

    Partial Derivatives. Why and when to avoid the quotient rule?

    Hello PH, This is my first post. I came here while studying partial derivatives and after clicking here and there for over 4hrs for an answer. While practicing the derivatives rules i came across the hideous quotient rule. I've solved around 20 fractional problems trying to find a decision...
  31. M

    Partial derivatives in scientific analysis

    The idea of varying one thing but keeping others constant is central in scientific analysis. People want to know, other things constant, the effect of taking vitamins, smoking or drinking alcohol, just as examples. Is the idea of the partial derivative analogous to scientific empiricism's...
  32. R

    Is there any meaning to higher order derivatives?

    We know that the first derivative represents the slope of the tangent line to a curve at any particular point. We know that the second derivative represents the concavity of the curve. Or, the first derivative represents the rate of change of a function, and the second derivative represents the...
  33. P

    Derivatives with Quotient Law Help

    Derivatives with Quotient Law Help! I have a test tomorow, any help is much appreciated! :) Homework Statement Dervive using the quotient rule: [(2-x)^3] / [(x+1)^2] My attempt: = [(x+1)^2 (3(2-x)^2)]-[(2-x)^3 2(x+1)] When I try expanding I get the wrong answer. The...
  34. P

    Question involving higher derivatives

    Homework Statement Which of the following satisfy (f^k)(x) = 0 for all k >= 6? a) f(x) = 7x^4 + 4 + x^-1 b) f(x) = sqrt(x) c) f(x) = x^(9/5) d) f(x) = x^3 - 2 e) f(x) = 1 - x^6 f) f(x) = 2x^2 + 3x^5 Homework Equations None, but given as a problem in a chapter where the topic is higher...
  35. Petrus

    MHB Derivatives of implicit realtionships.

    Hello, Today did me and my friend talked of derivate and he asked about some help. Then he asked me is it possible to derivate $y^3+5x^2=5x-2y$ and i was clueless how i derivate that. Is this difficoult to derivate?is it possible to do it?
  36. V

    Proving equation involving limits without derivatives

    Homework Statement This is not really a homework or a coursework question. But I got a warning that I should submit my post in this section of the website.. I'm just saying this because I don't know if the answer to my question is at all achievable. And if it is how I should go about trying to...
  37. G

    Express derivatives most intuitively

    I have given a function g(t)=∇(f(x(t))) , f: IR³->IR and x: IR-> IR³ and want to express the first 3 derivatives with respect to time most simply. I thought that g'(t)=Hessian(f(x(t)))dx/dt but how do I get the further derivatives. is there any chance to express those in terms of taking the...
  38. J

    MHB Connecting the Keystone Equation and Derivatives

    Here's a question. This formula seems to be the keystone of calculus. That seems to be what the calculus books say, and it makes sense to me, as a rank beginner. This equation is what makes the seeming magic of defining the slope of a dimensionless point on a curved slope possible. And...
  39. E

    Use of derivatives to find coordinates

    Homework Statement Find the coordinates of the point(s) on the following curves where the second derivative is as stated. Homework Equations y= \frac{x^3}{12} and \frac{d^{2}y}{dx^{2}} = 1.5 The Attempt at a Solution I'm used to working with the first derivative. Would I need to...
  40. C

    Partial derivatives of a strong solution are also solutions?

    Homework Statement For the heat equation u_{t}=\alpha^{2}u_{xx} for x\in\mathbb{R} and t>0, show that if u(x,t) is a strong solution to the heat equation, then u_{t} and u_{x} are also solutions. Homework Equations u_{t}=\alpha^{2}u_{xx} The Attempt at a Solution I've considered...
  41. C

    Higher Order Partial Derivatives and Clairaut's Theorem

    Homework Statement general course question Homework Equations N/A The Attempt at a Solution fx is a first order partial derivative fxy is a second order partial derivative fxyz is a third order partial derivative I understand that Clairaut's Theorem applies to second order...
  42. R

    Derivatives Please, help me, I have an exam tomorrow

    f(x)=5x^3+6x^2-3x+lnx (lnx)`=1/x f(x)=2x^4+3x^2+ cosx (cosx)`=-sinxI know that if I only have x, like 3x, then x disappears (correct me if I'm wrong). So what happens with lnx if x disappears? Same thing with cosx. The lesson is extreme values of functions and i saw critical points mentioned...
  43. J

    Partial derivatives of 3D rotation vectors

    I am utilitizing rotation vectors (or SORA rotations if you care to call them that) as a means of splitting 3D rotations into three scalar orthogonal variables which are impervious to gimbal lock. (see SO(3)) These variables are exposed to a least-squares optimization algorithm which...
  44. Mandelbroth

    What Do Partial Derivatives Tell Us in Thermodynamics and Beyond?

    In a thermodynamics question, I was recently perplexed slightly by some partial derivative questions, both on notation and on physical meaning. I believe my questions are best posed as examples. Suppose we have an equation, (\frac{\partial x(t)}{\partial t}) = \frac{1}{y}, where y is a...
  45. T

    Plasma Fluid Mechanics - Convective Derivatives

    Homework Statement Use the continuity and momentum conservation equations for a single species to construct the following "convective derivative" equation for the fluid velocity: \frac{\partial\vec{v}}{\partial t}+\vec{v}\cdot\nabla\vec{v}=\vec{g}-\frac{1}{\rho}\nabla p...
  46. T

    Compute the following derivatives

    Homework Statement compute the following derivatives using the product rule and quotient rule as necessary, without using chain rule. Homework Equations d/dx ((sin(x))^2) The Attempt at a Solution =(sin(x))(sin(x)) =(cos(x))(sin(x))+(sin(x))(cos(x)) =2(sin(x))(cos(x))
  47. Hysteria X

    Why do some functions not have Anti derivatives?

    Why do some functions not have Anti derivatives?? as the title says why are some functions like ## √cotx##(root cotx) not integratable??
  48. P

    Are Lie Brackets and Their Derivatives Equivalent in Vector Field Calculations?

    Hi i have two questions: 1) When asked to prove \mathcal{L}_{u}\mathcal{L}_{v}W - \mathcal{L}_{v}\mathcal{L}_{u}W = \mathcal{L}_{[u,v]}. I achieved [u,v]w = \mathcal{L}_{[u,v]}. This was found by appliying a scalar field <b> to the LHS and simplifying and expanding using + and scalar...
  49. B

    Lie derivatives and Lipschitz contants

    Hello all, I have a problem with an inequality. Let Is the following proof valid? from which, taking the norm to both sides yields where L is the Lipschitz constant of f w.r.t. x. Thus, can I conclude that Is it correct? Thanks :)
  50. D

    Maxwell-Boltzmann speed distribution derivatives

    Hi everyone, Molecules move into a vacuum chamber from an oven at constant T. The molecules then pass through a slit. They reach two rotating discs before finally reaching a detector. Show that a molecule that passes through the first slit will...
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