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. F

    Prove limit x approaches 0 of a rational function = ratio of derivatives

    1. The problem statement, all variables and given/known dat If f and g are differentiable functions with f(O) = g(0) = 0 and g'(O) not equal 0, show that lim f(x) = f'(0) x->0 g(x) g'(0) The Attempt at a Solution I know that lim as x→a f(a) = f(a) if function is continuous. since its...
  2. P

    Partial Derivatives multivariable

    I am quite new to the topic of multivariable calculus. I came across the concept of "gradient" (∇), and although the treatment was somewhat slapdash, I think I got the hang of it. Consider the following case: ##z = f(x,y,t)## ##∇z = \frac{∂z}{∂t} + \frac{∂z}{∂y} + \frac{∂z}{∂x}## If we're...
  3. D

    Difference between these 4-vector derivatives?

    Hey everyone, So I've come across something in my notes where it says that these two Lagrangian densities are equal: \mathcal{L}_{1}=(\partial_{mu}\phi)^{\dagger}(\partial^{\mu}\phi)-m^{2}\phi^{\dagger}\phi \mathcal{L}_{2}=-\phi^{\dagger}\Box\phi - m^{2}\phi^{\dagger}\phi where \Box =...
  4. admbmb

    Conceptual trouble with derivatives with respect to Arc Length

    Hi, So I'm working through a bunch of problems involving gradient vectors and derivatives to try to better understand it all, and one specific thing is giving me trouble. I have a general function that defines a change in Temperature with respect to position (x,y). So for example, dT/dt would...
  5. BondKing

    Directional derivatives and the gradient vector

    If the unit vector u makes an angle theta with the positive x-axis then we can write u = <cos theta, sin theta> Duf(x, y) = fx(x,y) cos theta + fy(x,y) sin theta What if I am dealing with a function with three variables (x, y, z)? How can I find the directional derivative if I have been given...
  6. I

    Partial Derivatives of x^2-y^2+2mn+15=0

    x^2 - y^2 +2mn +15 =0 x + 2xy - m^2 + n^2 -10 =0 The Question is: Show that del m/ del x = [m(1+2y) -2 x n ] / 2 (m^2 +n^2) del m / del y = [x m+ n y] / (m^2 +n^2) note that del= partial derivativesMy effort on solving this question is Fx1=2x Fm1=2n Fx2 =2y Fm2 =-2m del m /del x =...
  7. K

    Derivatives and the relation to limits

    I'm in calc 1 and want to make sure I'm understanding the reason that we find derivatives. From what I understand, a derivative is simply an equation for the rate of change at any given point on the original function. Is that correct? And the tangent line at point (x,y) is obtained by using...
  8. J

    On derivatives of higher order

    let's assume that f(t) is a real function on [a,b], n is a positive integer (n-1)th derivative of f is continuous on [a,b], (n)th derivative exists for all t in (a,b) 1. Then can we say that (n-2)th derivative of f is continuous on [a,b]? 2. (n-2)th derivative of f is defined on a and b?
  9. MexChemE

    Thermodynamics, manipulating partial derivatives

    Hello PF! It's been a while since I last posted here. I have come across a problem in my textbook, which asks me to find expressions for V as a function of T and P, starting from the coefficients of thermal expansion and compressibility. \alpha = \frac{1}{V} \left(\frac{\partial V}{\partial T}...
  10. S

    How can I solve for average velocities using derivatives and integrals?

    Stressed first year university student here, fresh out of high school. I took physics in both grade 11 and 12, and thought I had a pretty good grasp on it; that is until this week. Introduction to derivatives and integrals to get from x(t) to v(t) to a(t) and vice-versa. I have a pretty good...
  11. A

    Differentiation of Quotients and Higher Derivatives

    1) The line 2x+9=3 meets curve xy+y+2=0 at the points P and Q. Calculate the gradient of the curve at P and Q 2)Given that y=(x^2)/(x-2), find a) (d^2)y/dx^2 in its simplest form b)ther range of value for which dy/dx and (d^2)y/dx^2 are positive. I can't figure out either of the sums...
  12. A

    MHB How to find the derivative of the absolute value of a vector?

    Why is \frac{d}{dt}\left | r(t) \right | =\frac{1}{\left | r(t) \right |} r(t)\cdot r'(t) ? A hint is given to me, saying \left | r(t)^2 \right | = r(t)\cdot r(t) . I think it's something to do with differentiating both side of the equation given as 'hint', but I have no idea how to execute...
  13. I

    Exploring Derivatives of |x| and \sgn(x)

    The wikipedia article on \sgn (x) (http://en.wikipedia.org/wiki/Sign_function) states that, \frac{d}{dx}\vert x\vert = \sgn(x) and \frac{d}{dx}\sgn(x) = 2\delta(x). I'm wondering why the following is not true: \begin{align*} \vert x\vert &= x\sgn(x)\\ \Longrightarrow \frac{d}{dx}\vert x...
  14. X

    MATLAB Matlab vs Mathematica for computing derivatives

    I'm trying to do a calculation which involves taking the derivative of a term that is millions of terms long. I'm using a supercomputer right now with Mathematica to do it but it's taking to long. Does Matlab have a faster algorithm for computing derivatives analytically?
  15. ShayanJ

    Calculus of variations and integrands containing second derivatives

    You know that the problem of calculus of variations is finding a y(x) for which \int_a^b L(x,y,y') dx is stationary. I want to know is it possible to solve this problem when L is a function of also y'' ? It happens e.g. in the variational method in quantum mechanics where we say that choosing...
  16. O

    Leibnitz's notation and derivatives of implicit functions

    First off: I think I understand the chain rule and how it derives from \lim_{h \to 0} \frac{ f(x+h)-f(x)}{h} and how to apply the chain rule when taking the derivative of an implicit function. The textbook I am reading Applied Calculus (by B. Rockett) uses the following example on...
  17. A

    MHB Derivatives and Inverse Trigonometry

    Hey guys, I have a couple of questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For 1a, using inverse trigonometric derivative identities should work, right? I got y' = 1/sinØ + 1/cosØ and multiplied by the common...
  18. Hepth

    Mathematica [Mathematica] Bug in Integrate with derivatives of a delta function

    Integrate[f[qs] DiracDelta'[qs (1 - 1/x)], {qs, -\[Infinity], \[Infinity]}, Assumptions -> 0 < x < 1] Integrate[f[qs] DiracDelta'[qs - qs/x], {qs, -\[Infinity], \[Infinity]}, Assumptions -> 0 < x < 1] This is on Mathematica 8 for windows. The results differ by a sign. They are effective...
  19. davidbenari

    Clairauts “equality of mixed partial derivatives” theorem

    I know how to prove this via limits and I'm okay with that. What I want to understand is the interpretation of the theorem and specifically a visualisation of why what the theorem states must be the case. My guess is that this theorem is saying that change is symmetrical. But I don't know...
  20. C

    Multivariable Calculus - Partial Derivatives Assignment

    1. Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the blood increases most rapidly. Suppose that in a certain case, the concentration of blood at a point P(x; y) on the surface of the...
  21. phoenixXL

    How Does Differentiation Relate to Polynomial Inequalities?

    Homework Statement Suppose p(x)\ =\ a_0\ +\ a_1x\ +\ a_2x^2\ +\ ...\ + a_nx^n. Now if |p(x)|\ <=\ |e^{x-1}\ -\ 1| for all x\ >=\ 0 then Prove |a_1\ +\ 2a_2\ +\ ...\ + na_n|\ <=\ 1. 2. Relevant Graph( |e^{x-1}\ -\ 1| ) The Attempt at a Solution From the graph we can conclude that p(x)...
  22. C

    Derivatives, and a little bit of linear approxim

    Homework Statement I have to find the deriv of ##f(x)=xArctan(x^{3})## I just need an explanation of how the arctan works... So I understand the rest, but just the deriv of arctan itself is confusing for me. So the derivative of ##arctan(x)## just in general is ##\frac{1}{1+x^{2}}## But if...
  23. K

    What if it's not a unit vector in directional derivatives

    i came to this topic and they said Duf(x) = ||gradient vector|| * ||U|| * cos 0 if ||U|| were not a unit vector it would give different rate of change of f in any direction what would happens if used ||U|| = 10 ?
  24. V

    Proving derivatives of a parametrized line are parallel

    Homework Statement Show that if σ(t) for (t in I) is a parametrization of a line, then σ''(t) is parallel to σ'(t). Homework Equations The Attempt at a Solution I thought that if σ(t) is a parametrization of a line then it could be expressed as σ(t) = vt + a, but then σ'(t) = v...
  25. I

    Derivatives of singularity functions

    Homework Statement Hello, I'm having trouble understanding this, seemingly simple, concept. Any help or input is appreciated. Evaluate the following derivatives: $$\frac{d}{dt} u(t-1)u(t+1)$$ $$\frac{d}{dt} r(t-6)u(t-2)$$ $$\frac{d}{dt} sin(4t)u(t-3)$$Homework Equations The Attempt at a...
  26. E

    Intuition for positive third derivatives

    We can get a lot of good intuition for how first and second derivatives work by interpreting a sign restriction. Let ##I\subseteq \mathbb R## be an interval and ##f:I\to\mathbb R##. 1) If ##f## is differentiable, then ##f## is monotone iff ##f'\geq 0## everywhere. 2) If ##f## is twice...
  27. D

    Solving word problems using derivatives

    Hi Guys, I am revising for an exam i have this week, the last module on my subject was calculus. I did not understand it entirely. I have posted a pic below of a typical problem i can expect to encounter, would anybody be able to point me in the right direction to study material that...
  28. C

    A quick simple derivatives question

    I've never really noticed whether this is true but if I know that: dt/ds = k/(1-s/r) for example, how does one find ds/dt? Is it the inverse, as you would expect, or is there some other method?
  29. T

    Derivative of x^2sin(4x) + xcos^(-2x)

    Need to find the Derivative using the chain rule y = x2sin4(x) + xcos-2(x) I am not sure where to start. answer in book is 2xsin4(x) + 4x2sin3(x)cos(x) + cos-2(x) +2xcos-3(x) xsin(x)
  30. C

    Derivatives involving the Absolute Value of Functions

    Problem: y = ln|sec(x) + tan(x)| Attempted Solution: See Attachment I was hoping someone could identify my error and possibly write it out for me. At first I thought my steps were correct and everything was in order. However, when I checked my answer on WolframAlpha it gave me a slightly...
  31. U

    How do derivatives provide answers for optimization problems ?

    I' at a a very very very basic level of calculus and usually have to watch a video or read something basic just to understand the basics. I'm fascinated by optimization equations, for example what is the largest area that can made with 500m of fencing. So at some point in solving this we end...
  32. S

    Question about calculus, on derivatives

    I am posting this question, in order to make something clear, since i am confused by derivation of the exponential function. I'll post the formula i used, correct me if you find something wrong thank you: {\frac{d}{Dx}}\ e^x => {\frac{e^{x + Dx} - e^x}{Dx}}\ => Here i factored out...
  33. T

    Finding derivatives of functions

    Need to find derivative f(s) = [(√s) -1]/[(√s) + 1] answer in book is f'(s) = 1/[√s(√s+1)^2]
  34. S

    Question about covariant derivatives

    Why is it that the covariant derivative of a covariant tensor does not seem to follow the product rule like contravariant tensors do when taking the covariant derivatives of those? Here is a visual of what I mean: This is the covariant derivative of a contravariant vector. As you can see, it...
  35. A

    The multiple derivatives of position

    Hi all, I came across the term placement, and presentment and the likes. I was wondering what they may mean in physics. I tried a search but the answers were not very clear. I was also wondering what action means in physics. Any answers will be appreciated. Thanks
  36. T

    Find Derivative: dp/dq if p = 1/(√q+1)

    find the indicated derivative dp/dq if p = 1/(√q+1) I apologize ahead of time if you can't read my work. my work [(1/(√(q+h+1))) - (1/(√(q+1))] \divh [((√(q+1)) - (√(q+h+1)))/((√(q+h+1))(√(q+1)))] \divh [(q+1-q-h-1)/(((√q+h+1)(√q+1))((√q+1)+(√q+h+1)))]\div h...
  37. M

    Correct Partial Derivatives: Double Check my Answers

    Hello. Can someone check if my partial derivatives are correct? I am not so confident with my answers.
  38. DreamWeaver

    MHB Derivatives and Integrals of the Hurwitz Zeta function

    Initially, the purpose of this tutorial will be to explore and evaluate various lower order derivatives of the Hurwitz Zeta function. In each case, the Hurwitz Zeta function will be differentiated with respect to its first parameter. A little later on - although this will take some time! - these...
  39. N

    Analytic Functions and Equality of their Complex Derivatives

    Homework Statement If f and g are both analytic in a domain D and if f'(z)=g'(z) for every z\in{D}, show that f and g differ by a constant in D Homework Equations Cauchy-Riemann Equations Possibly Mean Value Theorem The Attempt at a Solution I'm pretty sure I'm making a...
  40. C

    Unraveling the Mystery of Directional Derivatives

    I've come across a question which as really stumped me because I thought I knew how to do this but apparently not. The question is that we have a tangent vector to a level curve of a function of two variables f(x,y) at a point P is (2,1). Hence by my logic this means grad of f x unit vector...
  41. I

    Partial derivatives, change of variable

    Given V=xf(u) and u = \frac{y}{x} How do you show that: x^2 \frac{\partial^2V}{\partial x^2} + 2xy\frac{\partial^2V}{\partial x\partial y} + y^2 \frac{\partial^2V}{\partial y^2}= 0 My main problem is that I am not sure how to express V in terms of a total differential, because it is a...
  42. C

    Why Does the Scalar k Equal 4/9 in the Second Partial Derivatives Problem?

    Homework Statement Suppose z=ψ(2x-3y), Show that the second partial derivative of z with respect to x, is equal to the second partial derivative with respect to y multiplied by a scalar k. Homework Equations The Attempt at a Solution I thought this was too simple to be correct...
  43. E

    Help with this problem - Proof with first and second derivatives

    Help with this problem -- Proof with first and second derivatives Homework Statement I'm stuck on this problem and I'm not sure what I'm missing. The problem states: Assume that |f''(x)| \leq m for each x in the interval [0,a] , and assume that f takes on its largest value at an interior...
  44. S

    Partial derivatives chain rule

    Suppose we have a function V(x,y)=x^2 + axy + y^2 how do we write \frac{dV}{dt} For instance if V(x,y)=x^2 + y^2, then \frac{dV}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} So, is the solution \frac{dV}{dt} = 2x \frac{dx}{dt} + ay\frac{dx}{dt} + ax\frac{dy}{dt} + 2y \frac{dy}{dt}
  45. A

    Is ∂x/∂f = 1/(∂f/∂x) a Valid Equation in Implicit Differentiation?

    I have some complicated function f of the variables x,y: f(x,y) Now I can't really invert this expression for f for x and y, but I want the derivative of x and y wrt f. How can I do that? Am I allowed to say: ∂x/∂f = 1/(∂f/∂x) I have seen physicists "cheat" by using this relation, though I...
  46. N

    Derivatives of contravariant and covariant vectors

    Can someone explain why the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector and the derivative with respect to a covariant coordinate transforms as a contravariant 4-vector.
  47. J

    Partial derivatives; Tangent Planes

    Hi guys, Question is: Find the slopes of the curves of intersection of surface z = f(x,y) with the planes perpendicular to the x-axis and y-axis respectively at the given point. z = 2x2y ...at (1,1). fx(x,y) = 4xy ∴ Slope = 4 fy(x,y) = 2x2 ∴ Slope = 2 Is this wrong? Answer...
  48. R

    MHB Using Sampled Data to Estimate Derivatives, Integrals, and Interpolated Values

    I think you may be interested in the Wolfram Demonstration, "Using Sampled Data to Estimate Derivatives, Integrals, and Interpolated Values"
  49. J

    Vector Derivatives Explained: Definition and Examples

    What means: ? This guy, ##\vec{\nabla}_{\hat{\phi}} \hat{r}##, for example, means: \\ \hat{\phi}\cdot\vec{\nabla}\hat{r} = \begin{bmatrix} \phi _1 \\ \phi _2 \\ \end{bmatrix} \begin{bmatrix} \frac{\partial r_1}{\partial x} & \frac{\partial r_1}{\partial y} \\ \frac{\partial...
  50. J

    Exploring Field Derivatives: Rotational and Translational Tendencies

    Given a vector field f, I can compute the rotational tendency in the direction n (∇×f·n), the translational tendency in the direction t (∇f·t) and the divergence (∇·f) too. So, given a scalar field f, why I can compute only the directional derivative (translational tendency (∇f·t)) in the wanted...
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