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. chwala

    Find rate at which the liquid level is rising in the problem

    I was able to solve it using, ##\dfrac{dV}{dt} = \dfrac{dV}{dh}⋅\dfrac{dh}{dt}## With, ##r = \dfrac{h\sqrt{3}}{3}##, we shall have ##\dfrac{dV}{dh} = \dfrac{πh^2}{3}## Then, ##\dfrac{dh}{dt}= \dfrac{2×3 ×10^{-5}}{π×0.05^2}= 0.00764##m/s My question is can one use the ##\dfrac{dV}{dt} =...
  2. M

    Can anyone please verify/confirm these derivatives?

    Note that ## \frac{\partial F}{\partial x}=\frac{2x}{2\sqrt{x^2+y'^2}}=\frac{x}{\sqrt{x^2+y'^2}}, \frac{\partial F}{\partial y}=0, \frac{\partial F}{\partial y'}=\frac{2y'}{2\sqrt{x^2+y'^2}}=\frac{y'}{\sqrt{x^2+y'^2}} ##. Now we have ## \frac{dF}{dx}=\frac{\partial F}{\partial x}+\frac{\partial...
  3. M

    Are these the correct expressions for ## dF/dy' ##?

    a) ## dF/dy'=\frac{1}{4}(1+y'^2)^{\frac{-3}{4}}\cdot 2y' ## b) ## dF/dy'=cos (y') ## I just took the derivatives above and found out these expressions, but may anyone please check/verify to see if these expressions for ## dF/dy' ## are correct? Also, I do not understand part c). What does 'exp'...
  4. L

    Linear first-order differential equation with an initial condition

    Hi, unfortunately I have problems with the task d and e, the complete task is as follows: I tried to form the derivative of the equation ##f(x)##, but unfortunately I have problems with the second part, which is why I only got the following. $$\frac{d f(x)}{dx}=f_0 g(x) \ exp\biggl(...
  5. C

    h'(x) of h(x) = 3f(x) + 8g(x)

    For part(a), The solution is, However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule? For example if ##f(x) = \sin(x^2)## Many thanks!
  6. F

    I Partial derivatives of the function f(x,y)

    Hello, Given a function like ##z= 3x^2 +2y##, the partial derivative of z w.r.t. x is equal to: $$\frac {\partial z}{\partial x} = 6x$$ Let's consider the point ##(3,2)##. If we sat on top of the point ##(3,2)## and looked straight in the positive x-direction, the slope The slope would be...
  7. C

    Finding formula for nth derivatives of some functions

    For part(a), The solution is, However, I am having trouble understanding their finial formula. Does anybody please know what the floating ellipses mean? I have only seen ellipses that near the bottom like this ##...## I am also confused where they got the ##2 \cdot 1## from. When solving...
  8. C

    I Lacking intuition with partial derivatives

    Hello everyone, I seem to be majorly lacking in regards to intuition with partial derivatives. I was studying the Euler-Lagrange equations and realized that my normal intuition with derivatives seems to lead me to contradictory or non sensical interpretations when reading partial derivatives...
  9. MatinSAR

    Not understanding these manipulations involving Partial Derivatives

    Can someone please help me to find out what happened here ?
  10. H

    I Lars Olsen proof of Darboux's Intermediate Value Theorem for Derivatives

    Here is Lars Olsen's proof. I'm having difficulty in understanding why ##y## will lie between ##f_a (a)## and ##f_a(b)##. Initially, we assumed that ##f'(a) \lt y \lt f'(b)##, but ##f_a(b)## doesn't equal to ##f'(b)##.
  11. V

    Limit question to be done without using derivatives

    I am confused by this question. If I try applying the theorem under Relevant Equations then it seems to me that the theorem cannot be applied since the limit of the denominator is zero. This question needs to be done without using derivatives since it appears in the Limits chapter, which...
  12. H

    I Explanation of all "its linearly independent derivatives"

    I'm studying Differential Equations from Tenenbaum's, and currently going through non-homogeneous second order linear differential equations with constant coefficients. Method of Undetermined Coefficients is the concerned topic here. I will put forth my doubt through an example. Let's say we are...
  13. H

    A Polar Fourier transform of derivatives

    The 2D Fourier transform is given by: \hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy In terms of polar co-ordinates: \hat{f}(\rho,\phi)=\int_{0}^{\infty}\int_{-\pi}^{\pi}rf(r,\theta)e^{-ir\rho\cos(\theta-\phi)}drd\theta For Fourier transforms in cartesian co-ordinates, relating the...
  14. N

    Using derivatives in physics

    Hi, I'm trying to calculate my own physics problem but didn't get it something. When I'm trying to calculate the impulse of the object when it's hit by F=10N force in the smallest possible time, then should I write: dP/dt = Fnet => dP = Fnet*dt ? Another question: In general, if I calculate...
  15. C

    I Carroll GR: Tangent Space & Partial Derivatives

    He draws an n-manifold M, a coordinate chart φ : M → Rn, a curve γ : R → M, and a function f : M → R, and wants to specify ##\frac d {d\lambda}## in terms of ##\partial_\mu##. ##\lambda## is the parameter along ##\gamma##, and ##x^\mu## the co-ordinates in ##\text{R}^n##. His first equality is...
  16. Ahmed1029

    I Can I always think of derivatives this way?

    In physics, the differential is always treated as a little change or a tiny element of something, be it volume, area, etc. However, when I differentiate a function with respect to another, I always of it as a change divided by a change, not an element divided by an element: like when volume is...
  17. L

    A Maxwell equation. Commuting time and spatial derivatives.

    When we can use relation? \mbox{curl}(-\frac{\partial \vec{B}}{\partial t})=-\frac{\partial}{\partial t}\mbox{curl}\vec{B}
  18. AL107

    Derivatives and the chain rule

    I originally thought you’d have to use the chain rule to get h’, as in: f’(g(x))*g’(x). Plugging in 1 for x, I got an answer of 10. An online solution, however, said that you only had to get f(g(1)), which was f(-1), then look up f’(-1) in the table. Both approaches seem logical to me, but they...
  19. F

    I Time derivatives in variational calculus

    Taking the variation w.r.t f(x) of the integral over some x domain of F[f(x), f'(x), df(x)/dt], why doesn't df(x)/dt need to be taken a variational derivative and is treated as if it were constant?
  20. K

    I Derivations and Derivatives

    Hello! According to the attached proposition on ##C^\infty## manifold space of derivations ##D_m M## is isomorphic to Tangent space ##T_m M##. Cited here another proposition (1.4.5) states the following 1. For constant function ##D_m(f)=0## 2. If ##f\vert_U=g\vert_U## for some neighborhood...
  21. MrNotknowinganything

    Derivatives & L'Hôpital's Rule Explained

    Tried to use the information to put it in the definition of derivative and lopital but I couldn't get to anything
  22. docnet

    I Where do these time derivatives of Pauli matrices come from?

    Wolfgang Pauli's matrices are $$\sigma_x=\begin{bmatrix}0& 1\\1 & 0\end{bmatrix},\quad \sigma_y=\begin{bmatrix}0& -i\\i & 0\end{bmatrix},\quad \sigma_z=\begin{bmatrix}1& 0\\0 & -1\end{bmatrix}$$ He introduces these equations as "the equations of motion" of the spin in a magnetic field. $$...
  23. L

    A Laplace transform of derivatives

    I have a question regarding Laplace transforms of derivatives \mathcal{L}[f'(t)]=p\mathcal{L}[f(t)]−f(0^−) Can anyone explain me why ##0^-##?
  24. M

    I Time Derivatives: Hi Guys, Am I on the Right Track?

    Hi Guys I just want to make sure that I am on the right track, with regards to time derivatives. I have been out of university for many years and I have become a bit rusty. Please refer to the attached image and let me know if I am on the right track.
  25. L

    Prove eigenvalues of the derivatives of Legendre polynomials >= 0

    The problem has a hint about finding a relationship between ##\int_{-1}^1 (P^{(k+1)}(x))^2 f(x) dx## and ##\int_{-1}^1 (P^{(k)}(x))^2 g(x) dx## for suitable ##f, g##. It looks they're the weighting functions in the Sturm-Liouville theory and we may be able to make use of Parseval's identity...
  26. F

    A Dissipative and dispersive derivatives

    Why are even order derivatives dissipative and odd order derivatives dispersive?
  27. Haorong Wu

    I Derivatives for a density operator

    Hi. Suppose I have a state ##\left | \psi (0)\right >=\sum_m C_m \left | m\right >## evolving as $$\left | \psi (0+dz)\right>=\left | \psi (0)\right >+dz \sum_iD_i\left | i\right >=\sum_m C_m \left | m\right >+dz \sum_iD_i\left | i\right >=\sum_m( C_m+dz D_m)\left |m\right >.$$ Then the density...
  28. Delta2

    I From a proof on directional derivatives

    Given that the partial derivatives of a function ##f(x,y)## exist and are continuous, how can we prove that the following limit $$\lim_{h\to 0}\frac{f(x+hv_x,y+hv_y)-f(x,y+hv_y)}{h}=v_x\frac{\partial f}{\partial x}(x,y)$$ I can understand why the factor ##v_x## (which is viewed as a constant )...
  29. N

    B Derivatives vs. Fractions

    I don't understand the logic behind why derivatives can be treated like fractions in solving equations: ## \frac {du}{dx} = 2 ## simplified to ## du = 2dx ## I keep seeing this done with the explanation that "even though ## \frac {du}{dx} ## is not a fraction, we can treat it like one". Why...
  30. M

    I Summation notation and general relativity derivatives

    Does $$\partial^\beta(g_{\alpha\beta}A_\mu A^\mu)$$ mean the same as $$\frac {\partial (g_{\alpha\beta}A_\mu A^\mu)}{\partial A^\beta} ?$$ If not could someone explain the differences?
  31. brotherbobby

    The first and second derivatives at various points on a drawn graph

    Problem statement : The function ##y = f(x)## is given above. Question 1 : Locate the points at which the ##\text{first derivative}## of ##y## with respect to ##x## is ##\text{non-zero}##.##\\[5pt]## At points of extrema, like A, C and D, the derivative is zero. Hence the derivative is non...
  32. K

    B Second derivatives and inflection points

    Hi there. I'm having some trouble wrapping my head around some ideas of inflection points as they relate to the second derivative. I know that an inflection point occurs when f''(x)=0 in most cases. This makes sense to me because at this inflection point the slopes of the tangent change from...
  33. A

    Directional derivatives vs Partial derivatives

    Good day I just want to confirm if a function f(x,y) who has directional derivatives has automatically partial derivatives (even though the function itself is not necessarly differentiable)? Can we consider that partial derivatives are special cases of directional derivatives? Thank you in advance!
  34. Like Tony Stark

    Partial derivatives of enthelpy and Maxwell relations

    I've attached images showing my progress. I have used Maxwell relations and the definitions of ##\alpha##, ##\kappa## and ##c##, but I don't know how to continue. Can you help me?
  35. Mayan Fung

    Relating the entropy of an ideal gas with partial derivatives

    It looks very easy at first glance. However, the variable S is a variable in the given expression. I have no clue to relate the partial derivatives to entropy and the number of particles.
  36. Arman777

    A Lagrange with Higher Derivatives (Ostrogradsky instability)

    In class our teacher told us that, if a Lagrangian contain ##\ddot{q_i}## (i.e., ##L(q_i, \dot{q_i}, \ddot{q_i}, t)##) the energy will be unbounded from below and it can take any lower values (in other words be unstable). In this type of systems can we show that the energy is conserved ? Or in...
  37. E

    Help with a few annoying exterior derivatives

    I'm not very comfortable with these computations, so go easy on me! :smile: $$ \begin{align*} [d(d\omega)]_{\mu_1 \dots \mu_{p+2}} &= (p+2) \partial_{[\mu_1} (d\omega)_{\mu_2 \dots \mu_{p+2}]} \\ &= (p+1)(p+2) \partial_{[\mu_1} \partial_{\mu_2} \omega_{\mu_3 \dots \mu_{p+2}]} \end{align*}$$The...
  38. Leo Liu

    Minimization problem using partial derivatives

    a) ONLY The common way to solve this problem is minimizing the two-variable equation after using the substitution ##z^2=1/(xy)##. Yet I wondered if it is possible to optimize the distance equation with three varibles. So I wrote the following equations: Distance: $$f(x,y,z)=s^2=x^2+y^2+z^2$$...
  39. Kaguro

    Verifying Chain Rule for Partial Derivatives

    I have no answer or solution to this. So I'm trying to seek a confirmation of whether this is correct or not: ##df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial t}dt ## ##\frac{df}{dt} = \frac{\partial f}{\partial x} \dot x + \frac{\partial f}{\partial t} ## Therefore, ##...
  40. M

    MHB Calculate derivatives

    Hey! :giggle: I want to calculate the derivatives of the below functions. 1. $\displaystyle{f(x)=x^n\cdot a^x}$, $\in \mathbb{N}_0, x\in \mathbb{R},a>0$ 2. $\displaystyle{f(x)=\log \left [\sqrt{1+\cos^2(x)}\right ]}$,$x\in \mathbb{R}$ 3. $\displaystyle{f(x)=\sqrt{e^{\sin \sqrt{x}}}}$, $x>0$...
  41. Shackleford

    A Estimates on derivatives

    This is from Evans PDE page 29. Assume u is harmonic. 18. $$ |D^\alpha u(x_0)| \leq \frac{C_k}{r^{n+k}} \|u\|_{L^1(B(x_0,r))} $$ 19. $$ C_0 = \frac{1}{\alpha(n)}, \qquad C_k = \frac{(2^{n+1}nk)^k}{\alpha(n)}, \qquad (k=1,...). $$ 20. \fint is the average integral. $$ \begin{gather*}...
  42. A

    Solving Derivatives with the Chain Rule

    Hello! Now this is not really a physics problem of the usual kind but I'd say you could consider it one.Still I'd like to post my problem here because here I always get great help and advice.Now for this problem in particular,it is in the section of the book that deals with derivatives so I...
  43. sahilmm15

    B An intuitive understanding for derivatives

    Given, $$y=5x+3$$. We need to find how ##y## would change when we would make a very small change in ##x##. So, if we assume the change in ##x## to be ##dx## the corresponding change in ##y## would be ##dy##.So, $$y+dy = 5(x+dx)+3$$ From here we get $$\frac{dy}{dx}=5$$ From mathematical point of...
  44. E

    I Are Derivatives of the Metric Different in Flat Spacetime?

    The general metric is a function of the coordinates in the spacetime, i.e. ##g = g(x^0, x^1,\dots,x^{n-1})##. That means that in the most general case we can't simplify an expression like ##\partial g_{\mu \nu} / \partial x^{\sigma}##. But, what about the special case of the flat spacetime...
  45. D

    MHB Using Chain rule to find derivatives....

    y = (csc(x) + cot(x) )^-1 Find dy/dx
  46. Ugnius

    Oscillating body derivatives

    I know it is a quite simple task. p = mv and F=ma. All i need to do is find the normal and double derivatives of s(t). But here's the problem , i have the answers and they state that first derivative is v = -Awcoswt and second is -Aw^2sinwt. Everything is quite clear to me, but I am wondering...
  47. docnet

    Tangent vector fields and covariant derivatives of the 3-sphere

    This week, I've been assigned a problem about a 3-sphere. I am confused how to approach this problem and any comments would be greatly appreciated. (a) - would I be correct to assume the metric G is simply the dot product of two vector fields with dx^2 dy^2 du^2 and dv^2 next to their...
  48. docnet

    I Covariant derivatives, connections, metrics, and Christoffel symbols

    Is a connection the same thing as a covariant derivative in differential geometry? What Is the difference between a covariant derivative and a regular derivative? If you wanted to explain these concepts to a layperson, what would you tell them?
  49. anemone

    MHB First, second and third derivatives of a polynomial

    Let $p(x)$ be a polynomial with real coefficients. Prove that if $p(x)-p'(x)-p''(x)+p'''(x)\ge 0$ for every real $x$, then $p(x)\ge 0$ for every real $x$.
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