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. 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 $x$.
  2. U

    Finding the horizontal tan() lines of this equation

    I've been able to find the tangent line with most equations, but I don't have any idea how to do it with a range of values instead of being given a singular value.
  3. M

    I Derivatives of Standard Functions

    I can't help but feeling these days that I don't actually understand where most of the maths I use comes from. Unfortunately, I can't remember whether this is due to the fact that I didn't take my studies seriously until the end of undergrad, or rather that these things were never actually...
  4. Haorong Wu

    I Lorentz transformation for derivatives

    Hello again. I am sorry I got another problem when learning QFT regarding the Lorentz transformation of derivatives. In David Tong's notes, he says I am not sure how to transform the partial derivatives. Explicitly, should ##\frac {\partial} {\partial x ^{\mu}}## transform to ##\frac...
  5. S

    A Fractional Calculus - Variable order derivatives and integrals

    Does anyone know any good research on this topic? I'm basically looking for information on what would be solving integral and differential equations in which the unknown you need to solve for is the level of a integral or derivative in the equation. For example F'1/2(u)+F'x(u)=F'1/3(u) where the...
  6. Ishika_96_sparkles

    I Directional Derivatives of a vector ----gradient of f(P)----

    Definition: Let f be a differentiable real-valued function on ##\mathbf{R}^3##, and let ##\mathbf{v}_P## be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the tangent vector $$\mathbf{v}_p[\mathit{f}]=\frac{d}{dt} \big( \mathit{f}(\mathbf{P}+ t...
  7. R

    Can we take the partial derivatives of φ and ψ here?

    I research about coordinate systems and I found the following informations about transformation. Now, if I replace arctan (x/y) (according to the picture above) to φ, I think I can solve. But if I can do this, then what will be replaced to ψ? I mean, I know just taking partial derative about...
  8. A

    I The Ratio of Total Derivatives

    If we have two functions C(y(t), r(t)) and I(y(t), r(t)) can we write $$\frac{\frac{dC}{dt}}{\frac{dI}{dt}}=\frac{dC}{dI}$$?
  9. T

    MHB Derivatives with Quadratic Functions.

    Slightly confused at what it wants me to do here?
  10. T

    Higher order derivatives using the chain rule

    Mentor note: Fixed the LaTeX in the following I have the following statement: \begin{cases} u=x \cos \theta - y\sin \theta \\ v=x\sin \theta + y\cos \theta \end{cases} I wan't to calculate: $$\dfrac{\partial^2}{\partial x^2}$$ My solution for ##\dfrac{\partial^2}{\partial...
  11. Adesh

    How to find the curl of a vector field which points in the theta direction?

    I have a vector field which is originallly written as $$ \mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$(##r## is the distance from origin, ##\phi## is azimuthal angle and ##\theta##...
  12. iya

    This problem is a combination of optics and derivatives

    According to Cartesian second law, we can get this formula. but wo don't know how to do next. because we just know the sinr and sini.
  13. ttpp1124

    Calculus and Vectors - Limits and Derivatives

    if someone can concur that'd be great; also, is there any way for me to check myself in the future?
  14. Physics lover

    Chemistry Stability of the derivatives of cyclopropyl methyl carbocation

    Here is an image of the structure I know that cyclopropyl methyl carbocation is exceptionally stable due to an effect so called dancing resonance which takes place because of lot of strain in cyclopropyl ring and vacant p orbital of Carbon attached with the ring. So I think this is a similar...
  15. T

    I Differentiability assumptions of Wirtinger derivatives

    In defining the Wirtinger (aka Cauchy-Riemann) linear operators, often used in signal analysis and in proofs of complex derivatives and the Cauchy-Riemann equations, one assumes differentiability in the real sense. This assumption is usually seen as obvious in the complex analysis setting...
  16. P

    Partial derivatives of thermodynamic state functions

    I'm in a first-year grad course on statistical mechanics and something about multivariable functions that has confused me since undergrad keeps popping up, mostly in the context of thermodynamics. Any insight would be much appreciated! This is a general question, but as an example imagine...
  17. Tony Hau

    I Problems on partial derivatives

    So in my lecture notes on Differential Equations, it states that a first order ODE is exact if A(x,y)dx + B(x,y)dy = 0 and ∂A/∂y = ∂B/∂x. Okay I accept this definition. Then, there is a sentence like this: Our goal is to find the function V(x,y) satisfying Adx + Bdy = dV = ∂V/∂x(dx) +...
  18. P

    A Calculating Functional Derivatives: -1≤xₒ≤1 vs -1<xₒ<1

    ##\frac {\delta I[f]} {\delta f(x_o)} = \int_a ^b \delta(x-x_o) \, dx## with a=-1 and b=+1 ## -1 \leq x_o \leq +1 ## vs ## -1 \lt x_o \lt +1 ##, 0 otherwise. Which is correct and does it matter when doing integration by parts?
  19. redtree

    I Chain rule for denominator in second order derivatives

    Given ## \frac{d^2x}{dy^2} ##, what is the chain rule for transforming to ##\frac{d^2 x}{dz^2} ##? (This is not a homework question)
  20. Z

    Expanding Brackets with Partial Derivatives

    Hi, I just need some (hopefully) quick calculus help. I have the following: ##(y\frac {\partial } {\partial z}(z\frac{\partial f} {\partial x}))## After it is expanded this is the solution: ##(yz\frac {\partial^2 f} {\partial z \partial x} + y\frac{\partial f} {\partial x} \frac{\partial z}...
  21. D

    I Derivatives of Energy: 2nd Time Rate of Change, Useless in Physics?

    I have seen the usage of a term which corresponds to changing energy per time and that would be power. Are there any such corresponding terms that can be described as 2nd time rate of change of energy, a sort of acceleration of energy. Is such a notion non existent in physics and completely...
  22. Math Amateur

    MHB Differentials/Total Derivatives in R^n .... Browder, Proposition 8.12 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ... I need some help in formulating a proof of Proposition 8.12 ... Proposition 8.12 reads...
  23. E

    Expanding a bracket of derivatives

    I am wondering why the two methods below give different answers. If I multiply z through the second bracket I get $$(\frac{d}{dx} +x)(-\frac{dz}{dx} + xz)$$which, on expansion, yields $$-\frac{d}{dx}\frac{dz}{dx} -x\frac{dz}{dx} + \frac{d(xz)}{dx} + x^{2}z = -\frac{d^{2}z}{dx^{2}} + x^{2}z +...
  24. Anonymous_

    I Is There a Way to Override Reserved Symbols in Webpage Code?

    You basically just take the second derivative of the given function and multiply it by the original then multiple everything by m. I just don’t understand how the second derivative would be negative.
  25. The black vegetable

    I Functional Derivatives: Overview & Tips

    Hi In the last sentence I mean you do include constant terms like I have done when taking the product above?
  26. karush

    MHB 9.1.317 AP calculus exam multiple choice derivatives of sin wave

    ok just posted an image due to macros in the overleaf doc this of course looks like a sin or cos wave and flips back and forth by taking derivatives looks like a period of 12 and an amplitude of 3 so... but to start I was not able to duplicate this on desmos altho I think by observation alone...
  27. S

    A thought I had about inertia, and derivatives of acceleration

    Well, I just had this thought earlier, and I want to share it. Here it is. So, we all know about inertia, right? The resistance to acceleration, or change in motion. Well, there is also a concept about derivatives of acceleration, mainly jerk and yank. If you don't know, jerk is said to be the...
  28. T

    A Derivatives of Lagrangian Terms: Why We Lower?

    In Lagrangians we often take derivatives (##\frac{\partial}{\partial (\partial_{\mu}\phi)}##) of terms like ##(\partial_{\nu}\phi \partial^{\nu}\phi)##. We lower the ##\partial^{\nu}## term with the metric and do the usual product rule. My question is why do we do this? Isn't...
  29. Math Amateur

    MHB Complex Derivatives .... Palka, Examples 1.1 and 1.2, Chapter III, Section 1.2 ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need help with some aspects of Examples 1.1 and 1.2, Section 1.2, Chapter III ... Examples 1.1 and 1.2, Section 1.2...
  30. I

    Time Derivatives of Expectation Value of X^2 in a Harmonic Oscillator

    I can show that ##\frac{d}{dt} \langle \psi (t) \vert X^2 \vert \psi (t) \rangle = \frac{1}{m} \langle \psi (t) \vert PX+XP \vert \psi (t) \rangle##. Taking another derivative with respect to time of this, I get ##\frac{d^2}{dt^2} \langle \psi (t) \vert X^2 \vert \psi (t) \rangle = \frac{i}{m...
  31. EchoRush

    I What can we learn from higher derivatives in calculus?

    As I have said before, I am in calculus class for the first time. I am doing really well in the class, however because of how my mind works, I’m always asking questions to know more, even when it’s too advanced for me. I just like to ponder and think about “what if” I know it’s probably not good...
  32. J

    I How do charts on differentiable manifolds have derivatives without a metric?

    I was reading about differentiable manifolds on wikipedia, and in the definition it never specifies that the differentiable manifold has a metric on it. I understand that you can set up limits of functions in topological spaces without a metric being defined, but my understanding of derivatives...
  33. EchoRush

    I Question about the quotient rule of derivatives

    Now, I understand how to use the quotient rule for derivatives and everything. I do not struggle with using it, my question is mostly about the formula itself...I very much enjoy WHY we do things in math, not just “here’s the formula, do it”...Here is the formula for the quotient rule of...
  34. KF33

    B How do I differentiate vectors with derivatives and properties?

    Homework Statement: The homework problem is included below, but I am looking at the derivatives of vectors. Homework Equations: I have the properties of derivatives below, but not sure they help me here...
  35. K

    Understanding Iterative Derivatives of Log Functions

    I have never used induction to justify the derivative to a function, so I don't know where to start. Does anyone have some tips?
  36. karush

    MHB 242 Derivatives of Logarithmic Functions of y=xlnx-x

    $\tiny{from\, steward\, v8\, 6.4.2}$ find y' $\quad y= x\ln{x}-x$ so $\quad y'=(x\ln{x})'-(x)'$ product rule $\quad (x\ln{x})'=x\cdot\dfrac{1}{x}+\ln{x}\cdot(1)=1+\ln{x}$ and $\quad (-x)'=-1$ finally $\quad \ln{x}+1-1=\ln{x}$...
  37. E

    B Difficulty with derivatives using the Lorentz transformations

    Two frames measure the position of a particle as a function of time: S in terms of x and t and S', moving at constant speed v, in terms of x' and t'. The acceleration as measured in frame S is $$ \frac{d^{2}x}{dt^{2}} $$ and that measured in frame S' is $$ \frac{d^{2}x'}{dt'^{2}} $$My question...
  38. balaustrada

    How to compute the variation of two covariant derivatives?

    I'm working with modfied gravity models and I need to consider the perturbation of field equations. I have problems with the term were I have two covariant derivatives, I'm not sure if I'm doing it right. I have: $$\delta(\nabla_\rho \nabla_\nu \left[F'(G)R_{\mu}^{\hphantom{\mu} \rho}\right])$$...
  39. berlinspeed

    A Two Covariant Derivatives (Chain Rule)?

    Summary: Failed find information on the internet, really appreciate any help. Can someone tell me what is ∇ϒ∇δ𝒆β? It seems to be equal to 𝒆μΓμβδ,ϒ+(𝒆νΓνμϒ)Γμβδ. Is this some sort of chain rule or is it by any means called anything?
  40. D

    I What is the Purpose of Calculating the Christoffel Symbols in Curved Spacetime?

    Calculating the christoffel symbols requires taking the derivatives of the metric tensor. What are you taking derivatives of exactly? Are you taking the derivatives of the inner product of the basis vectors with respect to coordinates? In curvilinear coordinates, for instance curved spacetime in...
  41. E

    Compute the energy for 2D wave function with discontinuous derivatives

    I have calculated the normalization constant, but I'm struggling with the discontinuities in the derivatives of the wave function. Due to the symmetry, it should suffice to consider the first two cases. The results should be (according to WolframAlpha): \left( \frac{\partial^{2}}{\partial...
  42. RicardoMP

    Derivatives on tensor components

    This was my attempt at a solution and was wondering where did I go wrong: -\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=-\frac{\partial}{\partial p_\mu}[\gamma^\nu p_\nu]^{-1}=\gamma^\nu\frac{\partial p_\nu}{\partial p_\mu}[\gamma^\sigma...
  43. KristinaMr

    How Do I Apply L'Hopital's Rule to Exponential Derivatives?

    I encountered this problem is Hopital section...how do I even apply it?
  44. Boltzman Oscillation

    How can I solve for these partial derivatives given a set of variables

    I am given the following: $$u = (x,t)$$ $$\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0$$ and $$E = x + ct$$ $$n = x - ct$$ I need to solve for $$\frac{\partial^2 u}{\partial x^2}$$ and $$\frac{\partial^2 u}{\partial t^2}$$ using the chain rule.How would I even...
  45. Q

    A Functional Derivatives in Q.F.T.

    I'm can't seem to figure out how to functionally differentiate a functional such as Z(J)= e^{\frac{i}{2} \int \mathrm{d}^4y \int \mathrm{d}^4x J(y) G_F (x-y) J(x)} with respect to J(x) . I know the answer is \frac{\delta Z(J)}{\delta J(x)}= -i \int \mathrm{d}^4y J(y) G(x-y) but I'm struggling...
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