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. Math Amateur

    I Directional & Partial Derivatives .... working from the definition

    I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an element of the proof of Kantorovitz's Proposition on pages 61-62 ... Kantorovitz's Proposition on pages 61-62 reads as follows: I am...
  2. Math Amateur

    MHB Directional and Partial Derivatives .... working from the definitions ....

    I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an element of the proof of Kantorovitz's Proposition on pages 61-62 ... Kantorovitz's Proposition on pages 61-62 reads as...
  3. M

    Why can Nature can be modeled with only the 1st and 2nd derivatives?

    It´s not a technical question, is about why the classic mechanics and even quantum mechanics equations are first or second order? ¿Exist any model with up order derivates?
  4. S

    I Euler Lagrange formula with higher derivatives

    I was trying to Extrapolate Eulers formula , after deriing the basic form I wanted to prove: ∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2 = 0 Here is my attempt but I get different answers: J(y) = ∫abF(x,yx,y,yxx)dx δ(ε) = J(y+εη(x)) y = yt+εη(x) ∂y/∂ε = η(x) ∂yx/∂ε = η⋅(x)...
  5. V

    Partial derivatives and thermodynamics

    Hi all. Suppose I have the ideal gas law $$P=\frac{RT}{v}$$If I'm asked about the partial derivative of P with respect to molar energy ##u##, I may think "derivative of P keeping other quantities (whatever those are) constant", so from the formula above I get $$\frac{\partial P}{\partial...
  6. E

    MHB Derivatives of trigonometric equation

    Could I please get help with the following question? f(x)=(2cos^2 x+3)^5/2 Any help would be very much appreciated:)
  7. S

    I Lie Derivatives vs Parallel Transport

    Hello! In my GR class we were introduced to the parallel transport as the way in which 2 tensors can be compared with each other at different points (and how one reaches the curvature tensor from here). I was wondering why can't one use Lie derivatives, instead of parallel transport. As far as I...
  8. Y

    MHB Need help finding derivatives and concavity.

    Hi, I am having some trouble with this problem. I have completed part a but I am stuck on part b and c. I used the quotient rule to try and find the first derivative, but I am unsure if I have done so correctly. This is my work for part b so far...
  9. maistral

    Thermodynamic second derivatives?

    This is for research purposes. I am aware that first derivatives in thermodynamics always occur (a no-brainer). Do second derivatives occur in thermodynamics commonly as well?
  10. M

    Second derivatives when pouring juice into a cup

    Its a question about volume increase (in units cm^3) and height increase (cm) when pouring juice into a cup. Its stated that the volume of the juice in the cup increases at a constant rate, so I know the volume derivatives are zero. But the shape of the cup is inconsistent and there is a lot...
  11. L

    I Higher order derivatives rules

    mod: moved from homework Does anyone know why and when this equation holds? I have searched online but cannot find the reason or the rules for the higher order derivatives.
  12. 4

    MHB Higher order derivatives calculation

    Does anyone know why this is true?
  13. Velo

    MHB Equation of the Tangent Line? (Derivatives)

    So, I can't wrap around my head of why the Equation of the Tangent Line is: y = f(a) + f'(a)(x - a) I get it that it's the equation of a line, and so it should be something like y = mx + b. I also understand why f(a) = b (since it's a point in that line) and why f'(a) = m (since it's the slope)...
  14. M

    Question about Vector Fields and Line Integrals

    Homework Statement (a) Consider the line integral I = The integral of Fdr along the curve C i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C? ii) What is the value of I if the vector field F is is a unit vector...
  15. S

    A How to find the partial derivatives of a composite function

    Hello, dear colleague. Now I'm dealing with issues of modeling processes of heat and mass transfer in frozen and thawed soils. I am solving this problems numerically using the finite volume method (do not confuse this method with the finite element method). I found your article: "Numerical...
  16. K

    Graphing Derivatives: How to Find Maxima, Minima, and Points of Inflection

    Homework Statement Only 15 Homework Equations First derivative=maxima/minima/vertical tangent/rising/falling Second derivative=points of inflection/concave upward-downward The Attempt at a Solution $$x=y^3+3y^2+3y+2~\rightarrow~1=3(y^2+2y+1)y'$$ $$y'=\frac{1}{3(y+1)^2}>0,~y\neq...
  17. tomdodd4598

    I Problem with Commutator of Gauge Covariant Derivatives?

    Hi there, I have just read that the gauge field term Fμν is proportional to the commutator of covariant derivatives [Dμ,Dν]. However, when I try to calculate this commatator, taking the symmetry group to be U(1), I get the following: \left[ { D }_{ \mu },{ D }_{ \nu } \right] =\left( {...
  18. mertcan

    A Christoffel symbols expansion for second derivatives

    Hi, I really wonder how these second derivatives can be written in terms of christofflel symbols. I have made so many search but could not find on internet What is the derivation of equations related to second derivatives in attachment?
  19. K

    Prove the Leibnitz rule of derivatives

    Homework Statement Homework Equations Newton's binomial's: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n## The Attempt at a Solution I use induction and i try to prove for n+1, whilst the formula for n is given: $$\frac{d^{n+1}(uv)}{dx^{n+1}}=\frac{d}{dx}\frac{d^{n}(uv)}{dx^n}=$$ The...
  20. A

    B Quick question about calculus (derivatives)

    I thought Differentiation is all about understanding it in a graph. Every time I solve a question on differentiation I visualise it as a graph so it's more logical. After all, that IS what the whole topic is about, right? Or am I just wrong? But when you look at these questions...
  21. Biscuit

    Calculate Instantaneous Velocity at t=2s

    Homework Statement Homework EquationsThe Attempt at a Solution I tried to find the slope of the tangent line, but this gave me 3.66 and the answer is 3.8 how do I find this?
  22. rhdinah

    Polar Partial Derivatives - Boas Ch 4 Sect 1 Prob 13

    Homework Statement If ## z=x^2+2y^2 ##, find the following partial derivative: \Big(\frac{∂z}{∂\theta}\Big)_x Homework Equations ## x=r cos(\theta), ~y=r sin(\theta),~r^2=x^2+y^2,~\theta=tan^{-1}\frac{y}{x} ## The Attempt at a Solution I've been using Boas for self-study and been working on...
  23. davidge

    I Solving Covariant Derivatives: Minkowskian Metric

    How does one solve a problem like this? Suppose we have $$(e_\theta + f(\theta)e_\varphi) (e_\theta + f(\theta)e_\varphi)$$ What is the result of the above operation? As I remember it from the theory of covariant derivatives, the above relation would look like this $$e_\theta[e_\theta] +...
  24. Vectronix

    I Stress tensor and partial derivatives of a force field

    If F = Fxi + Fyj +Fzk is a force field, do the following derivatives have physical significance and are they related to the components of the stress tensor? I notice they have the same dimensions as stress. ∂2Fx / ∂x2 ∂2Fx / ∂y2 ∂2Fx / ∂z2 ∂2Fx / ∂z ∂y ∂2Fx / ∂y ∂z ∂2Fx / ∂z ∂x ∂2Fx / ∂x...
  25. D

    I Why can't I use the partial derivatives method to solve this problem correctly?

    Hi. If I have a function f ( x , t ) = x - 6t with x ( t ) = t2 and I take the partial derivative of f with respect to x I get the answer 1 as t acts as a constant so its derivative is zero. But if I substitute t with x1/2 I get the answer 1 - 3x-1/2 which is obviously different and wrong , I...
  26. DavideGenoa

    I Differentiating a particular integral (retarded potential)

    Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...
  27. Oats

    I Must functions really have interval domains for derivatives?

    Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so...
  28. C

    A Question about derivatives of complex fields

    https://arxiv.org/pdf/1705.07188.pdf Equation 5 in this paper states that $$\frac{\partial F}{\partial p_i} = 2Re\left\lbrace\frac{\partial F}{\partial x}\frac{\partial x}{\partial p_i}\right\rbrace$$ Here, p_i stands for the i'th element of a vector of 'design parameters' \mathbf{p}. These...
  29. Jess Karakov

    Simplifying this derivative....

    Homework Statement Evaluate the derivative of the following function: f(w)= cos(sin^(-1)2w) Homework Equations Chain Rule The Attempt at a Solution I did just as the chain rule says where F'(w)= -[2sin(sin^(-1)2w)]/[sqrt(1-4w^(2)) but the book gave the answer as F'(w)=(-4w)/sqrt(1-4w^(2))...
  30. binbagsss

    General relativity, geodesic, KVF, chain rule covariant derivatives

    Homework Statement To show that ##K=V^uK_u## is conserved along an affinely parameterised geodesic with ##V^u## the tangent vector to some affinely parameterised geodesic and ##K_u## a killing vector field satisfying ##\nabla_a K_b+\nabla_b K_a=0## Homework Equations see above The Attempt at...
  31. kupid

    MHB What are partial derivatives ?

    Its about functions with two or more variables ? How do you keep this x and y constant , i don't understand .
  32. Blockade

    B Is dy/dx of x2+y2 = 50 the same as dy/dx of y = sqrt(50 - x2)?

    For implicit differentiation, is dy/dx of x2+y2 = 50 the same as y2 = 50 - x2 ? From what I can take it, it'd be a no since. For x2+y2 = 50, d/dx (x2+y2) = d/dx (50) --- will eventually be ---> dy/dx = -x/y Where, y2 = 50 - x2 y = sqrt(50 - x2) dy/dx = .5(-x2+50)-.5*(-2x)
  33. Saracen Rue

    B Domain of Derivatives: Is f'(x)<=f(x)?

    Just a quick question - Is it true that the domain of ##f'(x)## will always be less than or equal to the domain of the original function, for any function, ##f(x)##?
  34. H

    MHB Partial Derivatives of Functions

    I am having some trouble solving the problem shown below. Can anyone point me in the right direction? or provide the location of a worked example? The volume V of a cone of height h and base radius r is given by V=1/3 πr^2 h. The rate of change of its volume V due to stress expansions with...
  35. S

    Velocity, momentum and energy values for a Pendulum swing

    Homework Statement This is my 'carrying out a practical investigation' assignment for Maths. I've attached the coursework (what I've wrote up to now) and my main concern is whether I've got the right differential equation to find 3 new velocity values throughout the pendulum trajectory...
  36. Ken Gallock

    I What does it mean: "up to total derivatives"

    Hi. I don't understand the meaning of "up to total derivatives". It was used during a lecture on superfluid. It says as follows: --------------------------------------------------------------------- Lagrangian for complex scalar field ##\phi## is $$ \mathcal{L}=\frac12 (\partial_\mu \phi)^*...
  37. S

    Partial Derivatives of U w.r.t. T and ##\mu## at Fixed N

    Homework Statement Show that ##\frac{\partial U}{\partial T}|_{N} = \frac{\partial U}{\partial T}|_{\mu} + \frac{\partial U}{\partial \mu}|_{T} \frac{\partial \mu}{\partial T}|_{N} ## (Pathria, 3rd Edition, pg. 197) Homework Equations ##U=TS + \mu N - pV## The Attempt at a Solution I tried to...
  38. Const@ntine

    Comp Sci Function Derivatives & Sines (C++)

    Homework Statement Okay, I'm going to "cheat" a bit and add two programs here, but I don't want to clutter the board by making two threads. Anyways, here goes: (1) The value of the sine of an angle, measured in rads, can be found using the following formula: sin(x) = x - x3/3! + x5/5! - ...
  39. fresh_42

    Insights The Pantheon of Derivatives - Part V - Comments

    fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part V Continue reading the Original PF Insights Post.
  40. fresh_42

    Insights The Pantheon of Derivatives - Part IV - Comments

    fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part IV Continue reading the Original PF Insights Post.
  41. fresh_42

    Insights The Pantheon of Derivatives - Part III - Comments

    fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part III Continue reading the Original PF Insights Post.
  42. fresh_42

    Insights The Pantheon of Derivatives - Part II - Comments

    fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part II Continue reading the Original PF Insights Post.
  43. fresh_42

    Insights The Pantheon of Derivatives - Part I - Comments

    fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part I Continue reading the Original PF Insights Post.
  44. K

    I The fractional derivative operator

    I've been thinking about it since yesterday and have noticed this pattern: We have, the first order derivative of a function ##f(x)## is: $$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} ...(1)$$ The second order derivative of the same function is: $$f''(x)=\lim_{h\rightarrow...
  45. M

    I Understanding the Difference Between Partial and Full Derivatives

    Hi PF! Regarding derivatives, suppose we have some function ##f = y(t)x +x^2## where ##y## is an implicit function of ##t## and ##x## is independent of ##t##. Isn't the following true, regarding the difference between a partial and full derivative? $$ \frac{df}{dt} = \frac{\partial f}{\partial...
  46. O

    A What Are the Different Types of Derivatives in Calculus?

    Derivatives in first year calculus Gateaux Derivatives Frechet Derivatives Covariant Derivatives Lie Derivatives Exterior Derivatives Material Derivatives So, I learn about Gateaux and Frechet when studying calculus of variations I learn about Covariant, Lie and Exterior when studying calculus...
  47. K

    B Is the theory of fractional-ordered calculus flawed?

    Let's talk about the function ##f(x)=x^n##. It's derivative of ##k^{th}## order can be expressed by the formula: $$\frac{d^k}{dx^k}=\frac{n!}{(n-k)!}x^{n-k}$$ Similarly, the ##k^{th}## integral (integral operator applied ##k## times) can be expressed as: $$\frac{n!}{(n+k)!}x^{n+k}$$ According...
  48. K

    B Average angle made by a curve with the ##x-axis##

    The average angle made by a curve ##f(x)## between ##x=a## and ##x=b## is: $$\alpha=\frac{\int_a^b\tan^{-1}{(f'(x))}}{b-a}$$ I don't think there should be any questions on that. Since ##f'(x)## is the value of ##\tan{\theta}## at every point, so ##tan^{-1}{(f'(x))}##, should be the angle made by...
  49. K

    I Taylor series to evaluate fractional-ordered derivatives

    Can the Taylor series be used to evaluate fractional-ordered derivative of any function? I got this from Wikipedia: $$\frac{d^a}{dx^a}x^k=\frac{\Gamma({k+1})}{\Gamma({k-a+1})}x^{k-a}$$ From this, we can compute fractional-ordered derivatives of a function of the form ##cx^k##, where ##c## and...
  50. maxhersch

    I Entries in a direction cosine matrix as derivatives

    This is a somewhat vague question that stems from the entries in a directional cosine matrix and I believe the answer will either be much simpler or much more complicated than I expect. So consider the transformation of an arbitrary vector, v, in ℝ2 from one frame f = {x1 , x2} to a primed...
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