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. J

    Understanding Partial Derivatives with Multiple Variables

    Se a function f(x(t, s), y(t, s)) have as derivative with respect to t: \frac{df}{dt}=\frac{df}{dx} \frac{dx}{dt}+\frac{df}{dy} \frac{dy}{dt} And, with respect to s: \frac{df}{ds}=\frac{df}{dx} \frac{dx}{ds}+\frac{df}{dy} \frac{dy}{ds} But, how will be the derivative with respect to...
  2. J

    Derivatives with multiple variable, help

    Homework Statement This is an Optimization Problem, find the maximum value. P(R)=(E^2*R)/(R+r)^2 Homework Equations P'(R)=? The Attempt at a Solution I have the solution to this problem, and I can solve it, I just don't understand some parts. I tend to think that the...
  3. J

    Integrating derivatives in matrix elements

    First a little warm up problem. Suppose g:\mathbb{R}^N\to\mathbb{C} is some fixed function, and we want to find f:\mathbb{R}^N\to\mathbb{C} such that g(x) = u\cdot\nabla_x f(x) holds, where u\in\mathbb{R}^N is some constant. The problem is not extremely difficult, and after some work...
  4. lonewolf219

    Derivatives and equilibrium position of a spring

    I determined the equilibrium point of a spring by setting the potential energy function U(r) equal to zero and solving for r. But I just looked at the guided solution, and they took the derivative of U(r) first, then solved for r. Is my approach correct? Can we solve for the equilibrium...
  5. 1

    Functions with increasing derivatives

    Consider a function f(x), such that for all points x0 in the domain, the nth derivative of f evaluated that x0 is less than the n+1th derivative of f evaluated at x0. A quick example is f(x) = e^(ax) where a > 1, what others are there (not including just changing e to something else)? Is...
  6. H

    Functional derivatives worksheet

    Greetings, I want to become more fluent using functional derivatives. Does anyone have a link to sets of problems involving functional derivatives or anything like that (e.g., a worksheet from a class where they were used or something)? The lengthier the better, and ideally the solutions...
  7. J

    Two Functions intersect and have equal derivatives at x=a and x=b

    Sorry, wasn't sure how to describe the problem in the title. Homework Statement Okay, this problem has really been bugging me for a while. It's a question that I thought of when I was daydreaming in class, but now I can't stop thinking about it. I've been going crazy for days Here is the...
  8. P

    How do you find the second derivative of a function using the chain rule?

    Homework Statement Find the second derivative of $$9x^2+y^2=9$$ Homework Equations Chain rule The Attempt at a Solution I find the first derivative first. $$18x+2y\frac{dy}{dx}=0$$ $$\frac{dy}{dx}=-9\frac{x}{y}$$ I then find the second derivative...
  9. P

    Derivatives of Trigonometric Functions with the Chain Rule

    Homework Statement Find the derivative of $$y=cos(\frac{1-e^{2x}}{1+e^{2x}})$$ Homework Equations Chain rule The Attempt at a Solution $$y=cosu$$ $$\frac{dy}{du}=-sinu$$ $$u=\frac{1-e^{2x}}{1+e^{2x}}$$ $$ \frac{du}{dx}=(1-e^{2x})(-(1+e^{2x})^{-2})+(1+e^{2x})^{-1}(-2e^{2x})$$...
  10. P

    Derivative of y=cos(a3+x3)What is the derivative of y=cos(a3+x3)?

    Homework Statement Find the derivative of y=cos(a3+x3) Homework Equations Chain rule The Attempt at a Solution y=cosu \frac{dy}{du} = -sinu u=a3+x3 \frac{du}{dx} = 3a2+3x2 \frac{dy}{dx} = -3sin(a3+x3)(a2+x2). The answer is supposed to be -3x2sin(a3+x3). What did...
  11. P

    Finding the Derivative of y=xe-kx using the Chain Rule

    Homework Statement Find the derivative of y=xe-kx Homework Equations Chain rule The Attempt at a Solution y = xeu \frac{dy}{du} = xeu+eu u = -kx \frac{du}{dx} = -k \frac{dy}{dx} = (xe-kx+e-kx)(-k) = e-kx(x+1)(-k) = e-kx(-kx-k) The answer is e-kx(-kx+1)...
  12. P

    Derivatives, chain rule

    1. Find the derivative of y=e\sqrt{x} Homework Equations Chain rule The Attempt at a Solution y=eu \frac{dy}{du}= ueu-1 u=\sqrt{x} \frac{du}{dx}= \frac{1}{2}x-1/2 \frac{dy}{dx}= \sqrt{x}e\sqrt{x}-1 × \frac{1}{2}x-1/2 = \sqrt{x} \frac{e^\sqrt{x}}{e} ×...
  13. P

    Derivatives and rate of change

    1. A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to θ when...
  14. Q

    What Are the Properties of First and Second Derivatives?

    Homework Statement http://i3.minus.com/j7uTkNLAl2aBy.png Homework Equations Extrema occur at critical points; critical points are either where the first derivative fails to exist or equals 0. Horizontal tangent lines occur where the first derivative is 0. Points of inflections occur...
  15. C

    Clarification on Lie Derivatives

    The standard definition of the lie derivative of X along Y is just $$(*) \mathcal{L}_YX = \lim_{t\to 0} \frac{X_{\phi(t)} - \phi_{t*}X}{t}$$ where ##\phi_t## is the flow generated by Y. I.e. the limit of the difference between a pushforward of X along Y and X evaluated at a point...
  16. snoopies622

    Finding the operators for time derivatives of observables

    Looking through this matrix approach to the quantum harmonic oscillator, http://blogs.physics.unsw.edu.au/jcb/wp-content/uploads/2011/08/Oscillator.pdf especially the equations m \hat{ \ddot { x } } = \hat { \dot {p} } = \frac {i}{\hbar} [ \hat {H} , \hat {p} ] I'm getting the impression...
  17. K

    Derivatives of a higher order - Satisfying the equation

    Homework Statement Show that y= xex satisfi es A(d2y)/dx2 + B(dy/dx) + Cy = 0 for suitably chosen values of the constants A, B, and C. Homework Equations Y=xex The Attempt at a Solution Please see the attachment. I get to a point where I need to find the value of A, B...
  18. L

    Coordinate systems. Derivatives

    ## \vec{r}=\rho \cos \varphi \vec{i}+\rho \sin \varphi \vec{j}+z\vec{k} ## we get \vec{e}_{\rho}=\frac{\frac{\partial \vec{r}}{\partial \rho}}{|\frac{\partial \vec{r}}{\partial \rho}|} \vec{e}_{\varphi}=\frac{\frac{\partial \vec{r}}{\partial \varphi}}{|\frac{\partial \vec{r}}{\partial...
  19. T

    Directional Derivatives and Commutation

    Homework Statement I need to prove that directional derivatives do not commute. Homework Equations Thus, I need to show that: (\vec{A} \cdot \nabla)(\vec{B} \cdot \nabla f) - (\vec{B} \cdot \nabla)(\vec{A} \cdot \nabla f) = (\vec{A} \cdot \nabla \vec{B} - \vec{B} \cdot \nabla...
  20. J

    Propogation of Coefficient Standard Deviations using Derivatives

    Hi all, This is my first post here. As a bit of background I have a keen interest in mathematics, but I'm not necessarily strong in it... Today I was working with a dataset that models difference in elevation values (dH) over time (x). I have approximately 20 samples, of which each dH value...
  21. M

    Questions about vectors and derivatives.

    Homework Statement I failed my math exam last year (first year of college) because I basically had no math classes during my last two years of high school. So I would like to learn the specific materials to be able to solve these problems. So I guess this isn't a traditional question, but I do...
  22. P

    Angle Calculation for Particle Velocity and Z-Axis at Given Time t1

    Homework Statement A point particle has a position vector r⃗ (t) as a function of time t, given by r(t)=(1−t2)x^−2t(t+5)y^+8(t+2)z^. where distances are in meters, and time t is in seconds. Now, let t=t1= 23 s. What is the (smaller) angle between the velocity vector at time t1 and the z^...
  23. M

    Small question about derivatives and gradients.

    Is the grad(\frac{\partial f}{\partial t}) the same as \frac{\partial}{\partial t}(gradf)? Thank you.
  24. Kelsi_Jade

    Partial Derivatives for an Ideal Gas

    The question is: a) Find explicit expressions for an ideal gas for the partial derivatives: (∂P/T)T, (∂V/∂T)P and (∂T/∂P)V b) use the results from a) to evaluate the product (∂P/V)T*(∂V/∂T)P*(∂T/∂P)V c) Express the definitions of V(T,P) KT(T,P)an BT(T,V) in terms of the indicated independent...
  25. C

    MHB Understanding Derivatives: Step-by-Step Guide to Finding f'(x) of f(x)=2

    I don't understand this concept very well. So suppose I want to find the derivative of the function f(x)=2. is the answer just zero? what if I want to find f'(x)=2x, or f'(x)=2x3? please explain how to get the answer step by step, is there some equation I use or something? I just don't get it!
  26. Saitama

    Comparing Derivatives: Solving for g'(x) and f'(x) with Two Given Points

    Homework Statement Let ##f(x)## and ##g(x)## be two differentiable function in R and f(2)=8, g(2)=0, f(4)=10 and g(4)=8 then A)##g'(x)>4f'(x) \forall \, x \, \in (2,4)## B)##3g'(x)=4f'(x) \, \text{for at least one} \, x \, \in (2,4)## C)##g(x)>f(x) \forall \, x \, \in (2,4)##...
  27. Y

    Nth Derivatives and Taylor Polynomials

    Homework Statement Show that if f^{(n)}(x_0) and g^{(n)}(x_0) exist and \lim_{x \rightarrow x_0} \frac{f(x)-g(x)}{(x-x_0)^n} = 0 then f^{(r)}(x_0) = g^{(r)}(x_0), 0 \leq r \leq n . Homework Equations If f is differentiable then \lim_{x \rightarrow x_0}\frac{f(x)-T_n(x)}{(x-x_0)^n}=0 ...
  28. S

    Deriving Partial Derivatives of Z with Respect to r and θ

    I am given Z = f (x, y), where x= r cosθ and y=r sinθ I found ∂z/∂r = ∂z/∂x ∂x/∂r + ∂z/∂y ∂y/∂r = (cos θ) ∂z/∂x + (sin θ) ∂z/∂y and ∂z/∂θ = ∂z/∂x ∂x/∂θ + ∂z/∂y ∂y/∂θ= (-r sin θ) ∂z/∂x + (r cos θ) ∂z/∂y I need to show that ∂z/∂x = cos θ ∂z/∂r - 1/r * sin θ ∂z/∂θ and ∂z/∂y = sin...
  29. M

    Finding the nth Derivatives of cos^12x & a-x/a+x

    Homework Statement how to find nth derivatives cos^12x and a-x/a+x Homework Equations The Attempt at a Solution
  30. W

    Properties of mixed partial derivatives

    Hi, I am sort of hung up with a particular step in a derivation, and this has caused me to ponder a few properties of partial derivatives. As a result, I believe I may be correct for the wrong reasons. For this example, the starting term is (\frac{\partial}{\partial x}\frac{\partial...
  31. Mandelbroth

    Challenging Summations, Limits, and Derivatives

    I'm going to talk with someone at a local university tomorrow to see if I can get out of AP Calculus. Essentially, I would like to be prepared for our meeting tomorrow. I'm good with integrals, so that shouldn't be a problem. However, I'm not quite as confident with derivatives, limits, and...
  32. C

    Derivatives of functions of products of variables

    Is there a general formula for (total) derivatives of functions of the form f(xy(x)+z(x)? I tried the most simple function of that form f(xy(x)+z(x))=xy(x)+z(x) and the formula I got was \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} \frac{dy}{dx}+\frac{\partial f}{\partial...
  33. S

    MHB Derivatives of $\frac{a}{1-r}$: Exploring the Rules

    I am having a little trouble remembering the rules with derivatives. $\frac{a}{1-r}$ I know that it should be (derivative of the top*bottom - top*derivative bottom) / (bottom squared). $\frac{d}{dr}\frac{a}{1-r}$ I tried this got the answer wrong, and looked up how to do this and they showed...
  34. 1

    Cubes, spheres, and derivatives.

    The volume of a sphere with radius r is v = \frac{4}{3}\pi r ^{3} It makes sense that its derivative with respect to radius is the surface area of the sphere. \frac{dv}{dr} = a = 4\pi r ^{2} The volume of a cube with side length n is n^{3} The derivative of this is just 3n^(2)...
  35. N

    Having Issues with Solving Derivatives

    Homework Statement Use the definition of a derivative to compute the derivative of the function at the given value. P(t)=t^3-4, t=-2 Homework Equations Please help me solve it in this form: f(a+h)-f(a)/h lim h→0 The Attempt at a Solution...
  36. S

    Notation for partial derivatives using indexes

    Is there a standard notation for partial derivatives that uses indexes instead of letters to denote ideas such as the 3 rd partial derivative with respect to the the 2nd argument of a function? As soon as a symbol gets superscripts and subscripts like \partial_{2,1}^{3,1} \ f the spectre of...
  37. N

    Geometrical meaning of derivatives

    I don't understand what the derivatives really mean?I know that they are the slope of the tangents drawn to a function.But see for example we have a function f(x)=x2 The derivative of this gives us '2x'. But what does '2x' mean?If i draw a graph of f(x)=2x what does it give me?what should i...
  38. G

    Find a 2-arguments function from six constraints on its derivatives

    Hello, I need to find a two-arguments function u(x,y) which satisfies six constraints on its derivatives. x and y are quantities so always positive. 1&2: On the first derivatives: du/dx>0 for all x & du/dy>0 for all y (so u is increasing in x and y) 3&4: On the second derivatives...
  39. D

    Quotient Rule for Derivatives

    I am a little confused about when to use the quotient rule. When you have one function over another function, and are taking the derivative, are you required to use this technique? I thought you were, but then I was watching this video on Khan Academy...
  40. L

    Multipole expansion. Problems with understanding derivatives

    Hi everyone Homework Statement I want to find the multipole expansion of \Phi(\vec r)= \frac {1}{4\pi \epsilon_0} \int d^3 r' \frac {\rho(\vec r')}{|\vec r -\vec r'|} Homework Equations Taylor series The Attempt at a Solution My attempt at a solution was to use the Taylor series. I...
  41. L

    A question on notation of derivatives

    I was doing a proof on why the derivative of an even function is odd and vice versa. Now, the way I did the problem was by using the chain rule to rewrite the derivative of f(-x), and the proof worked out perfectly fine. But I had a thought that I can't quite wrap my around, and I think it's...
  42. C

    Can it be argued that derivatives should be undefined?

    I understand derivatives and I am not trying to be like a stickler or anything, but before manipulating the equation to arrive at a form where we can find a real answer for a derivative, we are left with [f(x+h)-f(x)]/h (where h is delta x I guess as most people write it). Before evaluating...
  43. K

    MHB Partial Derivatives of the cosine rule.

    Partial Derivatives Hi all I was wondering if anyone could help me with this problem. I have a triangle that has a = 13.5m, b = 24.6m c, and theta = 105.6 degrees. Can someone remind me of what the cosine rule is? Also (my question is here) From the cosine rule i need to find: the...
  44. Mandelbroth

    Integrating derivatives of various orders

    I'm trying to figure out the general solution to the integral ##\int \frac{d^ny}{dx^n} \, dy##, where n is a positive integer (Meaning no fractional calculus. Keeping things simple.). So far, I have been working with individual cases to see if I can establish a general pattern and then try a...
  45. J

    Finding f'(x) for Logarithm Derivatives Homework with Product and Quotient Rules

    Homework Statement Find f'(x): (abs(((x^2)*((3x+2) ^(1/3)))/((2x-3)^3)) <- Not sure why it''s not showing up but the 1/3 is an exponent to just the (3x+2).Homework Equations Product Rule and Quotient Rule for DifferentiatingThe Attempt at a Solution So I thought I should split it into two...
  46. I

    Relating limits to derivatives, as x approaches non zero number?

    Homework Statement Suppose that f' (2) = 3. Find the limit as x approaches 2 of [f(x)−f(2)]/[sqrt(x) - sqrt(2)] Answer: 6*sqrt 2 Homework Equations The Attempt at a Solution f'(x) = lim h->0 = [f(a +h) - f(a)]/h = slope [f(x)-f(2)]/ x-2 a = 2 i would think that the limit =...
  47. Mandelbroth

    Trying to understand derivatives in terms of differential forms

    Suppose we have a curve, formed by a function f that maps real numbers to real numbers, such that f is everywhere smooth over a subset D of its domain. Let's suppose that, for all x in D, there is a vector space that contains all vectors tangent to the curve at that point, called the tangent...
  48. O

    !Understanding Partial Derivatives of Coordinate Transformation

    Hi Everyone, I was studying coordinate transformation and I came across this equation, that I couldn't understand how it came up. Let me put it this way: x = rcosθ Then if I want to express the partial derivative (of any thing) with respect to x, what would be the expression? i.e. ∂/∂x=...
  49. C

    Pulling out partial derivatives?

    I'm reading through the book Quantum Mechanics (Second Edition) by David J. Griffiths and it got to the part about proving that if you normalise a wave function, it stays normalised (Page 13). That part that I don't get is how they say: ## \dfrac{i \hbar}{2m} \left( \Psi^* \dfrac{\partial^2...
  50. 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...
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