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. 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. 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. 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'...

24. 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^-##?
25. 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.
26. 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...
27. A Dissipative and dispersive derivatives

Why are even order derivatives dissipative and odd order derivatives dispersive?
28. 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...
29. 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 )...
30. 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...
31. 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?
32. 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...
33. 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...
34. 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!
35. 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?
36. 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.
37. 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...
38. 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...
39. 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: $$f(x,y,z)=s^2=x^2+y^2+z^2$$...
40. 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, ##...
41. 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. $\displaystyle{f(x)=\sqrt{e^{\sin \sqrt{x}}}}$, $x>0$...

45. 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...
46. MHB Using Chain rule to find derivatives....

y = (csc(x) + cot(x) )^-1 Find dy/dx
47. 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...
48. 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...
49. 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?
50. 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 $x$.