# Differentiation Definition and 61 Discussions

The cluster of differentiation (also known as cluster of designation or classification determinant and often abbreviated as CD) is a protocol used for the identification and investigation of cell surface molecules providing targets for immunophenotyping of cells. In terms of physiology, CD molecules can act in numerous ways, often acting as receptors or ligands important to the cell. A signal cascade is usually initiated, altering the behavior of the cell (see cell signaling). Some CD proteins do not play a role in cell signaling, but have other functions, such as cell adhesion. CD for humans is numbered up to 371 (as of 21 April 2016).

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1. ### Differential position vector

I had an equation. $$T=\frac{1}{2}m[\dot{x}^2+(r\dot{\theta})^2]$$ Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$ I was thinking how they had derived it. The equation is looking like, they had differentiate "something". Is it just an...
2. ### Point on a graph nearest to the origin

The Attempt at a Solution I know the answer is supposed to be ##(-1,0)##. However when I differentiate the above expression I get. $$2x+{\frac 5 2}$$ Then the shortest distance would be when the expression equates to 0. $$2x+{\frac 5 2}=0$$ I should be getting ##x=-1## but solving for ##x##...
3. ### Implicit Differentiation

The working I've tried is in the attachment.
4. ### 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 $$\frac{\frac{dC}{dt}}{\frac{dI}{dt}}=\frac{dC}{dI}$$?
5. ### A Trick with "Functional" derivative to evaluate commutators between between diagonal Hamiltonian and creation Fermionic operator

I found a theorem that states that if A and B are 2 endomorphism that satisfies $$[A,[A,B]]=[B,[A,B]]=0$$ then $$[A,F(B)]=[A,B]F'(B)=[A,B]\frac{\partial F(B)}{\partial B}$$. Now i'm trying to apply this result using the creation and annihilation fermionics operators $$B=C_k^+$$ and $$A=C_k$$...
6. ### A Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w))

Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w)) Hello to my Math Fellows, Problem: I am looking for a way to calculate w-derivative of Fourier transform,d/dw (F{x(t)}), in terms of regular Fourier transform,X(w)=F{x(t)}. Definition Based Solution (not good enough): from...
7. ### Derivative of a series

I first solved the first two terms and then i solved the resulting term with the third term and so on.At last i was left with x^n/((x-a1)(x-a2)...(x-an)) .Thrn i took log on both sides and then differentiated both sides with respect to x.I got 1/y dy/dx=n/x -1/(x-a1)-1/(x-a2)...-1/(x-an).But now...
8. ### Trivial question on differentiating logarithms

My Question : Shouldn't differentiating ##-log B## give ##\frac{-\delta B}{B}##? (Note : A, B and Z are variables not constants) By extension for ##Z=A^a \,B^b\, C^c## where ##c## is negative, should ##\frac{\Delta Z}Z=|a|\frac{\Delta A}A+|b|\frac{\Delta B}B-|c|\frac{\Delta C}C##?
9. ### Derivative of a term within a sum

Homework Statement [/B] From the Rodrigues’ formulae, I want to derive nature of the spherical Bessel and Neumann functions at small values of p. Homework Equations [/B] I'm going to post an image of the Bessel function where we're using a Taylor expansion, which I'm happy with and is as far...
10. ### I Total Derivative of a Constrained System

Hi all, I was working on a problem using Euler-Lagrange equations, and I started wondering about the total and partial derivatives. After some fiddling around in equations, I feel like I have confused myself a bit. I'm not a mathematician by training, so there must exist some terminology which...
11. ### Thermodynamics Pathria Eq.4.5.9

I'm confused about the mathematics that led to the equation 4.5.9. Specifically, I'm confused about what the variables that describe U are. From the equation I think temperature T(through beta), chemical potential (through alpha), V (through E_s) and N (through... restriction on the...
12. ### Problem in calculating work done?

Homework Statement See question number 3 Homework Equations Work Done="integral" F*ds The Attempt at a Solution I tried to solve this question using integration, I have replaced F with "1" and ds with "t^2+2t". So I am stuck in at that step. Please help me differentiate it further or solve...
13. ### Problem in applying the Chain Rule

Homework Statement I am facing problem in applying the chain rule. The question which I am trying to solve is, " Find the second derivative of " Homework Equations The Attempt at a Solution So, differentiated it the first time, [BY CHAIN RULE] And now to find the second derivative I...
14. ### B Understanding this graph

Could someone explain to me how from this graph you can deduce that ##\tan(\theta) = \frac {df} {dx}##. Thanks
15. ### Point on the curve closest to (18,1)

Question: Question: Find the point on the curve y=x^2 +1 that is closest to the point (18,1). Please see the image and that’s where I’m stucked- after taking the first derivate. Please solve it further step by step completely. It’d mean a lot.
16. ### Critical Points of a Parameter Dependent Integral

1. The problem statement, all variables, and given/known data Find and categorize extremes of the following function: $$F(y)=\int_{y}^{y^{2}}\frac{1}{\ln^{2}x}dx$$ for ##y>1##. Homework Equations $$\frac{d}{dx}\int_{a}^{b}f(x,y)dy=\int_{a}^{b}\frac{\partial}{\partial x}\left(f(x,y)\right)dy$$...
17. ### I Need a help in solving an equation (probably differentiation

I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following...
18. ### Determine the stability of a fixed point of oscillations

Homework Statement I have a system of coupled differential equations representing chemical reactions and given certain initial conditions for the equations I can observe oscillation behaviour when I solved the equations numerically using Euler's Method. However, then it asks to investigate the...
19. ### I Help please with biocalculus question involving differentiation

Hi, I was just wondering how one would arrive at the answers to these questions. I have the solution for parts a and b, but not for part c. Suppose that antibiotics are injected into a patient to treat a sinus infection. The antibiotics circulate in the blood, slowly diffusing into the sinus...
20. A

### Finding the approximate change in the perimeter of a circle

Homework Statement The radius of a circle increases from 3 to 3.01 cm. Find the approximate change in its perimeter. Here's a link to the actual question, in case you need the answer for 6(a) to solve 6(b) http://imgur.com/a/nQt6M Homework Equations Perimeter of circle = 2πr Area of circle =...
21. 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...
22. ### Show that this function is differentiable

Homework Statement [/B] 2. The attempt at a solution I'm not really sure where to start. We just want to show that ##\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = 0##. I see that ##\lim_{x \to c} (x - c)^2 = 0##. I feel that this may be a simple trick of inequalities, but I am having a complete...
23. ### Find the equation of a cubic graph

Homework Statement ##x^3 - 4x^2 + ax + b## tangent to x-axis at x = 3 Homework Equations The Attempt at a Solution if the graph tangent at x = 3, means at x =3, y = 0 my questions is, is at x = 3 the graph's gradient (slope) = 0 ? if yes why? if yes then means dy/dx = 0 ##3x^2 - 8x + a =...
24. ### Finding dy/dx for a circle

Homework Statement Hello I have this circle with the equation : [/B] (x-a)^2+(y-b)^2=r^2 I want to find dy/dx for it 2. Homework Equations (x-a)^2+(y-b)^2=r^2 The Attempt at a Solution I am looking on the internet and it appears that I should use what is called "Implicit differentiation"...
25. ### Alternate way of differentiating x^y

Homework Statement : [/B]find the dy/dx of xy=a constant Homework Equations : basic differentiation formulae[/B] The Attempt at a Solution :[/B] I know we can use logarithmic differentiation for differentiating x y..But can we differentiate it using chain rule and get answer as...

29. ### A Differentiation cost function

Could someone please help me work through the differentiation in a paper (not homework), im having trouble finding out how they came up with their cost function. The loss function is L=wE, where E=(G-Gest)^2 and G=F'F The derivative of the loss function wrt F is proportional to F'(G-Gest)...
30. ### Insights Partial Differentiation Without Tears - Comments

andrewkirk submitted a new PF Insights post Partial Differentiation Without Tears Continue reading the Original PF Insights Post.
31. ### Partial derivatives and chain rule

Homework Statement a. Given u=F(x,y,z) and z=f(x,y) find { f }_{ xx } in terms of the partial derivatives of of F. b. Given { z }^{ 3 }+xyz=8 find { f }_{ x }(0,1)\quad { f }_{ y }(0,1)\quad { f }_{ xx }(0,1) Homework Equations Implicit function theorem, chain rule diagrams, Clairaut's...
32. ### I Difficulty with function dependencies f(u,x)

If you have a function x = x(u,t) then does u necessarily depend on x and t? so u = (x,t) For example, if x(u,t)=u^2 t it seems that because t=x/u^2 , t=t(x,u) I am having difficulty working out the general equation for dz \over dx if z=z(x,y,t) x=x(u,t) y=y(u,v,t) The chain rule...
33. ### Simple Harmonic Motion - Getting Acceleration from Velocity

Homework Statement I am doing an experiment where I am measuring the force a speaker is exerting when it is driven by a certain voltage and frequency, so my voltage and frequency values are known. I am assuming the speaker is undergoing SHM and I am measuring its peak to peak velocity...
34. ### I How to Convert this Derivative?

I have a derivative of a function with respect to ##\log \left(r\right)##: \begin{equation*} \frac{dN\left(r\right)}{d \log\left(r\right)} = \frac{N}{\sqrt{2\pi} \log\left(\sigma\right)} \exp\left\{-\frac{\left[\log \left(r\right) - \log\left(r_M\right)\right]^2}{2...
35. ### Find expression in terms of time for a particle's velocity?

Homework Statement A particle moves along a straight line so that its acceleration t seconds after passing a fixed point O on the line is (2 - 2t) cm/s2. Three seconds after passing O, the particle has a velocity of 5 cm/s. Find and expression, in terms of t, for the velocity of the particle...
36. ### Differentiate trigonometric equation

Homework Statement a) Differentiate the following equation with respect to: 1) θ 2) Φ 3) ψ (Ua - Ub)' * C * r where: C is a 3 x 3 rotation matrix: [ cos θ cos ψ, -cos Φ sin ψ + sin Φ sin θ cos ψ, sin Φ sin ψ + cos Φ sin θ cos ψ] [ cos θ sin ψ, cos Φ cos ψ + sin Φ sin θ sin...
37. ### I Related Rates (+Trig) Question

I'm a bit iffy with the whole of the 'related rates' topic of my calculus course. I've tried coming up with a question of my own to see if I can solve it. The question is as follows: The distance between a point on the ground and the bottom of a pole is 26m. The angle of inclination from that...
38. ### I Magnitude of the Second Derivative

So to find the x values of the stationary points on the curve: f(x)=x3+3x2 you make f '(x)=0 so: 3x2+6x=0 x=0 or x=-2 Then to find which of these points are maximum or minimum you do f ''(0) and f ''(-2) so: 6(0)+6=6 6(-2)+6=-6 so the maximum has an x value of -2 and the minimum has an x value...
39. ### I Differentiating twice under the integral (with assumptions)

Let ##k:\mathcal{O}\times\mathbb{R}^n\to\mathbb{R}##, with ##\mathcal{O}\subset\mathbb{R}^m## open, be such that ##\forall x\in\mathcal{O}\quad k(x,\cdot)\in L^1(\mathbb{R}^n) ##, i.e. the function ##y\mapsto k(x,y)## is Lebesgue summable on ##\mathbb{R}^n##, according to the usual...
40. ### Did tissues arise from other cells or cell differentiation?

Hi I have a pretty specific question. It is in regards to tissues in multicellular organisms. Is there any information on how different cell grouping arose in multicellular organisms? I have some ideas from what I've so far read and learned: - Would this have happened because two different...
41. ### Surface area of a spherical cap by integration

hi guys, i have a question. i saw this picture, and i don't really understand how they derived with the formula. The aim is basically to find the formula for the surface area of a spherical cap. why do you differentiate the x=sqrt(rˆ2-yˆ2)? how does that help to find the surface? and then...
42. ### Minimizing surface area of a shaker

Hi, I have an mathematics assignment to do, and I wonder if the topic I have chosen is doable for me. I want to minimize the surface area of a cobbler cocktail shaker, and until now my plan was to get the curve equation for the side of it, and get the area equation from surface of revolution...
43. ### Rodrigues’ formula of Laguerre

Homework Statement I need to proof that Rodrigues’ formula satisfies Laguerre differential equation Homework Equations Rodrigues’ formula of Laguerre Laguerre differential equation The Attempt at a Solution first,I have to calculate = I tried to sum both terms and this is what I got...
44. ### Variation of parameters (1st order)

Homework Statement Find the general solution of the following equation: u(t): u' = u/t + 2t Homework Equations y' + p(x)y = Q(x).............(1) yeI = ∫ dx eIQ(x) + constant..............(2) The Attempt at a Solution I rearranged the equation to give: u' - u/t = 2t Then I considered the...
45. ### How to differentiate this potential energy function?

Homework Statement Given the potential energy function V(x,y)=V(ax-by) where a,b is an arbitrary constants differentiate with respect to x and y. Homework Equations Multivariavle differentiation The Attempt at a Solution The answer yields (d/dt)p1=-aV'(ax-by) (d/dt)p2=+bV'(ax-by). The right...
46. ### Implicit Differentiation: two different answers

Homework Statement with answers given: Homework Equations use implicit differentiation The Attempt at a Solution I always get this answer but not the second one PLs explain the second answer for I am very desperate. Thank You
47. ### Prove in case of projectile d^2 (v^2 ) / dt^2 = 2g^2

Homework Statement for a projectile show that d^2 (v^2) / dt^2 = 2g^2 2. The attempt at a solution =d/dt (d(v^2)/dt) =d/dt (2v)
48. ### Mean Value Theorem/Rolle's Theorem and differentiability

Homework Statement Let f(x) = 1 - x2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem? Homework Equations The Attempt at a Solution f(x) = 1 - x2/3. f(-1) = 1 - 1 = 0 f(1) = 1 - 1 = 0 f' = 2/3 x -1/3. I don't...
49. ### Functional derivative of normal function

I can't convince myself whether the following functional derivative is trivial or not: ##\frac \delta {\delta \psi(x)} \big[ \partial_x \psi(x)\big],## where ##\partial_x## is a standard derivative with respect to ##x##. One could argue that ## \partial_x \psi(x) = \int dx' [\partial_{x'}...
50. ### Derivatives in 3D and Dirac Delta

For a research project, I have to take multiple derivatives of a Yukawa potential, e.g. ## \partial_i \partial_j ( \frac{e^{-m r}}{r} ) ## or another example is ## \partial_i \partial_j \partial_k \partial_\ell ( e^{-mr} ) ## I know that, at least in the first example above, there will be a...