What is Differentiation: Definition and 1000 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. P

    How Do You Isolate dy/dx in Implicit Differentiation?

    Homework Statement Calculate the derivative with respect to x: x/y + y/x = 2yHomework Equations n/a The Attempt at a Solution I end up getting the right answer, but what I want to know is how to isolate dy/dx to one side after implicitly differentiating. I have tried differentiating the LHS...
  2. G

    Implicit Differentiation Problem

    Homework Statement The P(a,b) be a point on the curve √x + √y = 1. Show that the slope of the tangent P is -√b/a Homework Equations ? The Attempt at a Solution Apparently this is an implicit differentiation problem, however we haven't learned or discussed implicit...
  3. J

    Another question about implicit differentiation.

    Say you have x^2+y^2=100. why can't you just solve for y, so y=+- √(100-x^2) then use the chain rule to find the derivative. so y'= +- x/√(100-x^2). Then you can just deduce that y'= -x/y. What is the point of adding all the dy/dx in the equation? Seems like it just complicates it.
  4. M

    Differentiation with matrices/vectors

    Hello, I'm trying to understand this proof: http://en.wikipedia.org/wiki/Proofs_involving_ordinary_least_squares#Least_squares_estimator_for_.CE.B2 Can someone quickly talk me through the differentiation step, bearing in mind I've never learn how to differentiate with respect to a vector...
  5. I

    MHB A proof about maximum point, critical point and differentiation

    Let $E\subset\mathbb{R}^n$ and $f: E\rightarrow\mathbb{R}$ be a continuous function. Prove that if $a$ is a local maximum point for $f$, then either $f$ is differentiable at $x = a$ and $Df(a) = 0$ or $f$ is not differentiable at $x = a$. Deduce that if $f$ is differentiable on $E^o$, then a...
  6. W

    Differentiation with different variables

    Homework Statement I'm trying to take the derivative of the following integral \frac{d}{d V} \int_0^t{V(\tau)}d\tau Homework Equations FTC will probably be a part of it. The Attempt at a Solution I always get confused when I'm taking the derivative of an integral. I know the answer is...
  7. WannabeNewton

    Differentiation under integral sign - one parameter family

    Hi guys! Let \left \{ B_{t} \right \}_{t\in \mathbb{R}} be a one - parameter family of compact subsets of \mathbb{R}^{3} with smooth (manifold) boundary (e.g. one - parameter family of closed balls). In my context, each B_{t} belongs to a different constant time slice of Minkowski space - time...
  8. D

    MHB Differentiating $\mathcal{E}$: How to Reach $\dot{x}(m\ddot{x} + kx)$?

    $\mathcal{E} = \frac{1}{2}m\dot{x} + \frac{1}{2}kx^2$ The derivative is $$ \frac{d\mathcal{E}}{dt} = \frac{1}{2}m\ddot{x} + kx\dot{x} $$ but the solution is suppose to be $$ \frac{d\mathcal{E}}{dt} = \dot{x}(m\ddot{x} + kx). $$ How?
  9. M

    Implicit Differentiation for a 2nd Derivative

    Hello! As of right now (10:13 PM), I've tried 9 combinations of points to solve this problem. It's a WebWork-based problem that's due in about an hour in a half. Any help would be very, very appreciated. Homework Statement I was given this equation: ##ln(2y) = 2xy## and was asked to find...
  10. A

    How does differentiation under the integral sign

    I've read about it before and now I'm trying to learn it myself from Woods 'Advanced Calculus' (as well as other resources like http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf) In the pdf, it says the method concerns integrals that depend on a parameter...now couldn't we...
  11. L

    Using differentiation to find maximum length problems

    Homework Statement The line segment AB lies on a diameter of a circle of radius 1, and the angle BAC is a right angle. Find the greatest possible value of the sum of the lengths of AB and AC. Homework Equations The Attempt at a Solution I have no idea what parameters to use...
  12. C

    Using parametric differentiation to evaluate the slope of a curve - attempted

    Homework Statement x(t) = (t^2 -1) / (t^2 +1) y(t) = (2t) / (t^2 +1) at the point t=1 Homework Equations Line equation = y-y1 = m(x-x1) chan rule = (dy/dt) / (dx/dt) = dy/dx The Attempt at a Solution I find the y1 and x1 values by subing in t=1 to the x(t) and y(t)...
  13. B

    Mathematica Symbolic vector differentiation with Mathematica

    Hi, I am a new user of Mathematica (although I am reasonably familiar with MATLAB) and I am trying to differentiate a scalar wrt a vector Mathematica. ie, I want to check if \begin{equation} \phi = \textit{x}^{T} \textbf{A} \textit{y} \quad \mbox{where $\textit{x}$, $\textit{y}$ are vectors...
  14. D

    Nth derivative, differentiation

    Homework Statement Find f^n for f(x) = Ln(2x+1) Can anyone point me in the right direction with how to get the nth derivative of the above function please, I just cannot seem to work this out! Thank you
  15. J

    Review in Variable Differentiation

    Please see attached. I was looking for an explanation of the answer I have attached. Its been a little while and was just looking for the logic behind the differentiation shown for this problem. Its basically an optimization problem where I am looking for the minimum angle (theta) for the...
  16. N

    MHB Differentiation wrt integral boundaries

    Hi:Let $f,g$ and $h$ be continuous real valued function with domain $X \times Y$ where $X$ and $Y$ are compact sets. Let me define the set $S(x) = \{ y : h(y,x) \geq 0 \} $ and $\bar{S}(x) = \{ y : h(y,x) \leq 0 \} $ Also for any $(x,y) \in X \times Y $ such that $h(y,x)=0$, it holds...
  17. T

    Differentiation word problem (basic)

    A rectangle of length x, where x varies, has a constant area of 48cm2. Express the perimeter, y in terms of x. Find the least possible value of x. my problem is not the maths part i.e. the differentiation, but the equation to get things moving. I really have no idea where to start. I drew a...
  18. U

    Partial differentiation: prove this general result

    Homework Statement The function f(x,y,z) may be expressed in new coordinates as g(u,v,w). Prove this general result: The Attempt at a Solution df = (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz dg = (∂g/∂u)du + (∂g/∂v)dv + (∂g/∂w)dw df = dg since they are the same thing? but the...
  19. R

    Differentials and Implicit Differentiation

    Homework Statement I'm reviewing physics using Feynman's Lectures, and I'm finding that he frequently uses implicit differentiation in his lessons. This is unfortunate for me because I never got the hang of it beyond the simplest cases. I'm currently going through the proof that the...
  20. M

    Use logarithmic differentiation to find the derivative

    Hi .. Use logarithmic differentiation to find the derivative can please check my answer and How I can know if the question want answer by using logarithmic differentiation or not ?
  21. B

    Domain Differentiation question

    if a function f is differentiable of [0,2pi] can I integrate its derivative df on [-pi, pi]?
  22. D

    Use implicit differentiation to find dy/dx

    Homework Statement Use implicit differentiation to find dy/dx. Homework Equations xey - 10x + 3y = 0 The Attempt at a Solution = [xey + ey(y)'] - (10x)' + (3y)' = 0 = xey + ey(y') - 10 + 3(y') = 0 = y' (ey + 3) = 10 - xey = y' = 10 - xey/ ey + 3y However, my book says the answer: 10 -...
  23. R

    Differentiation using the quotient rule

    Homework Statement Use the quotient rule to differentiate y=(〖2x〗^4-3x)/(4x-1) Homework Equations y=(v du/dx-u dv/dx)/v^2 The Attempt at a Solution Please also find attached attempt as jpeg for clarity, and textbook supplied answer...
  24. Jalo

    Partial differentiation - Constants

    Homework Statement Consider the following equality: (\frac{∂S}{∂V})T = (\frac{∂P}{∂T})V If I rearrange the equality so that I write: (\frac{∂S}{∂P})? = (\frac{∂V}{∂T})? What variables will be constant in each side? I'm having some trouble in a few thermodynamics problems because...
  25. T

    Proof by Induction of the Power Rule of Differentiation

    Homework Statement Okay, the concept here is to use induction to prove that for n, (f1 x f2 x ... x fn-1 x fn)' = (f'1 x f2 x ... x fn) + (f1 x f'2 x ... x fn) + ... + (f1 x f2 x ... x f'n). 2. Homework Equations / 3. The Attempt at a Solution I solved the initial step, which was quite...
  26. M

    Double Root and Quotients in Differentiation of Polynomial Functions

    Homework Statement p(x)=vx^{n+1}+ux^{n}+1 Homework Equations 1) Find u and v so that 1 is a double root for p. 2) Conclude the quotient of p(x) over (x+1)^2. 3) For n=4 find u and v and find the quotient of p(x) over (x-1)^2. The Attempt at a Solution Can someone just tell me how to...
  27. R

    Finding the tangent line using implicit differentiation.

    Homework Statement The equations ##2x^3y+yx^2+t^2=0##, ##x+6+t-1=0## implicitly define a curve $$f(t) = \begin{pmatrix} x(t)\\y(t) \end{pmatrix}$$ that satisfies ##f(1)=\begin{pmatrix} -1\\1 \end{pmatrix}.## Find the tangent line to the curve when ##t=1##. Homework Equations The...
  28. N

    Differentiation Map of a Complex Transformation

    Homework Statement Find the eigenvectors and eigenvalues of the differentiation map C1(R) -> C1(R) from the vector space of differentiable functions to itself. Homework Equations The Attempt at a Solution Hi, I'm not entirely sure how to go about this, because would the...
  29. V

    Logarithmic Differentiation of (x+5)(x+9): Where Did I Go Wrong?

    Homework Statement Find the derivative using logarithmic differentiation: y=(x+5)(x+9) The Attempt at a Solution lny=ln(x^2+14x+45) lny=(2x+14)/(x^2+14x+45) y'=(x^2+14x+45)((2x+14)/(x^2+14x+45))However, I know the derivative of the function is actually 2x+14. So I am wondering what is wrong...
  30. C

    Mass and Energy Differentiation

    Hello friends: My Question: A massive object cannot move at the speed of light. Photons can move at the speed of light because they are massless. However, since energy and mass are equivalent, due to Einstein's famous equation E^2=(m(c^2))^2+(pc)^2, mass is energy by a conversion...
  31. D

    Numerical differentiation with change of variable

    Hi all I am trying to solve for an integral whose integrand is a derivative that has a change of variable inside of it. ∫ (dz/dx) * cos(θ) dθ between 0 and pi. I have a function for z(x), and also know the relation between of x and θ, but what I don't know is how to evaluate such...
  32. J

    Computational Physics (Making programs for interpolation and differentiation)

    Homework Statement Chapter 4 1. Write a program that implements the first order (linear) interpolation 2. Write a program that implemets n-point Lagrange interpolation. Trean n as an imput parameter. 3. Apply the program to study the quality of the Lagrange interpolation to functions...
  33. A

    Implicit differentiation help three variables

    Homework Statement I have a question. How in general would one differentiate a composite function like F(x,y,z)=2x^2-yz+xz^2 where x=2sint , y=t^2-t+1 , and z = 3e^-1 ? I want to find the value of dF/dt evaluated at t=0 and I don't know how. Can someone please walk me through this?Homework...
  34. U

    What is the Implicit Differentiation Equation for eysinx=x+xy?

    Equation: eysinx=x+xy I took the derivative of both sides. For the side with eysinx, I used the product rule and chain rule to get: ey*cosx + ey*sinx*y' For the side with x+xy, I used the sum and product rule to get 1+y+xy' So my resulting equation is: ey*cosx + ey*sinx*y'=1+y+xy', which...
  35. R

    Unusual partial differentiation equation

    Homework Statement Calculate ∂f/∂x and ∂f/∂y for the following function: yf^2 + sin(xy) = f The Attempt at a Solution I understand basic partial differentiation, but I have no idea how to approach the f incorporation on both sides of the equation nor what you would explicitly call this...
  36. D

    The derivative using logarithmic differentiation. Is this correct?

    1. Homework Statement [/b] Find the derivative of the given function. Homework Equations Chain rule and logarithmic differentiation. The Attempt at a Solution See attached .gif. I was just wondering if this seemed correct? Thanks!
  37. J

    Implicit Differentiation Problem

    1. Given that y^{2}-2xy+x^{3}=0, find \frac{dy}{dx} 2. (no relevant equations other than the problem statement) 3. So, I solved it like this, \frac{dy}{dx}y^{2}-2xy+x^{3}=0 2y\frac{dy}{dx}-2+3x^{2}=0 Solving for dy/dx I got... \frac{dy}{dx}=\frac{-3x^{2}+2}{2y}...
  38. J

    Implicit Differentiation, chain rule, and simplifying

    Okay so, I am having trouble figuring out what exactly to do in implicit differentiation and usage of the chain rule. Like, I keep getting the wrong answer somehow. See, from what I understand you have to find the derivative of both sides then use the chain rule or something and then solve for...
  39. J

    Implicit differentiation vs differential equations?

    Hello, I have recently started a little implicit differentiation and I have seen DEs before but I know that I still need to work on my differentiation and integration a little more before I am ready to tackle those. Anyway, I wish to ask, what distinguishes implicit differentiation from a...
  40. D

    Using implicit differentiation: Is this correct?

    Homework Statement I need to use implicit differentiation to find the derivative of y=sin(x+y). Homework Equations The Attempt at a Solution This is what I did: y=sin(x+y) y'=(sin(x+y))' y'=(1+y')(cos(x+y)) (by the chain rule) Now, what do I do? Is this correct...
  41. N

    Find the matrix representations of the Differentiation Map in the Basis

    Homework Statement Show that B = {x2 −1,2x2 +x−3,3x2 +x} is a basis for P2(R). Show that the differentiation map D : P2(R) → P2(R) is a linear transformation. Finally, find the following matrix representations of D: DSt←St, DSt←B and DB←B. Homework Equations The Attempt at a...
  42. M

    Solving ∫log(1+acosx) by Differentiation Under Integral Sign

    Homework Statement solve ∫log(1+acosx) by differentiation under integral sign (limits are 0 to ∏) Homework Equations The Attempt at a Solution =∫(1/1+acosx)cosxdx(by leibinitz by differentiating partially WRT a. Then how do I proceed,can anyone show me all the steps of...
  43. A

    Differentiation rate of flow from a cylinder

    1. Water flows out of a cylindrical tank under gravity via a tap, the height h(t) of the water column above the tap satisfies the differential equation in the form dh/dt = -2k√h where k is some positive constant. The water column has a height initially of 25m. The tap is turned on and...
  44. P

    Differentiation of a function in a domain

    Homework Statement Find the derivatives at an arbitrary point x in the domain of the following functions f_i: D_i → ℝ, where for 1 ≤ i ≤ 6 the domain D_i is the maximal subset of ℝ on which the mapping is defined - you don't have to determine the domains. Homework Equations a) f_1 (a) =...
  45. D

    What Calculations Determine When a Dropped Object Hits the Ground?

    Homework Statement The height "s" at time of a silver dollar dropped from a building is given by s(t) = -16t^2 + 1350, where "s" is measured in feet and "t" is measured in seconds [s'(t) = -32t] a) Find the average velocity on the interval [1,2]. ( I ALREADY SOLVED) b) Find the instantaneous...
  46. P

    How Do You Solve Implicit Differentiation for y = sin(xy)?

    1. y = sinxy Homework Equations 3. this was my attempt d/dx = (cosxy)(sinxy(d\dx))+(xy(d/dx) im getting stuck. i don't think I am starting it right. any suggestions.
  47. B

    Differentiation involving Sin(x) as a power

    Homework Statement Find the derivative of y=(x^2)^sinx; using the chain rule. Homework Equations No other relevant equations. The Attempt at a Solution I attempted to apply the Chain rule: dy/dx = dy/du X du/dx Subbing u for x^2, which made y = u^sinx I ended up with...
  48. Saitama

    Simple differentiation question

    Homework Statement Find \frac{dy}{dx}. y=\sin^{-1}(2x\sqrt{1-x^2}), \frac{-1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}Homework Equations The Attempt at a Solution I started with substituting x=sinθ. The expression simplifies to y=\sin^{-1}(\sin(2θ)) which is equal to y=2θ. Substituting back the value of...
  49. B

    Differentiation using chain/product rule

    Hi, Just a question on an example in a maths textbook. See attached image for question below. So, I understand that if you set u=sin(x) and v=e^-cos(x) f'(x)=u'.v + u.v' But I'm stuck looking at e^-cos(x), could it also be classified e^(w)? Also, the second step in differentiating...
  50. B

    Partial Differentiation with Indicial Notation (Ritz Method for FEM)

    Folks, I am stuck on an example which is partial differenting a functional with indicial notation The functional ##\displaystyle I(c_1,c_2,...c_N)=\frac{1}{2} \int_0^1 \left [ \left (\sum\limits_{j=1}^N c_j \frac{d \phi_j}{dx}\right )^2-\left(\sum\limits_{j=1}^N c_j \phi_j\right)^2+2x^2...
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