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. R

    A new Method of Differentiation (PURE ALGEBRA), what do you think?

    I have discovered this new method. It is a little harder mechanics than normal Calculus, but it is still a new method which I searched that Internet and many other mathematics textbooks for it but I did find nothing. I want to contact a math Proff about this, I have tried to Contact Two Stanford...
  2. S

    Implicit Differentiation concept help?

    Homework Statement Its not homework, i have the answer I am just having a hard time wrapping my head around the concept of differentiating implicitly defined functions. the question was: x^3+y^3=3xy, find the equation of the tangent line at the point (3/2,3/2). Homework Equations...
  3. J

    Integration and differentiation

    Why is antiderivative and area under the curve the same thing? Its not at all intuitive to me Derivative is the slope at a point and its opposite is area?? Can someone just explain me why when we are finding an antiderivative, we are actually finding area under the curve i don't buy the...
  4. C

    Differentiation Rules of Sinusoidal Functions

    Homework Statement f(x) = cos2x - sin2xThe Attempt at a Solution f'(x) = (2cosx)(-sinx) - (2sinx)(cosx) f'(x) = -2cosxsinx - 2sinxcosx This is what I think the answer should be, but the back of the book says otherwise. I need help identifying what I did wrong.
  5. B

    Implicit Differentiation Question

    Homework Statement 2x^{3}-3x^{2}y+2xy^{2}-y^{3}=2 Homework Equations The Attempt at a Solution 6x^{2}-(6xy+3x^{2}y')+(2y^{2}+4xyy')-3y^{2}y'=0 y'=\frac{-6x^{2}+6xy-2y^{2}}{-3x^{2}y+4xy-3y^{2}} My text's solution is the same answer but with every every term having the...
  6. H

    Showing differentiation is a linear map

    Homework Statement The Attempt at a Solution For part ii) I wrote it out as a matrix, getting \begin{array}{ccccccc} 0 & 0 & 0 & 0 & ... & 0 \\ 0 & 0 & 2 & 0 & ... & 0 \\ 0 & 0 & 0 & 6 & ... & 0 \\ . & . & . & . & . & . \\ 0 & 0 & 0 & 0 & ... & N(N-2) \end{array} So...
  7. C

    MHB Solving for the Derivative of a Tricky Equation

    Hi, I've been given the equation dP/dt = c log (P/M) P and I've solved it to find p(t) = M*exp(Aexp(c*t)) and I need to differentiate back in order to get it in the form of the original equation but I'm finding it extremely tricky and messy to achieve and it's really bugging me so I was...
  8. L

    Differentiation tricks/shortcuts

    This is not a specific HW/CW question, just a gap I have and want to fill. I came from a school in which calculus was only introduced in the last year so I learned only the basics. Now, I see more and more stuff like taking an expression, say A=B+C and simply making it to a dA=dB+dC. The...
  9. L

    Derivative of s(t) = (976(.835)^t -1) + 176t using log differentiation

    1. Homework Statement The derivative of s(t) = (976(.835)^t - 1) +176t I have to take ln of both sides to bring the t down from the exponent. But I never had to apply ln to an equation of this complexity. Here is my attempt, but it doesn't even look close to being on the correct...
  10. K

    Differentiation of a vektorfield

    hi! i flipped through my notes on a class on general relativity this morning and i found an expression which doesn't make sense to me. I am not sure if don't understand the last term in the last equality or it just dosn't make sense. i would appreciate your oppinion. a,b are abstract indicies...
  11. D

    Why is there a dx/dx in the Implicit Differentiation rule?

    I tried deriving this one on my own and I'm just not understanding where the dx/dx term comes from. I'm looking dy/dx. Starting with F(x,y) = 0: \frac{\partial{F}}{\partial{x}}\frac{dx}{dx} + \frac{\partial{F}}{\partial{y}}\frac{dy}{dx} = 0 It seems redundant to say dx/dx when it turns out to...
  12. L

    3(x^2 + y^2)^2 = 25(x^2 - y^2) implicit differentiation

    Use implicit differentiation to find the slope of the tangent line to the curve at the specified point. 3(x^2 + y^2)^2 = 25(x^2 - y^2) ; (2,1) This is where I'm stuck: I know how to get up to the first equation...and I know how to get to the final answer from the second equation, but I...
  13. P

    Differentiation and Stationary Points

    Hi, Q: By investigating the stationary points of f(x)= x3+3x2+6x-30 and sketching the curve y=f(x) show that the equation f(x)=0 has only one real solution. A: Well, I don't understand how I should use both. Plotting the graph, I can clearly spot a solution: x = 1.9319548 I know how to...
  14. A

    MHB Differentiation using first priciple

    Can anyone explain differentiation using first principle please......this is a question that i have no understanding on how to deal with it IF y=x^2 + 2 show that dy/dx using first principle equals to 2x
  15. A

    Finding Stationary Points on Implicitly Differentiated Curves

    Homework Statement Find the coordinates of the stationary points on the curve: x^3 + (3x^2)(y) -2y^3=16 Homework Equations Stationary points occur when the first derivative of y with respect to x is equal to zero The Attempt at a Solution I implicitly differentiated the...
  16. K

    Mathematica partial differentiation weirdness

    I am checking my homework with mathematica, but sometimes when I write stuff like D[(x/((x^2 - y^2)^0.5)), y] , which is supposed to give me the partial derivative of x/((x^2 - y^2)^0.5) with respect to y, i get answer like: (1. x y)/(x^2 - y^2)^1.5 which is right, except for the...
  17. S

    Partial differentiation question. Would all three methods work?

    I'm attaching the question and solution. I'm talking about the first part since the second part is the same just with different variables and stuff. I get what the solution is saying but: 1) What if I computed a Jacobian, with F = x^2 + xy + y^2 - z = 0 G = 2r + s - x = 0 H = r - 2s - y...
  18. A

    Vectors differentiation formulas for Dot and Box Product how?

    Let A ,B and C represent vectors. we have 1) d/dt (A . B) = A. dB/dt + dA/dt .B 2) d/dt [ A . (BxC) ] = A . (Bx dC/dt) + A . ( dB/dt x C) + dA/dt . (B xC) now the problem in these formulas is that we know that Dot product between two vectors and Scalar triple product of vectors is...
  19. S

    I'm having major difficulties with partial differentiation using the chain rule

    This is another problem than I've been stuck on for a long time and I tried reading and watching videos but I only find first order partial differentiation with more than two variables or higher order partial differentiation with only two variables. (I'm not calling f a variable but I am calling...
  20. T

    Chain rule for implicit differentiation

    I have derivative dx/dt = y(u(t)) * z(u(t)) + u(t) Now, what is dx/du ? I know the chain rule should help, but I am stuck :-(
  21. N

    How Do You Apply the Quotient Rule to Differentiate (x-1)^2/(x+1)^2?

    Homework Statement a) Differentiate y=((x-1)/(x+1))^2 b (non calculus simplification question): How would i simplify 10(5x+3)(5x-1)+5(5x-1)^2 to get 25(5x-1)(3x+1) should i expand all terms then combine and factor? Homework Equations The Attempt at a Solution a) i tried...
  22. Z

    Stationary Point after Partial Differentiation

    Homework Statement A question in a book has partially differentiated a function f(x,y) = x^2 + 8xy^2 + 2y^2 df/dx = 2x + 8y^2 = 0 at stationary point (eqn 1) df/dy = 16xy + 4y = 0 at stationary point (eqn 2) Homework Equations The Attempt at a Solution It then states...
  23. J

    Differentiation under the Integral Sign

    This is not actually a homework question but something I saw in a (economics) journal article and have been having trouble getting. It's very simple so I thought I'd post it here. Define the variable V (implicitly) by: V=b + \int^{rV}_{0} rV dF(s) + \int^{\infty}_{rV} s dF(s), where r...
  24. J

    Implicit exponential differentiation?

    Homework Statement Find an equation of the tangent line to the curve xe^y+ye^x=1 at point (0,1). Homework Equations I do not recall seeing the Implicit Function Theroem before, I even went back in my book (Stewart Calculus 6th) to check. I found this post but it does not help me...
  25. S

    Can you take a derivative with respect to the dependent variable?

    If you are given an equation y(x), where y is the dependent variable, and x is the independent variable, can you take a derivative with respect to the dependent variable? ex. y = 5x dy/dx = 5 dx/dy = 1/5 Thank you for any help
  26. S

    Interchanging summation with integral, differentiation with integral

    Hi. I've finished my undergraduate math methods courses. Many times we had problems where we had a summation and an integral both acting on the same term, and we'd switch the order of the two operations without thinking about it. The professor would always say, "I can interchange these two...
  27. 5

    Implicit partial differentiation functional det[]

    Hello! I was wondering how I could find the following derivatives from the given function using Jacobian determinants. f(u,v) = 0 u = lx + my + nz v = x^{2} + y^{2} + z^{2} \frac{∂z}{∂x} = ? (I believe y is constant, but the problem does not specify) \frac{∂z}{∂y} = ? (I...
  28. H

    Techniques of Differentiation: Applications of Derivatives

    Homework Statement We want to construct a box with a square base and we only have 10 m2 of material to use in construction of the box. Assuming that all the material is used in the construction process determine the maximum volume that the box can have. Homework Equations Chain rule...
  29. N

    [complex analysis] differentiation w.r.t. complex conjugate; does it make sense?

    Hello, Differentiability of f : \mathbb C \to \mathbb C is characterized as \frac{\partial f}{\partial z^*} = 0. More exactly: \frac{\partial f(z,z^*)}{\partial z^*} := \frac{\partial f(z[x(z,z^*),y(z,z^*)])}{\partial z^*} = 0 where z(x,y) = x+iy and x(z,z^*) = \frac{z+z^*}{2} and...
  30. H

    How do I use the chain rule to differentiate this function?

    Homework Statement Differentiate the functions using chain rule. 2(x3 −1)(3x2 +1)4 Homework Equations Chain Rule = f ' (g(x))g' (x) The Attempt at a Solution I don't know how to do using chain rule, but product rule is easier So using product rule, = f ' (x) g(x) + f (x)g'...
  31. C

    Logarithmic Differentiation for (1+x)^(1/x): Finding dy/dx

    Homework Statement Use logarithmic differentiation to find dy/dx for y=(1+x)^(1/x). Homework Equations dy/dx The Attempt at a Solution ln y = ln (1+x)^(1/x) = (1/x) ln (1+x) (dy/dx) /y = (-1/x^2)(ln(1+x) + (1/x)(1/(1+x) dy/dx = y [(-1/x^2) ( ln(1+x) + x/(1+x) ] or (-1/x^2)(...
  32. E

    Partial differentiation of cos (in vector calculus)

    Homework Statement So using standard spherical polar co-ordinates, my notes define a sphere as r(s,t) = aCos(s)Sin(t) i + aSin(s)Sin(t) j + aCos(t) k and the normal to the surface is given by the cross product of the two partial differentials: \partialr/\partials X \partialr/dt...
  33. D

    How Do You Calculate Max Relative Error in Projectile Motion?

    I'm trying to solve the problem but keep getting the wrong answer. Could someone help please? s = (v^2sin2t)/g where s = horizontal distance, v = initial velocity, t = angle, g = acc. gravity (9.81 m/s). Intended launch angle is 30 degrees, find max relative error in s if v and t are...
  34. T

    Problem with differentiation to find maximum points

    Homework Statement A sports stadium is lit by four floodlights standing at the four corners of a rectangle which contains the rectangular pitch placed symmetrically inside it. The length of the rectangle is 160 metres and the width is 64 metres. This question is concerned with finding the...
  35. S

    Can Termwise Differentiation Be Applied to the Series Ʃ(2+3i)^(2+i)n?

    Ʃ(2+3i)^(2+i)n Since there is no declared z (complex analysis) in this problem would we take this differentiation with respect to n? If so, my answer was Ʃ n(2+3i)^(2+i)n-1
  36. nomadreid

    Implicit differentiation gives too many stationary points

    If, with y a function of x, I have the equation x2-5xy+3y2 = 7, then by implicit differentiation, I get that dy/dx = (2x-5y)/(5x-6y). This equals zero everywhere on the straight line y=(2/5)x except at the origin. This would seem to indicate stationary points everywhere on that line, which is...
  37. J

    Trig Differentiation for tan2(4x): Solving 1+tan2(4x) using the Chain Rule

    Homework Statement how is the differential of 1+tan2(4x) 8*tan(4x)*sec2(4x) Homework Equations The Attempt at a Solution So 1 differentiates to 0 (we can now ignore this) tan2(4x) differentiates to 2*tan(4x)*sec2(4x)? BRING POWER FORWARD, DOWN POWER BY 1, DIFFERENTIATE TERM IN BRACKET by...
  38. D

    Possible mistake during differentiation? Please check

    For a function f(x), I have to determine intervals of increase/decrease, find local max(s)/min(s), and find intervals of concavity. The first thing I'm doing in this is to write out f'(x) and f''(x). f(x) = ln(x)/\sqrt{x} For f'(x), I used the quotient rule and received f'(x) =...
  39. W

    Differentiation, related rates?

    Homework Statement Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown? Homework Equations AB/AD = BC/DE The Attempt at a Solution I've been trying to do this problem. I looked online and saw an explanation without directly...
  40. D

    Differentiation of a polynomial function

    Hi guys I'm getting into a little trouble when differentiating polynomial functions. How do you differentiate f(x)=ax+b/cx+d ? Is there other ways of calculating this apart from the chain rule ? Thanks for any help.
  41. N

    Prove the differentiation rule: (cosx)' = sinx for any x.

    As the title says. I'm unsure what to use...limit definition? Thanks.
  42. A

    Does differentiation preserve linear independence?

    If we take the derivative of n functions that are linearly independent to each other and we write it down like c1f1(x) + c2f2(x) +...+ cnfn(x)=0, then would the linear independence be preserved if we differentiate the equation with respect to x?
  43. P

    Implicit Differentiation (for a normal derivative) with Multivariable Functions

    Homework Statement z^{3}x+z-2y-1=0 xz+y-x^{2}+5=0 Define z as a function of x, find z'. Homework Equations I guess the two equations above... The Attempt at a Solution Well, I just differentiated the first one with respect to x and got: 3z^{2}xz'+z^{3}3+1=0 z' = \frac{-1-z^{3}{3z^{2}x}...
  44. M

    Does g(x) = x|x| have an inflection point at (0;0) and g''(0) does not exist?

    Homework Statement show that g(x)= x|x| has an inflection point at (0;0) and g'' (0) Does not exist Homework Equations The Attempt at a Solution
  45. J

    Implicit Differentiation: How Can I Solve for y' in e^(x/y)=x-y?

    Homework Statement Find y' in e^(x/y)=x-y 2. The attempt at a solution I tried to differentiate it by changing it so that there would be a natural log (as seen in my attachment). However the end result is not the same as the answer key. How the answer key did it was they used the...
  46. T

    Differentiation of surface area of a right cylinder

    Homework Statement The total surface area of a right circular cylinder is given by the formula A = 2pir(r + h) where r is the radius and h is the height. a) Find the rate of change of A with respect to h if r remains constant. b) Find the rate of change of A with respect to r if h remains...
  47. T

    Implicit Differentiation and Orthogonal Trajectories

    Homework Statement Q 50: The ellipse 3x2 +2y2 = 5 and y3 = x2 HINT: The curves intersect at (1,1) and (-1,1) Two families of curves are said to be orthogonal trajectories (of each other) if each member of one family is orthogonal to each member of the other family. Show that the families of...
  48. E

    Differentiation from first principles

    Hey guys, I've got the following 2 mark question on a problem sheet, but I can't seem able to do it. I'd appreciate any help, thanks. Differentiate, from first principles, the following: y=\sqrt(a^2-x^2) I know I have to take the limit as δx tends to 0 of [(f(x+δx)- f(x)]/δx but can't seem to...
  49. J

    Differentiation by the chain rule

    Homework Statement Find the derivative of the following: Homework Equations Y= x^3(5x-1)^4 The Attempt at a Solution 4(3x^2(5x-1)^3)(4(3x^2(3(5x-1)^2)(2(5x-1)(5)
  50. L

    Differentiation and tangent line

    Homework Statement Function f(x) = KxL K= 1.78 L= -1.39 Problem 1: Find f'(x). ____________________ v= 0.89 w= 0.5 "v" and "w" are two points located on the x-axis. Problem 2: Calculate f'(v). ____________________ Problem 3: Find the equation of the tangent line of f(x) over the point "w". The...
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