What is Differentiating: Definition and 269 Discussions

Differentiated instruction and assessment, also known as differentiated learning or, in education, simply, differentiation, is a framework or philosophy for effective teaching that involves providing all students within their diverse classroom community of learners a range of different avenues for understanding new information (often in the same classroom) in terms of: acquiring content; processing, constructing, or making sense of ideas; and developing teaching materials and assessment measures so that all students within a classroom can learn effectively, regardless of differences in their ability. Students vary in culture, socioeconomic status, language, gender, motivation, ability/disability, learning styles, personal interests and more, and teachers must be aware of these varieties as they plan in accordance with the curricula. By considering varied learning needs, teachers can develop personalized instruction so that all children in the classroom can learn effectively. Differentiated classrooms have also been described as ones that respond to student variety in readiness levels, interests, and learning profiles. It is a classroom that includes and allows all students to be successful. To do this, a teacher sets different expectations for task completion for students, specifically based upon their individual needs.Differentiated instruction, according to Carol Ann Tomlinson, is the process of "ensuring that what a student learns, how he or she learns it, and how the student demonstrates what he or she has learned is a match for that student's readiness level, interests, and preferred mode of learning." Teachers can differentiate in four ways: 1) through content, 2) process, 3) product, and 4) learning environment based on the individual learner. Differentiation stems from beliefs about differences among learners, how they learn, learning preferences, and individual interests (Algozzine & Anderson, 2007). Therefore, differentiation is an organized, yet flexible way of proactively adjusting teaching and learning methods to accommodate each child's learning needs and preferences to achieve maximum growth as a learner. To understand how students learn and what they know, pre-assessment and ongoing assessment are essential. This provides feedback for both teacher and student, with the ultimate goal of improving student learning. Delivery of instruction in the past often followed a "one size fits all" approach. In contrast, differentiation is individually student centered, with a focus on appropriate instructional and assessment tools that are fair, flexible, challenging, and engage students in the curriculum in meaningful ways.

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  1. C

    Differentiating the equation for the mechanical energy of a spring

    Why when we differentiate ## E = \frac {1}{2}mv^2 + \frac {1}{2}kx^2 ## with respect to time the answer is ## \frac {dE}{dt} = mva + kxv ##? I though it would be ##\frac {dE}{dt} = ma + kv ##. Many thanks!
  2. ke7ijo

    B Differentiating Euler formula vs. multiplying by i

    I differentiated both sides of Euler's formula with respect to x : e^ix = sin x + i cos x => ie^ix = cos x - i sin x Then for comparison I multiplied both sides of Euler's formula by i: e^ix = sin x + i cos x => ie^ix = i sin x - cos x Each of these two procedures seems to yield the...
  3. moriah

    What Makes Human DNA Unique?

    What is in our DNA that separates us from other primates?
  4. cwill53

    I Maxwell’s Relations and Differentiating With Respect to ##N_i##

    I was wondering if anyone could write out Maxwell's relations for partial derivatives with respect to particle count ##N_i##. Starting from the fundamental thermodynamic relation, $$dU(S,V,N_i)=TdS-PdV+\sum_{i}\mu _idN_i$$ $$dU(S,V,N_i)=\left ( \frac{\partial U}{\partial S} \right )_{V,\left...
  5. murshid_islam

    I Gaussian integral by differentiating under the integral sign

    Hi, I have recently learned the technique of integration using differentiation under the integral sign, which Feynman mentioned in his “Surely You’re Joking, Mr. Feynman”. So, I decided to try it on the Gaussian Integral (I do know the standard method of computing it by squaring it and changing...
  6. qbar

    A Implicitly differentiating the vanishing real part of the hyperbolic tangent of one plus the square of the Hardy Z function

    Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to...
  7. agnimusayoti

    Differentiating $f(F,\theta)$: Why the Answers Differ?

    From the question, $$f(F,\theta)=F \cos \theta$$ 1. If I use: $$df=dF \cos{\theta} -F \sin {\theta} d\theta$$ and using radian, $$df=dF \cos{\theta} -F \sin {\theta} d\theta \frac {\pi}{180^\circ}=5.28 N$$ 2. But, if I take logarithm to both side: $$ln f=ln F+ln \cos{\theta}$$ differentiate...
  8. T

    I Differentiating the Dot Product's Evil Twin

    Hello, I found this link very useful: https://www.quora.com/Why-is-cosine-used-in-dot-products-and-sine-used-in-cross-products I understand all of Anders Kaseorg's discussion except for ONE PONT. At the very end, he writes: "[the evil twin of the dot product] is not differentiable at parallel...
  9. I

    Help please differentiating this equation from a mechanics textbook

    I tried doing it a few times and this is all I get: c(˙f1+˙f2)=a˙a2+a2˙a−3ca2+˙ha2+2ha+2˙af′2+2a˙f′2c(f1˙+f2˙)=aa˙2+a2a˙−3ca2+h˙a2+2ha+2a˙f2′+2af2′˙ Please let me know where I'm going wrong. Thanks
  10. H

    I Thermodynamics Differentiating logarithms

    <mentor: change title> In thermodynamics, there is a function which, for the three variables x, y, and z, can be given as ##G = xG_x+yG_y+zG_z + N[x\ln(x) + y\ln(y)+ z\ln(z)]+E(x,y,z)## where G_x, G_y, G_z, and N are some constants and E is some arbitrarily complicated term. There is a...
  11. V

    I Differentiating determinants

    i'll be grateful for any advice I've already tried using but it just gets massive and makes me get lost.
  12. L

    Differentiating with coordinate transformations

    T = (x+\frac{1}{\alpha}) sinh(\alpha t) X = (x+\frac{1}{\alpha}) cosh(\alpha t) - \frac{1}{\alpha} Objective is to show that ds^2 = -(1 +\alpha x)^2 dt^2 + dx^2 via finding dT and dX and inserting them into ds^2 = -dT^2 + dX^2 Incorrect attempt #1: dT= (dx+\frac{1}{\alpha})...
  13. J

    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##?
  14. Specter

    Differentiating a a rational function

    Homework Statement Find the first and second derivatives of ##\displaystyle f(x)=\frac {1} {x^2+6}## Homework EquationsThe Attempt at a Solution [/B] ##\displaystyle f(x)=\frac {1} {x^2+6}## ##\displaystyle f(x)=(x^2+6)^{-1}## ##\displaystyle f'(x)=-1(2x)(x^2+6)^{-2}## ##\displaystyle...
  15. Specter

    Differentiating this function

    Homework Statement Differentiate ##F(x)=4^{3x}+e^{2x}## Homework EquationsThe Attempt at a Solution I have an exam coming up and need some help with this problem. ##F(x)=4^{3x}+e^{2x}## ##f(x)=4^{3x}## ##G(x)=e^{2x}## First I need to find f'(x) and g'(x), which I thought I did correctly...
  16. Safinaz

    Differentiating a scalar potential

    Hello, I have this potential: ## V(\chi) = \frac { F ’’ (\chi) [ 2 F(\chi) - \chi F’ (\chi) ]}{ (F’(\chi))^3} ## How to get ## \frac{ d V(\chi)}{d \chi} = \frac{ \chi F’’ + F’ - F’ }{ F’^2} - 2 \frac{ \chi F’ -F }{ F’^3} F’’ ~~~~~~(*)## My trail, ## V( \chi) = 2 F F’’ F’^{-3} - \chi F’’...
  17. Mr Davis 97

    I Differentiating under the integral sign

    I am looking at a solution to an integral using differentiation under the integral sign. So let ##\displaystyle f(t) = \frac{\log (tx+1)}{x^2+1}##. Then, through calculation, ##\displaystyle f'(t) = \frac{\pi t + 2 \log (2) - 4 \log (t+1)}{4(1+t^2)}##. The solution immediately goes to say that...
  18. D

    I Differentiating vector dot product

    Hi. If I have a vector v , say for velocity for example then v.v = v2 and I differentiate wrt t v.v I get 2v.dv/dt but if I differentiate v2 I get 2v dv/dt but v.dv.dt is not the same as v dv/dt so what am I doing wrong ? Thanks
  19. Poetria

    Differentiating a power series

    Homework Statement [/B] Differentiate the power series for ##\frac 1 {1-x}## to find the power series for ##\frac 1 {(1-x)^2}## (Note the summation index starts at n = 1) 2. The attempt at a solution ##\sum_{n=1}^\infty n*x^{n-1}##
  20. Specter

    I'm having some trouble differentiating radical functions

    Homework Statement Find the derivative of the following functions. h(x)=8x2√x2+1 Homework EquationsThe Attempt at a Solution I just had a lesson on friday but it was a bit confusing to me. I am able to solve some problems but ones that contain multiple rules confuse me. If someone could...
  21. Y

    MHB Differentiating wrt constant to evaluate integral

    What is the justification for differentiating some integrals with respect to constants in order to obtain result, i.e. ∂/∂a[∫e^(− ax^2).dx] =∫-x^2.e^(-ax^2) dx?I mean what if we say "a" was 3 then differentiating wrt 3 would have no significance?How can we treat it like a multivariable function :/
  22. DavideGenoa

    I Differentiating a particular integral (retarded potential)

    Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...
  23. R

    Alternate way of differentiating x^y

    Homework Statement : [/B]find the dy/dx of xy=a constantHomework 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 yxy-1.dy/dx =0. ?
  24. SamRoss

    B Differentiating by two different variables -- when is it okay?

    I'm working through the problems in Mary Boas's Mathematical Methods text. Here's how she began solving one problem... "We take differentials of the equation 1/i + 1/o = 1/f (f=constant) to get -di/i2 - do/o2 = 0." So on the left side the first term was differentiated with respect to i and...
  25. Eri ep

    I Differentiating Biot-Savart Law

    Hello! I have the equation FM = qvBsinθ . As the end result, I am trying to figure out what B I need to change θ even a little bit. To do that, I was planning to find the minimum B by differentiating B=(μe/4π)(qv x R / R3) in terms of R and setting it equal to zero. . I am assuming that this...
  26. G

    I Differentiating Vectors and their Magnitudes in Physics

    I've taken up to Calc. III (vector/multi-variable calculus) but have not had any classes that used intensive calculus for a few semesters. However, I'm now in a quantum physics class (its a glorified title really--it's more or less "modern physics") and am looking to see if I am correct in my...
  27. T

    MHB Differentiating a power series

    I need to prove that for $-1 < x < 1$ $$\frac{1}{(1 - x)^2} = 1 + 2x + 3x^2 + 4x^3 ...$$ So, according to the textbook, the geometric series has a radius of convergence $R = 1$ (I'm not sure how this is true). In any case we can compare it to: $$\frac{1}{1 - x} =\sum_{n = 0}^{\infty} x^n$$...
  28. M

    MHB Differentiating (1+sin²x)³: A Step-by-Step Guide

    can someone go through the steps how to differentiate (1+sin²x)³
  29. C

    Differentiating an Implicit Function: a Circle

    Homework Statement Find the expression for the slope on the lower half of the circle y^2 + x^2 = 25. 2. Attempt at a solution. The text says you get 2x + 2y(dy/dx) = 0. I got this and then solved for dy/dx to get dy/dx = -2y - 2x. Then, I substituted for y the x value-expression for the...
  30. C

    Differentiating Vectors

    Homework Statement I was reading about the Hermite integration scheme for N-body simulations, as seen here: http://www.artcompsci.org/kali/vol/two_body_problem_2/ch11.html#rdocsect76 This scheme uses jerk, the time rate of change of acceleration. The problem is that I don't know how to...
  31. Trance-

    B Differentiating some simple harmonic equation

    So I was just trying to differentiate (for no good reason) the equation : x=x0sin(wt) (w= angular frequency, x0= maximum displacement, t=time) to obtain the expression : a= -w2x I differentiated twice with respect to time the initial expression for x and got: a= -w2x0sin(wt) I must have...
  32. DavideGenoa

    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...
  33. Z

    Differentiating an exponential with a complex exponent

    Hello, folks. I'm trying to figure out how to take the partial derivative of something with a complex exponential, like \frac{\partial}{\partial x} e^{i(\alpha x + \beta t)} But I'm not really sure how to do so. I get that since I'm taking the partial w.r.t. x, I can treat t as a constant term...
  34. I

    Differentiating Under Integral Sign

    I(α) = 0∞∫e-(x2+α/x2) dx Differentiating under the integral sign leads to: I(α) = 0∞∫-e-(x2+α/x2)/x2 dx Here I am supposed to let u = sqrt(a)/x, but the -x2 doesn't cancel out, Wolfram-Alpha tells me the answer is: e(-2 sqrt(α) sqrt(π))/(2 sqrt(α)). I understand where the sqrt(π))/(2)sqrt(α)...
  35. S

    Differentiating an integral and finding f(x)

    Homework Statement Find all f(x) satisfying: ∫f dx ∫1/f dx = -1 Homework EquationsThe Attempt at a Solution I solved for ∫1/f dx and differentiated both sides (using the quotient rule for the right side): ∫1/f dx = -1 / ∫f dx 1/f = f / (∫f dx)2 (∫f dx)2 = f2 ∫fdx = ±f f = ±f' Solving the...
  36. T

    MHB Differentiating tan2x: Steps & Solutions

    How do you differenciate tan2x and what are the steps?
  37. RUber

    Differentiating Integral with Green's function

    Hello, I am having trouble finding the proper justification for being able to pass the derivative through the integral in the following: ## u(x,y) = \frac{\partial}{\partial y} \int_0^\infty\int_{-\infty}^\infty f(x') K_0( \sqrt{ (x - x')^2 + (y-y')^2 } \, dx' dy' ## ##K_0## is the Modified...
  38. D

    Differentiating: Clarifying the Process

    Homework Statement Differentiate Homework EquationsThe Attempt at a Solution du/dx = cos(x) dv/du=cos(u) dg/dv=cos(v) dg/dx = dg/dv.dv/du.du/dx = cosx.cos(sinx).cos(sin(sinx)) I know the answer is correct but my issue is in the understanding of the...
  39. S

    Differentiating the relativistic charge wrt velocity

    Homework Statement Given that ##Q' = \int d^{3}x \bigg[ \rho \big( \frac{vx}{c^{2}},x,y,z \big) - \frac{v}{c^{2}} j^{x} \big( \frac{vx}{c^{2}},x,y,z \big) \bigg]##, show the following: ##\frac{dQ'}{dv} = \frac{1}{c^{2}} \int d^{3}x \bigg[ x \big( \frac{\partial \rho}{\partial t} -...
  40. M

    Differentiating maps between vector spaces

    I'm trying to understand the definition of maps between vector spaces (in normed vector spaces) listed in the following link: http://ocw.mit.edu/courses/mathematics/18-965-geometry-of-manifolds-fall-2004/lecture-notes/lecture3.pdf On the surface, this seems similar to what I expected from the...
  41. J

    Differentiating Lagrangian in Tensor Notation

    Homework Statement Hi all, I'm trying to learn how to manipulate tensors and in particular to differentiate expressions. I was looking at a Lagrangian density and trying to apply the Euler-Lagrange equations to it. Homework Equations Lagrangian density: \mathcal{L} = -\frac{1}{2}...
  42. Drakkith

    Implicit Differentiation: Differentiating in Terms of X

    I'm having some trouble with the terminology used in calculus. My book states: "Fortunately we don't need to solve an equation for Y in terms of X in order to find the derivative of Y. Instead we can use the method of implicit differentiation. This consists of differentiating both sides of the...
  43. D

    Adding resistors to a differentiating op-amp?

    I know how to derive the expression Vout = -CR*d(Vin)/dt for a circuit differentiator that consists of a capacitor and a resistor, how I can understand the attached circuit, which is part of practice exercise for my class. Why do we need to add R2 and R3? When I run the simulation Vout =...
  44. T

    Differentiating like vs. opposite charges

    Can experiments differentiate like vs. opposite charges? Two electrons repel, whereas an electron and positron attract. But for macroscopic observers, in the absence of annihilation, could anyone tell whether paths deflected due to attractions or repulsion? Or, is there always annihilation...
  45. M

    PDE and differentiating through the sum

    Hi PF! I'm reading my math text and am looking at the heat eq ##u_t = u_{xx}##, where we are are given non-homogenous boundary conditions. We are solving using the method of eigenfunction expansion. Evidently we begin by finding the eigenfunction ##\phi (x)## related to the homogenous...
  46. wolfspirit

    Differentiating cotx: Help Solving Wrong Answer

    I keep getting the wrong answer when i try to differentiate cotx.. this is what i get: cotx = 1/tanx =cosx/sinx=cosx ⋅ sin^-1 so by the product and chain rule we have: sinx⋅(sin x)^-1+cos⋅(-1sin^2 x)^-1 ⋅(cosx)^-1 = sinx/sinx - cosx/cosx ⋅ sin^2x =1-1/sin^2 x where as the correct answer...
  47. T

    MHB Differentiating a trignometric function

    So, I have $$\d{x}{t} = 20 sex^2∂ \d{∂}{t}$$ And the text goes to: $$ \d{∂}{t} = \frac{1}{20}cos^2 ∂ \d{x}{t}$$ I don't understand where the cos comes from? Is it a trigonometric identity? If so I can't find it.
  48. J

    Differentiating semi-classical Kubo trans. correlation functions wrt t

    I do not fully understand how non local correlation functions are derived from the differentiation of local ones. Specifically how in the following paper equation 23 is derived from 21 since the paper implies H and dpdq are invariant under time...
  49. S

    Differentiating complex exponential

    I asked to differentiate the given function using exponential function with sin(√3t + 1) I turned it into Im[e^(√3t+1)i] then I multiplied it by e^t which gave Im[e^t*e^(√3t +1)i] then I applied usual algebra to differentiate but I get a (t+√3ti +i) as the power of e when I try...
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