Differentiation Definition and 1000 Threads

  1. H

    MHB Implicit Differentiation to find equation of a tangent line

    I need urgent help. I have this question: Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \begin{equation} {x}^{2/3}+{y}^{2/3}=4 \\ \left(-3\sqrt{3}, 1\right)\end{equation} (astroid) x^{\frac{2}{3}}+y^{\frac{2}{3}}=4 My answer is...
  2. 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 =...
  3. 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...
  4. K

    Differentiation with respect to a complex expression

    Homework Statement Homework Equations $$(x-a)(x+a)=x^2-a^2$$ The Attempt at a Solution I have to express ##~\displaystyle x^2+16=f\left( \frac{x}{x-1} \right)## I guess it has to be ##~\displaystyle \left( \frac{x}{x-1} \right)^n-a~## or ##~\displaystyle \left( \frac{x}{x-1} \pm a...
  5. Oats

    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...
  6. H

    Find the equation of a cubic graph

    Homework Statement ##x^3 - 4x^2 + ax + b## tangent to x-axis at x = 3 Homework EquationsThe 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 = 0##...
  7. M

    What is Implicit Differentiation 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"...
  8. 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. ?
  9. DavideGenoa

    I Differentiation under the integral in retarded potentials

    Hello, friends! I know, thanks to @Hawkeye18 who proved this identity to me, that, if ##\phi:V\to\mathbb{R}## is a bounded measurable function defined on the bounded measurable domain ##V\subset\mathbb{R}^3##, then, for any ##k\in\{1,2,3\}##, $$\frac{\partial}{\partial r_k}\int_V...
  10. DavideGenoa

    I Laplacian of Retarded Potential: How to Derive the Equation Mathematically?

    Dear friends, I have found a derivation of the fact that, under the assumptions made in physics on ##\rho## (to which we can give the physical interpretation of charge density) the function defined by $$V(\mathbf{x},t):=\frac{1}{4\pi\varepsilon_0}\int_{\mathbb{R}^3}...
  11. Blockade

    B Is dy/dx of x2+y2 = 50 the same as dy/dx of y = sqrt(50 - x2)?

    For implicit differentiation, is dy/dx of x2+y2 = 50 the same as y2 = 50 - x2 ? From what I can take it, it'd be a no since. For x2+y2 = 50, d/dx (x2+y2) = d/dx (50) --- will eventually be ---> dy/dx = -x/y Where, y2 = 50 - x2 y = sqrt(50 - x2) dy/dx = .5(-x2+50)-.5*(-2x)
  12. P

    MHB Edin's question via email about implicit differentiation

    (a) Differentiate both sides of the equation with respect to x: $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \left[ y^3 + y + x\,y^2 \right] &= \frac{\mathrm{d}}{\mathrm{d}x} \left[ 10 + 4\sin{(x)} \right] \\ 3\,y^2\,\frac{\mathrm{d}y}{\mathrm{d}x} +...
  13. EEristavi

    Basic Calculus: Differentiation usage

    Problem: How fast is the area of a rectangle changing if one side is I0 cm long and is increasing at a rate of 2 cm/s and the other side is 8 cm long and is decreasing at a rate of 3 cm/s?I have 2 approach and I want to know which is correct, why and what am I missing
  14. binbagsss

    Differentiation / Chain rule - Splitting Logarithms

    Homework Statement Use the top line to get 1) and 2) Homework Equations above The Attempt at a Solution So for 2) split the log up using ##log (AB)=log (A) + log (B) ## and this is simple enough I think I may be doing something stupid with 1) though. I have ##\frac{\partial}{\partial...
  15. S

    Velocity, momentum and energy values for a Pendulum swing

    Homework Statement This is my 'carrying out a practical investigation' assignment for Maths. I've attached the coursework (what I've wrote up to now) and my main concern is whether I've got the right differential equation to find 3 new velocity values throughout the pendulum trajectory...
  16. A

    MHB Differentiation of infinite series

    Find derivative of y=✓{x+✓[y+✓(x+...)]}infinite. Here root comes for total inter terms
  17. awholenumber

    B Some doubts about these differentiation steps

    First of all there is an equation Then there is the derivative Then there is a point slope formula to find the equation of the tangent line Point slope formula to obtain the tangent line . y=3a2(x-a)+a3 Then Plug in the x coordinate into the derivative to get the slope f'(1) = 3(1)2 f'(1) = 3...
  18. kupid

    MHB Calculus & Diff. Eqn: Beginner Qs on Function, Derivative & Gradient

    I have some beginner doubts about Calculus and Differential equations . Is a function always a curve ? Doesn't a function already has a slope ? d/dx of a function gives the gradient of the curve between two points ? The derivative ,d/dx ,The gradient , is the rate of change of a...
  19. Z

    I Do normal differentiation rules apply to vectors?

    (I am not very sure if this is a high-school level question or a undergraduate level question. Sorry.) Does our normal differentiation rules, like the product rule and quotient rule apply to vectors? Say for example, differentiate ##r \times \dot r## ##r## is radius vector, ##\dot r## is the...
  20. binbagsss

    Statistical Mech- basically a differentiation /integral q

    Homework Statement See attached. to get ##p## I need to differentiate ##F## w.r.t ##V##, but I also have that the upper limit ##T_{D}## depends on ##V##, so I must take this into consideration when doing the differentiation. The solution looks as though it has done this without evaluating...
  21. S

    I Understanding the Notation in Lagrangian Differentiation

    I'm just in need of some clearing up of how to differentiate the lagrangian with respect to the covariant derivatives when solving the E-L equation: Say we have a lagrangian density field \begin{equation} \mathcal{L}=\frac{1}{2}(\partial_{\mu}\hat{\phi})(\partial^{\mu}\hat{\phi}) \end{equation}...
  22. F

    Algorithmic differentiation in unsteady heat equation

    Homework Statement Okey her we go I was given a base code called heat_equation_primer. The goal is to implement a optimizer into the program. The two methods that are going to be used is the Quasi-Newton and Steepest descent with search line. So I need gradients. So I tried to differentiate as...
  23. M

    A Time differentiation of fluid line integrals

    I am looking at a proof from a book in fluid dynamics on time differentiation of fluid line integrals - Basically I am looking at the second term on the RHS in this equation $$ d/dt \int_L dr.A = \int_L dr. \partial A / \partial t + d/dt \int_L dr.A$$ The author has a field vector A for a...
  24. Debaa

    B Implicit differentiation or just explicit?

    How do I figure whether to do implicit differentiation or just explicit?? Thanks for the answer.
  25. Mr Davis 97

    I Solving a definite integral by differentiation under the integral

    Say we have the following integral: ##\displaystyle \int_0^1 \frac{\log (x+1)}{x^2+1}##. I know how to do this integral with a tangent substitution. However, I saw another method, which was by differentiating ##f## under the integral with respect to the parameter ##t##, where we let...
  26. D

    Differential calculus ,Successive differentiation

    <Moved from a technical forum, therefore no template.> How is it coming (-1)^n(p+n-1)!/(p-1)! please help...!
  27. K

    I Can Taylor series be used to get the roots of a polynomial?

    I'm using this method: First, write the polynomial in this form: $$a_nx^n+a_{n-1}x^{n-1}+...a_2x^2+a_1x=c$$ Let the LHS of this expression be the function ##f(x)##. I'm going to write the Taylor series of ##f^{-1}(x)## around ##x=0## and then put ##x=c## in it to get ##f^{-1}(c)## which will be...
  28. FritoTaco

    Implicit Differentiation: How Do I Solve \dfrac{x^2}{x+y}=y^2+8?

    Homework Statement \dfrac{x^2}{x+y}=y^2+8 Homework Equations Quotient Rule: \dfrac{g(x)\cdot f'(x)-g'(x)\cdot f(x)}{(g(x))^2} Product Rule: f(x)\cdot g'(x)+g(x)\cdot f'(x) The Attempt at a Solution \dfrac{(x+y\cdot\dfrac{dy}{dx})(2x)-(1\cdot\dfrac{dy}{dx})(x^2)}{(x+y\cdot...
  29. Math Henry

    How Long Until a Cursed Civilization's Population Reaches Zero?

    Homework Statement [/B] Summarizing: two civilizations hate each other, one of the civilizations throws a curse at the second. The second civilization succumbs to chaos and has a change in Population each week of ΔP= -√P. That is: Pn = Pn-1-√Pn-1 Homework Equations [/B] Considering that the...
  30. Theia

    MHB What is the Differentiation Challenge?

    Let's have a snack challenge for a while. ^^ Let x and y be real numbers (with restrictions y \ne 0, \ y \ne -x) and \frac{x - y}{x + y} = \frac{x + y}{y}. Find \frac{\mathrm{d}y}{\mathrm{d}x} in whatever form you like most. I mean, for example forms \frac{\mathrm{d}y}{\mathrm{d}x} = f(x, y)...
  31. M

    A Deriving Expression for Differentiation and Summation in Special Case

    Dear Friends So, i have this special case where i have to do a differentiation and summation. Please check the following. Is it okay ?? Or, i how should i proceed with this ?
  32. J

    Implicit differentiation of many variables

    Homework Statement For the given function z to demonstrate the equality: [/B]As you see I show the solution provided by the book, but I have some questions on this. I don't understand how the partial derivative of z respect to x or y has been calculated. Do you think this is correct? I...
  33. Siddharth Rajvanshi

    I Ideal Gas Law and Differentiation

    Is it possible to calculate the rate of change of n with respect to rate of change of Pressure and rate of change of Temperature with V unknown but constant by PV = nRT? Rate of change of Pressure and rate of change of temperature can be measured. R and V are constants.
  34. I

    I Why Do Implicit Differentiation Results Differ When Multiplying by x or y?

    Hey, I found a thread about part of what I'm trying to ask long ago: https://www.physicsforums.com/threads/implicit-differentiation.178328/ Basically, I noticed that if you multiply by x or by y in an equation before implicitly deriving, you get two different answers. Unfortunately their whole...
  35. Dopplershift

    Partial Differentiation Laplace Equation Question

    Homework Statement Consider the Laplace Equation of a semi-infinite strip such that 0<x< π and y>0, with the following boundary conditions: \begin{equation} \frac{\partial u}{\partial x} (0, y) = \frac{\partial u}{\partial x} (0,\pi) = 0 \end{equation} \begin{equation} u(x,0) = cos(x)...
  36. E

    A Understanding the Cost Function in Machine Learning: A Practical Guide

    Could someone please help me work through the differentiation in a paper (not homework), I am 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)...
  37. R

    Implicit Differentiation Question

    Homework Statement I am told to find dy/dx by implicit differentiation where: e^(x^2 * y) = x + y Homework Equations The above equation and the ln of it.The Attempt at a Solution e^(x^2 * y) = x + y (x^2 * y)ln(e) = ln(x+y) x^2 * y = ln(x+y) x^2(dy/dx) + y(2x) = 1/(x+y) * (1 + dy/dx)...
  38. andrewkirk

    Insights Partial Differentiation Without Tears - Comments

    andrewkirk submitted a new PF Insights post Partial Differentiation Without Tears Continue reading the Original PF Insights Post.
  39. S

    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...
  40. U

    MHB Find the derivative using implicit differentiation (with inverse trig functions)

    Here is the question: This is the step I came to after taking the derivatives and doing some simplification: ^ I did the work myself on paper, I just couldn't type out the whole thing clearly so that anyone else can see what I'm referring too... so I used some online tool to show that...
  41. Q

    Second Derivative (Implicit Differentiation)

    Homework Statement Find y'' Homework Equations 9x^2 +y^2 = 9 The Attempt at a Solution y' 18x+2y(y')=0 y'=-18x/2y y'=9x/y For the second derivative, I get the correct answer (same as the book) up until the very last step. Here's where I'm left at: -9( (-9x^2 - y^2) / y^3 ) The book then...
  42. T

    Implicit Differentiation Question

    << Mentor Note -- thread moved from the technical math forums at OP request, so no Homework Help Template is shown >> x2y + xy2 = 6 I know we use the chain rule from here, so wouldn't that be: (d/dx)(x2y + xy2) = (d/dx)(6) so using the chain rule of g'(x)f'(g(x) and the d/dx canceling out on...
  43. B

    I What is the difference between d/dx and d/d(x^2) in differentiation notation?

    Hi members, see attached PdF file. What's the difference between d/dx and d/d(x^2) I don't understand this notation?? Thank You
  44. Clara Chung

    Rate of change problem (differentiation)

    Homework Statement Refer to the photo, please verify my answer Homework Equations calculus The Attempt at a Solution For c, can I do it by assuming Ah=V. A(dh/dt) + h(dA/dt) = dV/dt then find dA/dt?
  45. Q

    Implicit differentiation (beginner)

    Homework Statement Find y' ... X^2+y^2=25I understand (I think) implicit differentiation, but there is one issue which hangs me up. I've done this before and this is just a refresher as my last calculus course was four years ago. From what I understand, 2x+2y(y')=0 But why isn't it...
  46. M

    I What is the Explanation for Implicit Differentiation Equation?

    First of all thanks for the help, i have a problem finding a good explanation of de ecuation (d/dx)f=(∂f/∂x)+(∂f/∂y)*(dy/dx) could anyone write me a good explanation of this ecuation? thanks for the help
  47. D

    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...
  48. Romanopoulos Stelios

    Differentiation of unitary operator U(t,t') in Peskin and Schroeder

    How the authors came to the conclusion (eq. 4.25) that $$ U(t,t')=e^{iH_0(t-t_0)} e^{-iH(t-t')} e^{-iH_0(t'-t_0)} $$
  49. J

    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...
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