Differentiation Definition and 1000 Threads

An example of implicit differentation in Stewart, 6th ed, p 883, is given as follows: x^3 + y^3 + z^3 + 6xyz = 1 Differentiating to find dz/dx, 3x^2 + 3z^2(dz/dx) + 6yz + 6xy(dz/dx) = 0 where the product rule was used to differentiate 6xyz with respect to x. Why isn't the...

Homework Statement I've been reading a textbook on tensor analysis for a while. The book uses the conclusion of "covariant differentiation commutes with contraction" directly and I searched around and found most people just use the conclusion without proof.Homework Equations For example...

1. I'm pretty sure I've got this right.. just needing a check. Cheers! 2. x^2+2xy-y^2+x^2=2 3. 2x+2x(dy/dx)+2y-2y(dy/dx)+2x=0 2x(dy/dx)-2y(dy/dx)=-4x-2y dy/dx(2x-2y)=-4x-2y dy/dx=(-4x-2y)/(2x-2y)

Homework Statement I am using Crank–Nicolson to solve a logistic function, modeling population growth. To get the next time step, I have to solve a quadratic equation. The problem is that i get two solutions for y(i+1). Does it mean that I am doing it wrong? If not, can I just pick the...

Homework Statement The curve C has equation y=k x^3-x^2+x-5 where k is a constant. A) Find the derivative of the function with respect to x The point A with x-coordinate -\frac{1}{2} lies on C. The tangent to C at A is parallel to the line with equation 2 y-7 x+1=0 . Find...

Consider the power series (n=1 to infinity) \Sigma (x^n)/(n*3^n). (a) Find the radius of convergence for this series. (b) For which values of x does the series converge? (include the discussion of the end points). (c) If f(x) denotes the sum of the series, find f'(x) as explicitly as...

Find the derivative of the following: y = 1/ COS(t^2) Homework Equations Here is my attempt Using (Cos(t^2))^-1 SOlve using the chain rule: u = t^2 du = 2t dy/dt = -1.(-sin(u))^(-2) .(2t) = -2t(-sin(u))^(-2) = -2t/(Sin(t^2))^2 Is this correct...

If xlny - y = 5e then find the value of dy/dxat the point(6e,e) Homework Equations So here is my attempt at the solution, please any advice on where I am going wrong or if indeed it is correct greatly appreciated xlny - y - 5e Therefore dy/dx is: x/y + lny - 1 -5e Substituting...

Homework Statement What is the Importance of Cell Differentiation in the formation of tissue? Homework Equations N/A The Attempt at a Solution I know that cell differentiation is the process in which a generic cell develops into a specific type of cell, but could you please explain...

In the process of deriving .\nabla^2u(x,y). in polar coordinates I am confuse how to work through the steps. The first step from the book is to find \frac{\partial u}{\partial x} Standard partial differentiation is at follow: \frac{\partial u}{\partial x} = \frac{\partial u}{\partial r}...

f: (0,+inf)->R and f(x) is 0 if x is irrational 1/n if x is rational (n is positive integer) For each rational and irrational, i want to show continuity/discontinuity of f Intuitively, i think at each rational f is discontinuous, and at each irrational f is continuous, but i...

Homework Statement I have a few problems that are giving me some trouble: 1. Take the derivative of xe-4x 2. Find dy/dx and evaluate the slope for the curve ey^3 - 2x4 + y2 = 3 at (8,0) 3. Find dy/dx and evaluate the slope for the curve e-y - 4 = x2 + 1 at (-2,2) Homework Equations N/A...

Homework Statement Find the points on the lemniscate: 2( x^2 + y^2 )^2 = 25( x^2 - y^2 ) where the tangent is horizontal Homework Equations Horizontal tangent: y' = 0 The Attempt at a Solution I got the correct gradient of y' = [ 50x - 8x^3 - 8y^2 ] / [ (8x^2)y + 50y + 8y^3 ]...

Homework Statement If y= \ln \sqrt{xy} , find the value of dy/dx when y=1 Homework Equations The Attempt at a Solution \frac{dy}{dx} = \frac{1}{\sqrt{xy}} \cdot \frac{1}{2\sqrt{xy}}\cdot (x\frac{dy}{dx}+y) \frac{dy}{dx}=\frac{1}{x} , when y=1

Hi there If possible can someone please review my understanding of this question. Homework Statement At what point on the Parabola y= 2x^2-7x-15 will a normal to the parabola have a gradient of -1/2 Homework Equations The Attempt at a Solution We are given the gradient of the...

Homework Statement If x(t) = cos2t ; y(t) = t - tant Show that: (dy/dx)^2 = 1- x / 4(1+x)^3 The Attempt at a Solution dy/dx = dy/dt * dt/dx = (dy/dt) / (dx/dt) dy/dt = 1- sect^2 ; dx/dt = -2sin2t So (dy/dx)^2 = (1 - sect^2)^2 / (4 sin2t^2) From here I'm not too...

I am confuse with this: Let\;\; t=jx \;\;,\;\; j =\sqrt{-1} \Rightarrow \frac{dt}{dx}=j But if I use t^2=-x^2 \;\Rightarrow\; 2t\frac{dt}{dx}=-2x \Rightarrow \frac{dt}{dx}=-\frac{x}{t} What is wrong?

Homework Statement Differentiate the following function with respect to x, p(x) = (( x+5 )^2)*(( x+3 )^7) Homework Equations well the product rule is, p(x)=(f)*(g) p'(x)= (f')*(g)+(g')*(f) and general differentiation is, p'(x)=n(f)^(n-1)*n(g)^(n-1) The Attempt at a Solution well...

Homework Statement Determine dy/dx xy^-2 = 1 Homework Equations The Attempt at a Solution Here's what I tried: y^-2 - 2xy^-3(dy/dx) = 1 dy/dx = (y - y^3)/2x The answer in the book is y/2x. How do you do it?

Homework Statement Find the equation of normal to curve x=cos b , y=sin b at the point where b=pi/4 . Find the coordinates of the point on the curve where the normal meets the curve again . Homework Equations The Attempt at a Solution Gradient of normal : dx/db=-sin b...

Homework Statement [PLAIN]http://img408.imageshack.us/img408/7163/partialdifferent.jpg So this means differentiate w.r.t y first, so I want dz/dy, and then w.r.t x right? so I rearrange so that y=z3/3 + xz and differentiate w.r.t z to get dy/dz, and then do 1 over this which i get as...

Dear all, a question which has puzzled me for some days: (Assume that all are differentiable enough times): Calculate: \frac{\mathrm{d} }{\mathrm{d} x}\int_{g(x)}^{h(x)} f(x,t) dt

Homework Statement If siny=2sinx and (dy/dx)^2=1+3sec^2(y) show that: by differentiating 1+3sec^2(y) with respect to x, d^2y/dx^2=3sec^2(y)tan(y) Homework Equations The Attempt at a Solution

I am self studying for my AP Calc exam. I have just started on integration and I need a few pointers for a few questions. So here it is... y= (e^x)/(1+(e^x)^2) So I set u=e^x du=e^x then I am stuck. the second question is (btw [{( are all used as parentheses to make it abit less...

Hi there to all. I am stuck and i want some help to clear my view When you have defined the vector S=(S_x ,S_y ,S_z) and you have a function like E_h = k1 (S_x^2 S_y^2 + S_x^2 S_z^2 + S_y^2 S_z^2) + k2 S_z - k3 S_x S_y and you try to calculate -\frac{\partial E_h}{\partial S} what the...

Homework Statement This is from Spivak's Calculus on Manifolds, problem 2-12(a). Prove that if f:Rn \times Rm \rightarrow Rp is bilinear, then lim(h, k) --> 0 \frac{|f(h, k)|}{|(h, k)|} = 0 Homework Equations The definition of bilinear function in this case: If for x, x1, x2...

Hi, so first post on this forum so I hope I'm doing everything good-as-gold! I've been twisting my head around a derivation of the uncertainty principle and I'm a little stumped on something that I feel I really should know. Hope someone would like to help. Certain things are defined at...

hey there.. i've got unsolved solution here.. Solve initial value problem.. xy’ – y = 2x2y ; y(1) = 1 x dy/dx = y + 2x2y dy/dx = y/x + 2xy dy/dx = y[(1/x) + 2x] ∫1/y dy = ∫ (1/x) + 2x dx ln |y| = ln |x| + x2 + C apply exp to both side y = x + ex^2 + e^C...

hey there.. i've got unsolved question here.. Question: Solve the initial value problem. 2yy’ = x / √x^2 – 16 ; y(5) = 2 2yy’ = x / √x^2 – 16 2y dy/dx = x / √x^2 – 16 ∫2y dy = ∫ x / √x^2 – 16 dx y^2 = √x^2 – 16 y = (√x^2 – 16)^1/2 my answer was wrong. the answer is y = ((√x^2 –...

Homework Statement When I write 'd' I mean the partial differentiation symbol 'del' here... Find dz/dx and dz/dy of: z=f(xy) I'm guessing this isn't a case of simply pretending the 'f( )' isn't there, how do I approach this problem? Homework Equations The Attempt at a Solution

Homework Statement Find the derivative of y= x^2 (x^2 - 1)^3 (x+1)^4 using logaritmic differentiation. Homework Equations The Attempt at a Solution y= x^2 (x^2 - 1)^3 (x+1)^4 = 2 ln|x| + 3 ln |(x^2)-1|+ 4 ln |x+1| 2/x + (6x)/(x^2-1) + 4/(x+1) Is this right and if so...

Homework Statement Water flows from a tank of constant cross-sectional area 54 ft2 through an orifice of constant cross-sectional area 1.7 ft2 located at the bottom of the tank. Initially the height of the water in the tank was 20 and its height t sec later is given by the following...

Homework Statement Find y''(x) of the parametric equation 9x^2+y^2=9 using implicit differentiation. Homework Equations I already came up with y'(x) = -9x/y The Attempt at a Solution Here is what I have for y''(x) so far y''(x) = d/dx (-9xy^-1) =-9(d/dx)(xy^-1)...

I am in the middle of finding a general solution for an equation . However I am stuck here: c^2 ∫ dv/ (v^2 -c^2) = ∫ g dt I know Partial diiferentiation would be the best approach however I cannot really get started. Help appreciated Homework Equations The Attempt at a Solution

Homework Statement Could anyone explain that I got two different answers for this question: find dy/dx of \frac{x}{x+y}-\frac{y}{x}=4 Homework Equations The Attempt at a Solution 1. using quotient rule: \frac{x+y-(1+dy/dx)x}{(x+y)^{2}}-\frac{x\frac{dy}{dx}-y}{x^{2}}=0...

Could anyone explain that I got two different answers for this question: find dy/dx of \frac{x}{x+y}-\frac{y}{x}=4. 1. using quotient rule: \frac{x+y-(1+dy/dx)x}{(x+y)^{2}}-\frac{x\frac{dy}{dx}-y}{x^{2}}=0...

The function of a line is y^2 + x^3 = 9. I calculated the slope of its tangent to be -3x^2/2y. The question asked us to find a point(s) so that the equation of its tangent line is y + 6x = 13. So it's slope must be -6 at that point. I got (2,1) as a point. Are there more than one, or is that...

Hi I have an equation where x is a distance that varies with time, and i want to differentiate the expression again wrt to time, but I am struggling: \frac{dT}{dx^{.}}=Ir^{2}x^{.}\left[\frac{-(r^{2}+x^{2}-l^{2})2r-4rx^{2}}{\sqrt{1-(\frac{r^{2}+x^{2}-l^{2}}{2rx})4r^{2}x^{2}}}\right]^{2} i...

Homework Statement Use implicit differentiation to find an equation of the tangent line to the cardioid at the point (0, 0.5). x2 + y2 = (2x2 + 2y2 - x)2Homework Equations Derivative rules point slope formula The Attempt at a Solution I got y' = [16x3-4x2+16xy2-4y2-4y2-8x2+2x] / [2y -...

Homework Statement 2x^{2} - 3y^{2} = 4 Homework Equations We say that y is an implicit function of x if we are given an equation: \sigma(x,y) = \tau(x,y) Then to differentiate we do: \frac {d(\sigma(x,y))} {dx} = \frac {d(\tau(x,y))} {dx} The Attempt at a Solution...

Homework Statement Find the value of the following derivative at x=1: d/dx(2v/u) where u(1)=3, u'(1)=-4, v(1)=-2, v'(1)=5 The Attempt at a Solution Differentiating the equation: [(u)(2v') - (u')(2v)]/(u^2) plugging in the values i get 14/9 as the value. now my...

Homework Statement If z = f(x-y), show that dz/dx + dz/dy = 0 2. The attempt at a solution I thought: dz/dx = fx dz/dy = -fy which doesn't make sense really... because its not equal to 0. or maybe it should be: dz/dx = dz/df * df/dx = fx * ?? dz/dy = dz/df * df/dy = -fy * ??

Homework Statement Let V be the space of polynomials with degree \leq n (dimV=n+1) i. Let D:V->V be differentiation, i.e. D: f(x) -> f'(x) What are the eigenvalues of D? Is D diagonalisable? ii. Let T be the endomorphism T:f(x) -> (1-x)2 f''(x). What are the eigenvalues of T? Is...

I have a function z=f(xz+y) and I want to find the partial differential of z with respect to y (it's a general sort of question, I only need it in terms of the variables already given). My answer would be just partial df/dy but this isn't the right answer. I'm not too hot on partial...

Homework Statement Find f', where f(x) = 1+x^2)^(x^2) Homework Equations The Attempt at a Solution I attempted the question using the chain rule but I was told that you do not use the. That your meant to use logs. But I have no idea where to begin. Any help or tips would be much...

Homework Statement find the sum for \sum_{k=1}^{\infty} kx^{k} Homework Equations \sum_{k=0}^{\infty} x^{k} = \frac{1}{1-x}; -1 < x < 1 The Attempt at a Solution \sum_{k=1}^{\infty} kx^{k} = \sum_{n=0}^{\infty}(n+1)x^{n+1} = x\sum_{n=0}^{\infty} (n+1)x^{n} = x \frac{d}{dx}...

ive been given this question for a uni assignment: given the function: f (x) = 5(x^1.3) +1.5(7x − 3)+ 3(e^− x) + ln(2.5(x^3)) find the first derivative at all possible points within the interval [0, 6], with step length h = 1 for: forward difference aproximation, backward difference...

Homework Statement Find the derivative of \frac{sin x}{1 + cos x} Homework Equations Quotient rule \frac{gf' - fg'}{g^{2}} The Attempt at a Solution \frac{dy}{dx} = \frac{(1 + cos x)(\frac{d}{dx}(sin x)) - sin x(\frac{d}{dx}(1 + cos x)}{(1 + cos x)^{2}} simplify the...

In Wikipedia, http://en.wikipedia.org/wiki/Noether%27s_theorem#One_independent_variable You can see the proof of Noether's theorem for the system that has only one symmetry. I can't do the calculation of this, for \frac{dI'}{d\epsilon} = \frac{d}{d\epsilon} \int_{t_1+\epsilon...

Homework Statement I have to evaluate the numerical value of the derivative of the following integral for x=1 \int_{0}^{\ ln\ x}\ e^{\ -\ x\ (\ t^2\ -\ 2)}\ dt Homework Equations The formula for differentitation under integral sign. The Attempt at a Solution The upper limit...