Differentiation Definition and 1000 Threads

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...

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...

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...

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...

Homework Statement Find a function of the form f(x) = a + b \cos cx that is tangent to the line y = 1 at the point (0,1), and tangent to the line y = x + 3/2 - \pi /4 at the point (\pi /4 , 3/2). Homework Equations The Attempt at a Solution f(0) = a + b = 1, so a = 1 - b...

Homework Statement Use implicit differentiation to find dy/dx if y - sin(xy) = x^2. What I've got is dy/dx y - cos(xy)(y+x dy/dx) = 2x I don't know what I did and I don't know where to go from here.

Homework Statement The vector a depends on a parameter t, i.e. a=a(t)=a_x(t)i +a_y(t)j +a_z(t)k.. it satisfies the equation da/dt= j \times a show that d^2a_x/dt^2 =-a_x , da_y/dt=0 and d^2a_z/dt^2 =-a_z. For the vector a, find its value for t=pi if at t=0 a(0)=i+j and da/dt(0)=0k...

Homework Statement Hey >.< another simple problem; find \frac{\partial u}{\partial y} , \frac{\partial v}{\partial y}, \frac{\partial w}{\partial y} given that u - v + 2w = x + 2z 2u + v - 2w = 2x - 2z u - v + w = z - y The Attempt at a Solution I don't know, this question is really...

Homework Statement Given y = x^2 + 6 / x find x if dy/dx equals 0 zero Homework Equations dy/dx = v du/dx - u dv/dx / v^2 The Attempt at a Solution I have got as far as dy/dx = x^2 - 6 / x^2 However then zero must equal Sqrt 6, which 2.44... -...

Homework Statement Find \frac{\partial z}{\partial x} \frac{\partial z}{\partial y} where z=\left( [x+y]^3-4y^2 \right)^{\frac{1}{2}}Homework Equations -The Attempt at a Solution I know that \frac{\partial z}{\partial y}=\frac{3(x+y)^2-8y}{2\sqrt{(x+y)^3-4y^2}} but I am unsure whether...

Homework Statement find \frac{\mathrm{d}y}{\mathrm{d}x} where y is defined implicitly as a function of x Homework Equations x\sin(xy)=x The Attempt at a Solution x(\cos(xy)(x\frac{\mathrm{d}y}{\mathrm{d}x}+y))+\sin(xy)=1...

Homework Statement Prove that if F is a twice differentiable function with F(0) = 0 and F(1) = 1 and F'(0) = F'(1) = 0, then |F''(x)| >= 4 for x in (0,1). Hint: Prove that either F"(x) >= 4 for some x in (0,1/2) or else F"(x) <= -4 for some x in (1/2,1) Then, show that we...

Homework Statement Differentiate the following cos^2 x The Attempt at a Solution \frac{d(cos^2 x)}{dx} = 2 (-sin^{2-1} x).\frac{dx}{dx} = 2 (-sin x).1 = -2 sin x But the answer given is- -2 cos x sin x Can you please tell me what's wrong !

Homework Statement How do i differentiate a sum in MATLAB? So for an example, if I have: y = x^2 + \sum^{n}_{j=1}3x_{j}^4 I want to differentiate it with respect to x. It should return: 2x + \sum^{n}_{j=1}12x_{j}^3 Homework Equations The Attempt at a Solution If I want...

Homework Statement If x = yz and y = 2sin(y+z), find dx/dy and d^2x/dy^2 Homework Equations The Attempt at a Solution I am beyond confused at how to even start this one, the problem is not like any of the examples in my book. I know x = ydz + zdy but don't know how to deal...

The velocity of a particle moving in a plane in polar coordinates is {\bf{v}} = v_r {\bf{\hat r}} + r\omega \hat \theta where v_r = \frac{{dr}}{{dt}} and \omega = \frac{{d\theta }}{{dt}}. By differentiating w.r.t. time, show that the acceleration of the particle is {\bf{a}} =...

Homework Statement http://img94.imageshack.us/img94/3853/physicse.jpg The Attempt at a Solution I kept y fixed, and so I ended up with the following equation: Integ[dU/U] = Integ[x] And we end up with: U(x,y) = e^x * g(y) To solve g(y), we sub the solution into the 2nd PDE provided to...

Hello, I`m studying 1st year physics and I would really like to improve my basic maths. The thing is I don`t want to sit and first make up a random expression and then calculate the derivative, I would like to be able to print out hundreds and hundreds of small expressions with varying...

Prove that if P(a,b) is a point on the rotated ellipse C (whose equation is x^2 -xy +y^2=4 ), then so is Q(-a,-b), and that the tangent lines to C at P and Q are parallel. The equation of the line joining P and Q is y - b = m(x - a), where m= \frac{b-(-b)}{a-(-a)} = \frac{b}{a}, then the...

Homework Statement A swimming pool is 40 feet long, 20 feet wide, 4 feet deep at the shallow end, and 9 feet deep at the deep end. Water is being pumped into the pool at 10 cubic feet per minute. a. When the water is 3 feet deep at the shallow end, at what rate is the water level rising? b...