Differentiation Definition and 1000 Threads

So, as the title may have given away, I'm trying to figure out implicit differentiation in the multiple variable context. I thought a good practice would be the law of cosines, aka c^2 = a^2 + b^2 - 2abcosθ. So I'm trying to find ∂θ/∂a, ∂θ/db, ∂θ/dc. I tried solving for θ and then taking...

Homework Statement Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x^2+y^2=(2x^2+2y^2-x)^2 @ (0,(1/2)) Homework Equations The Attempt at a Solution So, I don't have a problem differentiating this. Which I get...

Homework Statement Calculate the derivative with respect to x: x/y + y/x = 2yHomework Equations n/a The Attempt at a Solution I end up getting the right answer, but what I want to know is how to isolate dy/dx to one side after implicitly differentiating. I have tried differentiating the LHS...

Homework Statement The P(a,b) be a point on the curve √x + √y = 1. Show that the slope of the tangent P is -√b/a Homework Equations ? The Attempt at a Solution Apparently this is an implicit differentiation problem, however we haven't learned or discussed implicit...

Say you have x^2+y^2=100. why can't you just solve for y, so y=+- √(100-x^2) then use the chain rule to find the derivative. so y'= +- x/√(100-x^2). Then you can just deduce that y'= -x/y. What is the point of adding all the dy/dx in the equation? Seems like it just complicates it.

Hello, I'm trying to understand this proof: http://en.wikipedia.org/wiki/Proofs_involving_ordinary_least_squares#Least_squares_estimator_for_.CE.B2 Can someone quickly talk me through the differentiation step, bearing in mind I've never learn how to differentiate with respect to a vector...

Let $E\subset\mathbb{R}^n$ and $f: E\rightarrow\mathbb{R}$ be a continuous function. Prove that if $a$ is a local maximum point for $f$, then either $f$ is differentiable at $x = a$ and $Df(a) = 0$ or $f$ is not differentiable at $x = a$. Deduce that if $f$ is differentiable on $E^o$, then a...

Homework Statement I'm trying to take the derivative of the following integral \frac{d}{d V} \int_0^t{V(\tau)}d\tau Homework Equations FTC will probably be a part of it. The Attempt at a Solution I always get confused when I'm taking the derivative of an integral. I know the answer is...

Hi guys! Let \left \{ B_{t} \right \}_{t\in \mathbb{R}} be a one - parameter family of compact subsets of \mathbb{R}^{3} with smooth (manifold) boundary (e.g. one - parameter family of closed balls). In my context, each B_{t} belongs to a different constant time slice of Minkowski space - time...

$\mathcal{E} = \frac{1}{2}m\dot{x} + \frac{1}{2}kx^2$ The derivative is $$ \frac{d\mathcal{E}}{dt} = \frac{1}{2}m\ddot{x} + kx\dot{x} $$ but the solution is suppose to be $$ \frac{d\mathcal{E}}{dt} = \dot{x}(m\ddot{x} + kx). $$ How?

Hello! As of right now (10:13 PM), I've tried 9 combinations of points to solve this problem. It's a WebWork-based problem that's due in about an hour in a half. Any help would be very, very appreciated. Homework Statement I was given this equation: ##ln(2y) = 2xy## and was asked to find...

I've read about it before and now I'm trying to learn it myself from Woods 'Advanced Calculus' (as well as other resources like http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf) In the pdf, it says the method concerns integrals that depend on a parameter...now couldn't we...

Homework Statement The line segment AB lies on a diameter of a circle of radius 1, and the angle BAC is a right angle. Find the greatest possible value of the sum of the lengths of AB and AC. Homework Equations The Attempt at a Solution I have no idea what parameters to use...

Homework Statement x(t) = (t^2 -1) / (t^2 +1) y(t) = (2t) / (t^2 +1) at the point t=1 Homework Equations Line equation = y-y1 = m(x-x1) chan rule = (dy/dt) / (dx/dt) = dy/dx The Attempt at a Solution I find the y1 and x1 values by subing in t=1 to the x(t) and y(t)...

Hi, I am a new user of Mathematica (although I am reasonably familiar with MATLAB) and I am trying to differentiate a scalar wrt a vector Mathematica. ie, I want to check if \begin{equation} \phi = \textit{x}^{T} \textbf{A} \textit{y} \quad \mbox{where $\textit{x}$, $\textit{y}$ are vectors...

Homework Statement Find f^n for f(x) = Ln(2x+1) Can anyone point me in the right direction with how to get the nth derivative of the above function please, I just cannot seem to work this out! Thank you

Please see attached. I was looking for an explanation of the answer I have attached. Its been a little while and was just looking for the logic behind the differentiation shown for this problem. Its basically an optimization problem where I am looking for the minimum angle (theta) for the...

Hi:Let $f,g$ and $h$ be continuous real valued function with domain $X \times Y$ where $X$ and $Y$ are compact sets. Let me define the set $S(x) = \{ y : h(y,x) \geq 0 \} $ and $\bar{S}(x) = \{ y : h(y,x) \leq 0 \} $ Also for any $(x,y) \in X \times Y $ such that $h(y,x)=0$, it holds...

A rectangle of length x, where x varies, has a constant area of 48cm2. Express the perimeter, y in terms of x. Find the least possible value of x. my problem is not the maths part i.e. the differentiation, but the equation to get things moving. I really have no idea where to start. I drew a...

Homework Statement The function f(x,y,z) may be expressed in new coordinates as g(u,v,w). Prove this general result: The Attempt at a Solution df = (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz dg = (∂g/∂u)du + (∂g/∂v)dv + (∂g/∂w)dw df = dg since they are the same thing? but the...

Homework Statement I'm reviewing physics using Feynman's Lectures, and I'm finding that he frequently uses implicit differentiation in his lessons. This is unfortunate for me because I never got the hang of it beyond the simplest cases. I'm currently going through the proof that the...

Hi .. Use logarithmic differentiation to find the derivative can please check my answer and How I can know if the question want answer by using logarithmic differentiation or not ?

if a function f is differentiable of [0,2pi] can I integrate its derivative df on [-pi, pi]?

Homework Statement Use implicit differentiation to find dy/dx. Homework Equations xey - 10x + 3y = 0 The Attempt at a Solution = [xey + ey(y)'] - (10x)' + (3y)' = 0 = xey + ey(y') - 10 + 3(y') = 0 = y' (ey + 3) = 10 - xey = y' = 10 - xey/ ey + 3y However, my book says the answer: 10 -...

Homework Statement Use the quotient rule to differentiate y=(〖2x〗^4-3x)/(4x-1) Homework Equations y=(v du/dx-u dv/dx)/v^2 The Attempt at a Solution Please also find attached attempt as jpeg for clarity, and textbook supplied answer...

Homework Statement Consider the following equality: (\frac{∂S}{∂V})T = (\frac{∂P}{∂T})V If I rearrange the equality so that I write: (\frac{∂S}{∂P})? = (\frac{∂V}{∂T})? What variables will be constant in each side? I'm having some trouble in a few thermodynamics problems because...

Homework Statement Okay, the concept here is to use induction to prove that for n, (f1 x f2 x ... x fn-1 x fn)' = (f'1 x f2 x ... x fn) + (f1 x f'2 x ... x fn) + ... + (f1 x f2 x ... x f'n). 2. Homework Equations / 3. The Attempt at a Solution I solved the initial step, which was quite...

Homework Statement p(x)=vx^{n+1}+ux^{n}+1 Homework Equations 1) Find u and v so that 1 is a double root for p. 2) Conclude the quotient of p(x) over (x+1)^2. 3) For n=4 find u and v and find the quotient of p(x) over (x-1)^2. The Attempt at a Solution Can someone just tell me how to...

Homework Statement The equations ##2x^3y+yx^2+t^2=0##, ##x+6+t-1=0## implicitly define a curve $$f(t) = \begin{pmatrix} x(t)\\y(t) \end{pmatrix}$$ that satisfies ##f(1)=\begin{pmatrix} -1\\1 \end{pmatrix}.## Find the tangent line to the curve when ##t=1##. Homework Equations The...

Homework Statement Find the eigenvectors and eigenvalues of the differentiation map C1(R) -> C1(R) from the vector space of differentiable functions to itself. Homework Equations The Attempt at a Solution Hi, I'm not entirely sure how to go about this, because would the...

Homework Statement Find the derivative using logarithmic differentiation: y=(x+5)(x+9) The Attempt at a Solution lny=ln(x^2+14x+45) lny=(2x+14)/(x^2+14x+45) y'=(x^2+14x+45)((2x+14)/(x^2+14x+45))However, I know the derivative of the function is actually 2x+14. So I am wondering what is wrong...

Hello friends: My Question: A massive object cannot move at the speed of light. Photons can move at the speed of light because they are massless. However, since energy and mass are equivalent, due to Einstein's famous equation E^2=(m(c^2))^2+(pc)^2, mass is energy by a conversion...

Hi all I am trying to solve for an integral whose integrand is a derivative that has a change of variable inside of it. ∫ (dz/dx) * cos(θ) dθ between 0 and pi. I have a function for z(x), and also know the relation between of x and θ, but what I don't know is how to evaluate such...

Homework Statement Chapter 4 1. Write a program that implements the first order (linear) interpolation 2. Write a program that implemets n-point Lagrange interpolation. Trean n as an imput parameter. 3. Apply the program to study the quality of the Lagrange interpolation to functions...

Homework Statement I have a question. How in general would one differentiate a composite function like F(x,y,z)=2x^2-yz+xz^2 where x=2sint , y=t^2-t+1 , and z = 3e^-1 ? I want to find the value of dF/dt evaluated at t=0 and I don't know how. Can someone please walk me through this?Homework...

Equation: eysinx=x+xy I took the derivative of both sides. For the side with eysinx, I used the product rule and chain rule to get: ey*cosx + ey*sinx*y' For the side with x+xy, I used the sum and product rule to get 1+y+xy' So my resulting equation is: ey*cosx + ey*sinx*y'=1+y+xy', which...

Homework Statement Calculate ∂f/∂x and ∂f/∂y for the following function: yf^2 + sin(xy) = f The Attempt at a Solution I understand basic partial differentiation, but I have no idea how to approach the f incorporation on both sides of the equation nor what you would explicitly call this...

1. Homework Statement [/b] Find the derivative of the given function. Homework Equations Chain rule and logarithmic differentiation. The Attempt at a Solution See attached .gif. I was just wondering if this seemed correct? Thanks!

1. Given that y^{2}-2xy+x^{3}=0, find \frac{dy}{dx} 2. (no relevant equations other than the problem statement) 3. So, I solved it like this, \frac{dy}{dx}y^{2}-2xy+x^{3}=0 2y\frac{dy}{dx}-2+3x^{2}=0 Solving for dy/dx I got... \frac{dy}{dx}=\frac{-3x^{2}+2}{2y}...

Okay so, I am having trouble figuring out what exactly to do in implicit differentiation and usage of the chain rule. Like, I keep getting the wrong answer somehow. See, from what I understand you have to find the derivative of both sides then use the chain rule or something and then solve for...

Hello, I have recently started a little implicit differentiation and I have seen DEs before but I know that I still need to work on my differentiation and integration a little more before I am ready to tackle those. Anyway, I wish to ask, what distinguishes implicit differentiation from a...

Homework Statement I need to use implicit differentiation to find the derivative of y=sin(x+y). Homework Equations The Attempt at a Solution This is what I did: y=sin(x+y) y'=(sin(x+y))' y'=(1+y')(cos(x+y)) (by the chain rule) Now, what do I do? Is this correct...

Homework Statement Show that B = {x2 −1,2x2 +x−3,3x2 +x} is a basis for P2(R). Show that the differentiation map D : P2(R) → P2(R) is a linear transformation. Finally, find the following matrix representations of D: DSt←St, DSt←B and DB←B. Homework Equations The Attempt at a...

Homework Statement solve ∫log(1+acosx) by differentiation under integral sign (limits are 0 to ∏) Homework Equations The Attempt at a Solution =∫(1/1+acosx)cosxdx(by leibinitz by differentiating partially WRT a. Then how do I proceed,can anyone show me all the steps of...

1. Water flows out of a cylindrical tank under gravity via a tap, the height h(t) of the water column above the tap satisfies the differential equation in the form dh/dt = -2k√h where k is some positive constant. The water column has a height initially of 25m. The tap is turned on and...

Homework Statement Find the derivatives at an arbitrary point x in the domain of the following functions f_i: D_i → ℝ, where for 1 ≤ i ≤ 6 the domain D_i is the maximal subset of ℝ on which the mapping is defined - you don't have to determine the domains. Homework Equations a) f_1 (a) =...

Homework Statement The height "s" at time of a silver dollar dropped from a building is given by s(t) = -16t^2 + 1350, where "s" is measured in feet and "t" is measured in seconds [s'(t) = -32t] a) Find the average velocity on the interval [1,2]. ( I ALREADY SOLVED) b) Find the instantaneous...

1. y = sinxy Homework Equations 3. this was my attempt d/dx = (cosxy)(sinxy(d\dx))+(xy(d/dx) im getting stuck. i don't think I am starting it right. any suggestions.

Homework Statement Find the derivative of y=(x^2)^sinx; using the chain rule. Homework Equations No other relevant equations. The Attempt at a Solution I attempted to apply the Chain rule: dy/dx = dy/du X du/dx Subbing u for x^2, which made y = u^sinx I ended up with...

Homework Statement Find \frac{dy}{dx}. y=\sin^{-1}(2x\sqrt{1-x^2}), \frac{-1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}Homework Equations The Attempt at a Solution I started with substituting x=sinθ. The expression simplifies to y=\sin^{-1}(\sin(2θ)) which is equal to y=2θ. Substituting back the value of...