Chain rule Definition and 511 Threads

1. Derive arcsin(1 - 2 e ^-t) 2. The derivative of arcsin is 1/√(1-x^2) 3. I tried using the chain rule for 1 - 2 e ^-t, but that didn't work out. What should I take the chain rule of?

hi does anyone know why the 2nd derivative chain rule is as such? i roughly know that if u = f(x,y) and x=rcos(T) , y = rsin(T) then du/dr = df/dx * dx/dr + df/dy * dy/dr but if i am going to have a second d/dr, then how does it work out?

Homework Statement Let \textbf{F}: \textbf{R}^m \rightarrow \textbf{R}^n and \textbf{G}: \textbf{R}^p \rightarrow \textbf{R}^m Prove that ({\textbf{F} \circ \textbf{G}})'(x) = {\textbf{F}}'(\textbf{G}(\textbf{x})) {\textbf{G}}'(\textbf{x}) Homework Equations Assume the single...

Homework Statement The movement of an object with a mass of 1500kg is given by v(x)=(4.0 [1/ms]) * x^2 Determine the net force acting on the object as a function of x. Homework Equations F=ma The Attempt at a Solution I know I'm supposed to use the chain rule to solve this but...

Homework Statement http://www.math.wvu.edu/~hjlai/Teaching/Tip-Pdf/Tip3-27.pdf Example 7. Not this question in particular, but it shows what I'm talking about. I understand how they get the first partial derivative, but I'm completely lost as how to take a second one. I have tried...

r' = 4r - rf f' = -3f + rf In this question, there was three parts: a) find all the critical points of this system. b) Derive the linearised system about each critcal point... c) Use the chain rule to derive the path equation of the trajectories in the phase plane. I managed to get a...

Homework Statement Find the derivative of: f(x)=\sqrt{2x} Homework Equations So using the chain rule: \frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dx} The Attempt at a Solution Isn't it just a simple matter of setting u=2x, therefore du/dx=2, and y=\sqrt{u}=u^1/2, therefore dy/du=1/2 *...

I just learned about chain rule in calculus, but I was wondering why exactly chain rule works. I understand how to use it, just not exactly why it works. Thanks in advance

Homework Statement Let z = z (x,y) be a function with x = x(s), y = y(t) satisfying the partial differential equation (Ill write ddz/ddt for the partial derivative of z wrt t and dz/dt for the total derivative of z wrt t, as I have no idea how to use Latex.) ddz/ddt +...

Homework Statement Say that f(x) is some function whose second derivative exists and say u(x, t)=f(x + ct) for c > 0. Determine \frac{\partial u}{\partial x} In terms of f and its derivatives. Homework Equations PD Chain rule. The Attempt at a Solution Say that x and y are...

I have a question about the variable changes in the proofs of Proposition 1.3.4 and Proposition 1.3.6. In the first one, it seems like the author does the variable change but once he applies the chain rule he doesn't do it completely. While in the second it seems like it does the variable...

If I have two functions, f(x,y,z) and g(x,y,z), do I use the chain rule to calculate df/dg? e.g. df/dg=df/dx df/dy df/dz Cheers! Romeo

Homework Statement Hi I'm currently trying to revise for a Calculus exam, and have very little idea of how to do the following: Let f be defined by f(x,y) = (y+e^x, sin(x+y)) Let g be of class C2 (twice differentiable with continuous second derivatives) with grad(g)(1,0) = (1,-1) and Hg(1,0)...

I'm looking at the proof of the multivariable chain rule & just a little bit curious about something. In the single variable chain rule proof the way I know it is that you take the derivative: f'(x) \ = \ \lim_{ \Delta x \to \infty} \frac{ \Delta y}{ \Delta x} and manipulate it as follows...

Homework Statement The function F is defined by F (r, θ) = f (x(r, θ), y(r, θ)), where f is twice continuously differentiable and x(r, θ) = r cos θ, y(r, θ) = r sin θ. Use the chain rule to find d2F/dθ2Homework Equations The Attempt at a Solution I know that dF/dθ = (df/dx)(dx/dθ) +...

Homework Statement I'm looking for a proof for the chain rule that is relatively easy to understand. Can someone show / link me one? Thanks. Homework Equations The Attempt at a Solution

Homework Statement Find dy/dx at x = 2. y = (1 + s)/(1 - s); s = t - 1/t; t = sqrt(x) Homework Equations I know if f = f(g(x)), then f1(x) = f1(g(x)) * g1(x) The Attempt at a Solution I think I may need to combine the chain rule and quotient rule, but all of the separate...

Homework Statement Apply the two cases of th change rule. For example: The voltage V in an electrical circuit is slowly decreasing as a battery wears out and the resistance R is slowly increasing as the resistor heats up Use Ohm's law V=IR to find how the current is changing (with respect to...

Problem with proof of Chain rule for f:R-->R Hi, Analysts: I am going over problems in Rosenlicht's Intro. Analysis book. In this problem , he asks one to find the flaw in this argument to the effect that (f(g(x))'=f'(g(x))g'(x). Unfortunately, author does not clearly state the...

Homework Statement Homework Equations The Attempt at a Solution a) (∂z/∂x)=-(∂f/∂x)*(∂z/∂f) i used that (AxB)=-(BxA) so i get (∂z/∂x)=-[-(∂z/∂f)(∂f/∂x)] =(∂z/∂x) is this correct if not can someone give me hints pls thanks

I have a function for velocity, V in terms of position, x. The equation is of the form V(x) = a*x2+b*x+c. Initial conditions are x=0, t=0. How do I change from V(x) to V(t)? It seems this would be an application of the chain rule, dy/dx = dy/du * du/dx, but I'm struggling to adapt it to...

Homework Statement Find: ∂f/∂x f(r,θ)=rsin^2(θ), x=rcosθ, y=rsinθ The Attempt at a Solution ∂f/∂r=sin^2θ ∂r/∂x=-cosθ/x ∂f/∂θ=2*r*cosθ*sinθ ∂θ/∂x=-1/sqrt(1-(x^2)/(r^2) ∂f/∂x = -sin^2θcosθ/x^2 + -2*r*cosθ*sinθ/sqrt(1-(x^2)/(r^2) ∂f/∂x = -y-sqrt(x^2+y^2) /...

I'm kind of confused about how to approach a function with the chain rule. For example in the equation ƒ(x) = sqrt(1-sin(x)) I know i simplify it to ƒ(x) = 1-sin(x)^(1/2) but I'm lost from there.

Problem Use the chain rule to proof \dot{A}=\partial_t A+v_j\partial_jA_i Attempt at Solution \dot{A}=\frac{dA_i}{dt} = \partial_t A_i+\frac{dr_i}{dt}\frac{\partial A_i}{\partial r_i} Obviously v_j = \frac{dr_j}{dt} I'm puzzled where the v_j and partial d_j come in

Homework Statement Reduce the order of a Cauchy-Euler Equation Homework Equations x = e^t \mbox{ and } \ln x = t The Attempt at a Solution \displaystyle \frac{d y}{d x} = \displaystyle \frac{d y}{d t} \displaystyle \frac{d t}{d x} = \displaystyle \frac{d y}{d t} \cdot...

Homework Statement A cubical block of ice is melting in such a way that each edge decreases steadily by 9.8 cm every hour. At what rate is its volume decreasing when each edge is 10 meters long? Homework Equations V(t) = (l(t))^3 m^3 l'(t) = 0.098 m/h The Attempt at a Solution...

Homework Statement One side of a triangle is increasing at a rate of 3cm/s and a second side is decreasing at a rate of 2cm/s. If the area of the triangle remain constant, at what rate does the angle between the sides change when the first side is 20cm long and the second side is 30cm, and...

Homework Statement Prove that (\frac{\partial u}{\partial x})^{2} + (\frac{\partial u}{\partial t})^{2} = e^{-2s}[(\frac{\partial u}{\partial s})^{2} + (\frac{\partial u}{\partial t})^{2}].Homework Equations u = f(x,y) x = e^{s}cost y = e^{s}sint The Attempt at a Solution I started out by...

This is strictly a math question but I figured that since it is something which would show up in QM, the quantum folks might be already familiar with it. Suppose we have an operator valued function A(x) of a real parameter x and another function f, both of which have well defined derivatives...

Homework Statement F(s) = ( s - \frac{1}{s^2})3 I have to calculate the derivative of this using chain rule everytime i try i get something way different than in the back of the book... my first move is 3( s - \frac{1}{s^2})2 X ( 1 + \frac{2}{s^3}) is this correct? then expand...

Homework Statement Let x=x^2ysin(u)tan(v), where x(u,v) and y(u,v) are smooth functions that, when evaluated at u=1 and v=-3 satisfy x=2.112, y=4.797, \partialx/\partialu = -3.491, \partialx/\partialv = -2.230 , \partialy/\partialu = 1.787 , \partialy/\partialv = 1.554. Then the...

\hbox { Let }\; u(x,y)=v(x^2-y^2,2xy) \;\hbox { and let }\; t=x^2-y^2,\;s=2xy u_x = 2xv_t \;+\; 2yv_s u_{xx} = 2v_t + 4x^2 v_{tt} + 8xyv_{ts} + 4y^2 v_{ss} The u_{yy} can be done the same way and is not shown here. According to Chain Rule: u_x = \frac{\partial v}{\partial x}...

I am currently learning calculus and just had my lecture on the chain rule. I noticed that we haven't learned how to take the derivative of a function like 2^2+x or 3^4+x. Any example works.. Is this something I will learn later as I progress through calculus or what?

Homework Statement I have a problem with the next exercise: Given de function f(x,y)=\begin{Bmatrix} \displaystyle\frac{xy^2}{x^2+y^2} & \mbox{ if }& (x,y)\neq{(0,0)}\\0 & \mbox{if}& (x,y)=(0,0)\end{matrix} with \vec{g}(t)=\begin{Bmatrix} x=at \\y=bt \end{matrix},t\in{\mathbb{R}} a) Find...

I am trying to differentiate ((5x-1)^4)((8x^2-5)^-3) but i am stuck at a certain point... Can you please help me fill in the blanks? Thank you so much:):) Work done: 1st step: (5x-1)^4 d/dx (8x^2-5)^-3 + (8x^2-5)^-3 d/dx(5x-1)^4 2nd step: (5x-1)^4 (-3)(8x^2-5)^-4 (16x)(8x^2-5)^-3...

I need some help with the chain rule...Thank you for helping me:) Question: G(y)=((x-1)^4)/(((x^2)+2*x))^7 I have no idea where to start.

Homework Statement I trying to find the second derivative of xe^x Homework Equations chain rule The Attempt at a Solution Two find the first derivative I use the chain rule. f'(y)g(y)+f(y)g'(y) so I get e^x+xe^x is the second derivative e^x+f'(y)g(y)+f(y)g'(y)...

Homework Statement If y=f((x2+9)0.5) and f'(5)=-2, find dy/dx when x=4 Homework Equations chain rule: dy/dx=(dy/du)(du/dx) The Attempt at a Solution In my opinion giving f'(5)=-2 is unnecessary as: y=f(u)=u, u=(x2+9)0.5 dy/dx= (dy/du)(du/dx) (dy/du)= 1 (du/dx)= x/((x2+9))0.5 dy/dx =...

Homework Statement See figure. Homework Equations The Attempt at a Solution Here's what I got, \frac{ \partial z}{\partial x} = \left( \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x} \right) + \left( \frac{\partial z}{\partial v} \cdot \frac{\partial...

Homework Statement tan^3(x) + tan(x^3) Homework Equations The Attempt at a Solution tan^3(x) + sec^2(x^3) + 3x^2 Im not sure how to do the tan^3(x) and not even sure I did the tan(x^3) right

Homework Statement y=cot^7(x^5) Homework Equations f(x)=f(g(x)) The Attempt at a Solution u=(x^5) y'=7(-csc^2)^6(x^5) * 5x^4

Homework Statement Find \frac{\partial z}{\partial y} [/itex] where z=F(u,v,y), u=f(v,x), v=g(x,y). The Attempt at a Solution If I remember multivariate calculus at all, this should be (please forgive the abuse of notation) \frac{\partial z}{\partial y} = \frac{\partial z}{\partial...

Hi I've just been reading something which is essentially how to work out what the deriviative of y=b^x is. Basically the explanation gets to the point which I understand and says \frac{dx}{dy} = \frac{1}{yln(b)} It then says because of the chain rule you can simply flip this to get...

Homework Statement Let f(x) = (x2)/sin(x)2. Find f'(x). Homework Equations Chain rule, quotient rule The Attempt at a Solution f'(x) = [2xsin(x)2 - x22cos(x)2]/(sin(x)2)2

Homework Statement express (\frac{\partial u}{\partial s})_{v} in terms of partial derivatives of u(s,t) and t(s,v) Homework Equations The Attempt at a Solution I'm pretty stuck with this problem. I know that dv = (\frac{\partial v}{\partial s})_{t} ds + (\frac{\partial...

Homework Statement The derivative of the function h(x) = sin((x2 + 1)2)Homework Equations Chain Rule The Attempt at a Solution h(x) = sin((x2 + 1)2) f(u) = sinu^2, f'(u)= 2ucosu^2 g(x) = x^2+1 g(x)= 2xI get lost putting this back together but: 2(sinu^2)[cos(sinu^2)^2](2x) ?

chain rule someone help please 1. let z=y^2-x^2cosy; x=t^3 y=cost, find dz/dt 2. let z=(x-y)^3;x=u+2v,y=2u-v,find dz/dvmy attempt: so i know the chain rule is (dz/dx)dx+(dz/dy)dy 1. should i substitute the x and y into t first or should i do the partial derivative first? 2. same thing what...

Homework Statement We say that a differentiable function f : \mathbb{R}^n \rightarrow \mathbb{R} is homogenous of degree p if, for every \mathbf{x} \in \mathbb{R}^n and every a>0, f(a\mathbf{x}) = a^pf(\mathbf{x}). Show that, if f is homogenous, then \mathbf{x} \cdot \nabla f(\mathbf{x}) = p...

Homework Statement If T is implicitly defined via the relationship f(x, y, z, T) = 0 to be a differentiable function of x, y and z, show that the first partial derivative of T with respect to z can be found using: \frac{\partial T}{\partial z} = -\frac{\partial f}{\partial z} / \frac{\partial...

Homework Statement Suppose the differentiable function f(x,y,z) has the partial derivatives fx(1,0,1) = 4, fy(1,0,1) = 1 and fz(1,0,1) = 0. Find g'(0) if g(t) = f(t2 + 1, t2-t, t+1).Homework Equations The Attempt at a Solution Ok I'm given the solution for this and I'm trying to work through it...