Chain rule Definition and 505 Threads

Homework Statement Let (u,v)=\mathbf{f}(x,y,x)=(2x+y^3,xe^{5y-7z}) Compute D\mathbf{f}(x,y,z),\;\partial (u,v)/\partial(x,y),\;\partial (u,v)/\partial(y,z)\text{ and }\partial (u,v)/\partial(x,z) Homework Equations -chain rule The Attempt at a Solution well I can get...

Homework Statement this is kinda funny I've been strugglnig with the proof for 3 hours straight today long story short(<--sorry couldn't find the best way t o say this ) i made it to this statement lim(x->a) g(x) if this is equal to g(x) then my proof is done lim(x->a) f(a) if this is equal...

Hello, I got problem again with chain rule and would like to have advice for this problem, $\frac{\displaystyle r} {\displaystyle \sqrt{r^2+1}}$ is it product rule I shall also use because I have rewrite it as $r(r^2+1)^{-0.5}$

Homework Statement Consider the transformation \mathbf{x}=G(\mathbf{u}), \text{ where } \mathbf{x}=(x_1,x_2,x_3),\:\mathbf{u}=(u_1,u_2,u_3) given byx_1=u_1+u_3^2x_2=u_3-u_1^2x_3=u_1+u_2+u_3 I need to compute the derivative of this transformation, and then show that the transformation is...

Homework Statement Find the derivative of the following cos(e^-θ^2) Homework Equations cos=-sin e^x=e^x power rule The Attempt at a Solution So I have gotten this far: -sin(e^-θ^2) * ... but then i don't know where to go. Would I treat the -θ^2 as the next step inwards? My...

Homework Statement Use chain rule to find the derivative of f(x)= sin(x)/(1+x^2) Homework Equations Chain Rule (f(g(x)))'*g'(x) The Attempt at a Solution y'(x)= cos (x)/(1+x^2)* (1-x^2)/((1+x^2)^2) I just want to make sure I am doing it correctly and this would be acceptable as a final answer.

If f is a differntiable function, find the expression for derivatives of the following functions. a) g(x)= x/ f(x) b) h(x) [f(x^3)]^2 c) k(x)= sqrt (1 + [f(x)]^2) First off, I am not even sure what they are asking. I am assuming that they want the derivative for each component of the...

How is the double derivative equal to that in the equation 2 in the attachment? =|

Homework Statement g:ℝ^{ 2 }\rightarrow ℝ is everywhere differentiable. For all (x,y) and for all t: g\left( tx,ty \right) =tg\left( x,y \right) . Prove g is linear (that there exist constants A, B such that for all (x,y): g\left( x,y \right) =Ax+By . I think my solution is correct, but the...

Homework Statement Please have a look at the picture attach, which shows the proof of the D'alembert's solution to the wave equation. If you can't open the open, https://www.physicsforums.com/attachment.php?attachmentid=54937&stc=1&d=1358917223 click onto this...

Homework Statement compute the gradient: ln(z / (sqrt(x^2-y^2)) Homework Equations ∇=(∂/(∂x)) + ... for y and z I just want to know how to do the first term with respect to x The Attempt at a Solution I am so rusty I don't know where to begin.

This guy relates the calculus chain rule to a popular mob movie. You should really check it out. This is one of the newer videos but people like the way this guy explains things. Here is the link: Ghetto Dude Relates Calculus Chain Rule To "THE MOB" - YouTube

Air is being pumped into a spherical balloon at a rate of 5 cm3/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm. So, to solve, I know HOW to do it, I just don't know WHY it's right. \frac{dv}{dr}=4pi r^{2}...

Homework Statement Show that: \frac{dx^\nu}{d \lambda} \partial_\nu \frac{dx^\mu}{d \lambda} = \frac{d^2 x^\mu}{d \lambda^2} The Attempt at a Solution Well, I could simply cancel the dx^nu and get the desired result; that I do understand. But what about actually looking at...

Homework Statement As part of a problem I am doing I am asked to show uβ∂βuα = aα where u is 4 velocity and a refers to 4 acceleration. The way to do this is not immediately obvious to me, especially since the problem implies there should be a chain rule step involved which I am not seeing. I...

Homework Statement If u=f(x,y) where x=escost and y=essint show that d2u/dx2+d2u/dy2 = e-2s[d2u/ds2+d2u/dt2 Homework Equations http://s11.postimage.org/sjwt1wkvl/Untitled.jpg The Attempt at a Solution ok i don't understand how they got to that i don't know what d/ds is...

Homework Statement Find the derivative of y = sin(πx)2 Homework Equations Chain Rule: y' = f'(u) * u' The Attempt at a Solution (See attached image) The answer according to the textbook is 2π2xcos(πx)2. What am I doing wrong here?

Homework Statement Suppose g(x,y)=f(x-y,y-s) Homework Equations Nothing else The Attempt at a Solution Find dg/dx + dg/dy

Homework Statement Ok I have this general homogeneous function, which is a C^1 function: f(tx,ty)=t^k f(x,y) And then I have to show that this function satisfies this Euler equation: x\frac{\partial f}{\partial x}(x,y)+y\frac{\partial f}{\partial y}(x,y)=k\cdot f(x,y) Homework...

Hi all, I've got a Calculus III Question Homework Statement Find the derivative zs and zt, where z=sin(x)cos(2y)Homework Equations x=s+t y=s-t The Attempt at a Solution I had a go at the solution and this was what I ended up getting for zs, I ended up getting (cosxcos2y)(1)-2sinxsin2y(1)...

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...

Find the derivative of 101-x2. 1.) So, I used power rule to get: (1-x2)(10-x2). 2.) Then I did chain rule and multiplied my previous answer with (d/dx)(-x2). 3.) The answer I get is: (-2x)(1-x2)(10-x2). When I use my calculator, the answer I get is: -20*ln(10)*10-x2. So, where does...

Homework Statement I'm curious to know if I'm actually doing this correctly. Suppose f(x,y) is a function where x = p(s,t) and y = g(s,t) so that w(s,t) = f(x,y). Compute ws and then wst Homework Equations Chain Rule. The Attempt at a Solution So! Let's compute ws first. Whenever I use a...

Let g(t) = f(tx, ty). Using the chain rule, we get g'(t) = (\frac{\partial f}{\partial x})(tx, ty) * x + (\frac{\partial f}{\partial y})(tx, ty) * y this was actually part of a proof and what i don't understand is that why didn't they write (\frac{\partial f}{\partial (tx)}) and...

Evaluate partial derivative. chain rule?? I would like to represent the term identified in the image as (term 1) in terms of those partial derviatives that are known. Unfortunatly I just can't seem to wrap my head around it at the moment. :bugeye: A prod in the right direction would be...

Homework Statement Given that f(x,y) = g(r,\theta), where x = r\cos\theta and y = r\sin\theta, find formulae for \frac{∂f}{∂x} and \frac{∂f}{∂y} expressed entirely in terms of r, \theta, \frac{∂g}{∂r} , \frac{∂g}{∂\theta} . The Attempt at a Solution I said \frac{∂f}{∂x} =...

$x = r\cos\theta$ and $y=r\sin\theta$ $$ \frac{\partial u}{\partial\theta} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial\theta} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial\theta} = -r\sin\theta\frac{\partial u}{\partial x} + r\cos\theta\frac{\partial u}{\partial y} $$...

Homework Statement Parametrize the upper half of the unit circle by x = cos(t), y = sin(t), for 0\leq t \leq\pi Let T = f(x,y) be the temperature at the point (x,y) on the upper half of the circle. Suppose that: \frac{\partial T}{\partial x} = 4x - 2y \frac{\partial T}{\partial y} = -2x +...

say you have a function f(x,y) \nablaf= \partialf/\partialx + \partialf/\partialy however when y is a function of x the situation is more complicated first off \partialf/\partialx = \partialf/\partialx +(\partialf/\partialy) (\partialy/\partialx) ( i wrote partial of y to x in case y was...

ok stupid question probably- take v(velocity) to be a function of x and x to be a function of t(time). then dv/dt=vdv/dx that's cool but in the hint in problem 2.12 classical mechanics by john r taylor he equates vdv/dx and 1/2(dv^2)/dx that is- vdv/dx=1/2(dv^2)/dx Could someone please...

Hi folks, I don't know if my experience is at all common (and I would like some feedback on this if possible), but I can't seem to nail down the properties of euler's number in the context of chain rule problems. Here is the nub of my difficulty: 1. $\text{If }f(x)=e^x \text{then }f'(x)=e^x$...

I have attached a pdf setting forth my question. This is a write up of a lesson i just had on yourtutor, in which i think the tutor might have made an error: this is a direct quote from the whiteboard: $Let g(x)=2x, f(y)=e^y\Rightarrow(fog)(x)=f(g(x))=f(2x)=e^{2x}$$\\Now...

Hi, I have been doing research in my spare time this summer on calculus proofs. I am working on a mathematics degree and I am working to understand calculus inside and out. It has been going really well but I have sort of hit a bump with the calc 1 chain rule. Here is my attempt: lim h -> 0...

Can someone help me out with what I believe who.d be a straight chain rule application? D/dx (dy/dx)^2 I would think, applying the chain rule, you get 2 (dy/dx) d/dx (dy/dx) = 2 (dy/dx) (d2y/dx2) But, from the papers I checked, the (dy/dx) in the answer goes away, leaving just 2...

If h(x) = ax, g(x) = bx and f(x) = g(h(x)). Wouldn't h'(x) = a? And g'(x) = b? And f'(x) = ab? But the chain rule says f'(x) must equal h'(x)g'(h(x)), so that means f'(x) = ab(ax) = (a^2)bx. Am I missing something obvious?

I have read a few sources regarding the chain rule, and a pervasive explanation that most of the sources share is this, which is way to sort of make sense of it: "Regard du/dx as the rate of change of u with respect to x, dy/du as the rate of change of y with respect to u, and dy/dx as the...

1. Find the derivative of the function 2. \left(y= x sin\sqrt{x}\right) 3. I started using the product rule and then proceeded to use the chain rule, but I am wondering if I should have used the chain rule twice rather than starting with the product rule. Since I know that x is the...

I was reading over a textbook that stated the following, where y(s,t) = v(x(s,t),t) \frac{\partial y}{\partial t} = \frac{\partial v}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial t} and \frac{\partial^2y}{\partial t^2} = \frac{\partial^2 v}{\partial x^2}\left (...

Homework Statement [SIZE="5"](x^{2}-x^{-1}+1)(x^{3}+2x-6)^{7} Homework Equations Chain Rule & Power RuleThe Attempt at a Solution [SIZE="5"](x^{3}+2x-6)^{6}[(x^{3}+2x-6)(2x+x^{-2})+7(3x^{2}+2)(x^{2}-x^{-1}+1)] This is the farthest I've gotten but when I do additional computation I do not...

Homework Statement s=\sqrt{(3x^2)+(6y^2)} Homework Equations None The Attempt at a Solution \stackrel{ds}{dt}=\stackrel{d}{dt}\sqrt{(3x^2)+(6y^2)} \stackrel{3x}{\sqrt{(3x^2)+(6y^2)}} The problem with that is its only d/dx if y is a set number. I don't know how to...

Any proof for the CHAIN RULE ?? Can somebody please show me the proof of the chain rule?? even though i have been applying that concept since i touch differentiation but i still have doubt and question on this concept!

Homework Statement If x=yz and y=2sin(y+z), find dx/dy Homework Equations Chain rule The Attempt at a Solution From y = 2sin(y+z) we get dz/dy= (1-2cos(y+z))/(2cos(y+z)) dz/dy=((1/2)sec(y+z) - 1) dx/dy = ∂x/∂y + ∂x/∂z dz/dy = z + y ((1/2)sec(y+z) - 1) = z...

Homework Statement Let x=ts^2 -1 and y=ln(s)-t Use the chain rule for functions of two variables to determine ∂f/∂t at (s,t)=(1,1) The Attempt at a Solution y=ln(s)-t ∂f/∂t= ∂f/∂s X ∂s/∂t -1 t=x+1/s^2 ∂t/∂s= -2(x+1)/s^3 ∂s/∂t=s^3/-2(x+1) ∴ ∂f/∂t= s^2/-2(x+1)...

Homework Statement Here is the problem: http://dl.dropbox.com/u/64325990/MATH%20253/help.PNG The Attempt at a Solution http://dl.dropbox.com/u/64325990/Photobook/Photo%202012-05-24%209%2037%2028%20PM.jpg This seems to be wrong... Since I have fx and fy which I cannot cancel out. Why...

Homework Statement Here is the problem with the solution: http://dl.dropbox.com/u/64325990/MATH%20253/Capture.PNG What I don't understand is circled in red, how did they combine dxdy with dydx? Is it with Clairaut's theorem? If it is can someone explain how it works in this case because...

$${x = r \cos \theta}$$, $${y = r \sin \theta}$$, $${r^2 = x^2 + y^2}$$ and $${\theta = \tmop{ \arctan} (y / x)}$$ (with some caveats for the last formula). Suppose $${u = u (x, y)}$$. Show that $${\frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \cos \theta +...

Hi, If I have the equation y' = ax - by where y = y(t) , x= x(t) and y' = \frac{dy}{dt} then what is \frac {d}{dy} y' = \frac {d}{dy}(ax - by) ? I think it would come out to \frac {dy'}{dy} = a \frac {dx}{dt}\frac {dt}{dy} - b Is that right? In general...

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Homework Statement I have vector R. I need to show the R dot dR/dt = 0 => 1/2 d/dt[R dot R] Homework Equations The Attempt at a Solution I guess I've never really applied the chain rule to dot products and its throwing me off. How does one go from R.dR/dt=0 to 1/2 d/dt[R.R] = 0. I...

Since Calculus has supposedly been around for a long time, when is there actual evidence of the chain rule first being used?