Chain rule Definition and 505 Threads

I am reading Barrett O'Neil's book: Elementary Differential Geometry ... I need help to get started on Exercise 4(a) of Section 1.1 Euclidean Space ... Exercise 4 of Section 1.1 reads as follows:Can anyone help me to get started on Exercise 4(a) ... I would guess that we need the chain rule...

Homework Statement If d/dx(f(x)) = g(x) and d/dx(g(x)) = f(x2), then d2/dx2(f(x3)) = a) f(x6) b) g(x3) c) 3x2*g(x3) d) 9x4*f(x6) + 6x*g(x3) e) f(x6) + g(x3) Homework EquationsThe Attempt at a Solution The answer is D. Since d/dx(f(x)) = g(x), I said that d/dx(f(x3)) should equal 3x2*g(x3), then...

Not sure if this is the correct place to post this. dy/dt = 0, find y(t) My professor told me that the chain rule is used to determine that (dy/dt)*dt = dy, but I just don't see it. Multiply both sides by dt. (dy/dt)*dt = 0dt (dy/dt)*dt = 0 dy = 0, then integrating both sides: y = C dy/dt is...

Here is a simple question : let f(g(x)) = h(x)*g(x). I want to calculate df/dx. If I use the product rule, I get g(x)h'(x) + h(x)g'x). Now if I use the composition/chain rule, I get df/dx = df/dg * dg/dx = h(x) * g'(x) which is different. I guess my df/dg = h is wrong, but I can't see what...

F(r,s,t,v) = r^2 + sv + t^3, where: r = x^2 +y^2+z^2 /// s = xyz /// v = xe^y /// t = yz^2 find Fxx i have 2 solutions for this and i am not sure what is the right one the first solution finds Fx then uses formula : Fxx = Fxr.Rx + Fxs.Sx + Fxv.Vx+Fxt.Tx the 2nd solution find Fx then uses the...

Homework Statement Homework Equations Chain rule, partial derivation The Attempt at a Solution dv/dt=dv/dx*dx/dt+dv/dy*dy/dt dx/dt=-4t -> evaluate at (1,1) =-4 dv/dt=-4dv/dx+4(-2) dv/dt=-4dv/dx-8 How can I find the missing dv/dx in order to get a value for dv/dt? Thanks!

Say I have a function of three variables, ##F=F(s_{12},s_{23},s_{13}) = F(s,t,-s-t)##, where ##s_{12}=s,s_{23}=t## and ##s_{13}=u = -s-t##. I want to compute the differential operators $$\frac{\partial}{\partial s}, \frac{\partial}{\partial t}\,\,\text{and}\,\,\frac{\partial}{\partial u}.$$...

Homework Statement Homework Equations dz/dt = dz/dx⋅dx/dt + dz/dy⋅dy/dt The Attempt at a Solution [/B] I am getting : =[-sin(x+7y) ⋅ 10t] + [-sin(x+7y) ⋅ 7 ⋅ (-1/ t2)] then changing x and y terms: =[-sin((5t2)+7(1/t)) ⋅ 10t] + [-sin((5t2)+7(1/t)) ⋅ 7 ⋅ (-1/ t2)]

Im stuck on theorem 5 where the book used chain rule then used product rule then again using the chain rule. How in the world does it work? I don't get product rule used and chain rule used after.

Homework Statement Find f'(x) if f(x) = 8^(sin^2(3x)) Hint: you will need to use the double angle formula for trig functions and your answer should only have one trig function in it. Homework Equations if y=a^u then y' = ln a * a^u * du sin(2x) = 2sinxcosx The Attempt at a Solution We're...

Hello Forum, I have a couple of kinematics questions. The position of a point object is given by the position vector x(t). Speed is v(t)=dx(t)/dt and the acceleration a(t)= dv(t)/dt. What if we wanted to know the velocity and/or the acceleration as a function of position, i.e v(x) or a(x)...

Hey all, I am reading Goldstein and I am at a point where I can't follow along. He has started with D'Alembert's Principle and he is showing that Lagrange's equation can be derived from it. He states the chain rule for partial differentiation: \frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial...

Homework Statement z = ƒ(x,y), x = rcos(θ), y = rsin(θ) Use the chain rule to show that: \frac{1}{r^{2}}\frac{\partial ^{2} z}{\partial \theta ^{2}} = sin^{2}(\theta)\frac{\partial ^{2} z}{\partial x^{2}}-2sin(\theta)cos(\theta)\frac{\partial ^{2} z}{\partial x \partial...

Homework Statement Homework EquationsThe Attempt at a Solution I have the solution to this problem and the issue I'm having is that I don't understand this step: Maybe I'm overlooking something simple but, for the red circled part, it seems to say that ∂/∂x(∂z/∂u) =...

Hi I'm learning about The chainrule, and I understand how to apply the chain rule on various problems, but there is a problems I don't understand how works: 1) The book I'm reading writes acceleration as a=v*(dv)/dt And IT argues that v=ds/dt and a=dv/dt (which i understand) So therefore by...

Homework Statement I'm unable to see fully how the following equality is determined: $$\frac {\partial f(y + \alpha \eta, \, y' + \alpha \eta', \, x)}{\partial \alpha} = \eta \frac {\partial f}{\partial y} + \eta' \frac {\partial f}{\partial y'}$$ where ##y = y(x)##, ##\eta = \eta (x)##, and...

Homework Statement [/B] hi could some body please help me factorise this please ? any chance of a few stages would be much appreciated Homework EquationsThe Attempt at a Solution my attempt , but my solutions say otherwise ? [/B]

Homework Statement Consider the function of two variables: u(x,y) = f(x-y) + g(x+(1/3)y) where f(s) and g(t) are each arbitrary functions of a single variable. Using the change of variables: s = x-y t = x-(1/3)y use the chain rule to determine the first and second derivatives of u with...

I'm in Calc 1 and the Chain Rule is giving me one hell of a rough time. I've spent about 10-12 hours over the last few days just on the homework problems in this one section (only getting about 15-20 problems done) and still feel like I barely understand it. Does anyone have any tips, tricks...

Homework Statement ##J=r^{2}\dot{\phi}## [1] ##\dot{r^{2}}=E^{2}-1-\frac{J^{2}}{r^{2}}+\frac{2MJ^{2}}{r^{3}}+\frac{2M}{r}##. [2] (the context is geodesic equation GR, but I'm pretty sure this is irrelevant). where ##u=r^{-1}## Question: From these two equations to derive...

Homework Statement Given that the surface x^7y^2+y^4z^6+z^8x^8+9xyz=12 has the equation z=f(xy) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives. df/dx (1,1) = ? d^2f/dx^2 (1,1) = ? Homework EquationsThe Attempt at a Solution df/dx (1,1) I got -24/23 or...

When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...

Homework Statement y = (tan^-1(6x))^2 Homework Equations Chain Rule, power rule? The Attempt at a Solution Okay, so I did power rule to bring it to 2(tan^-1(6x)) Then, I know to use the chain rule... I get 2(tan^-1(6x)*(1/1+x^2)... I know u = 6x so I play 6 into x^2 and I get 6x^2... I see...

Consider a function of several variables ##T=T(x_{1},...,x_{3N})## Let's say I have N vectors of the form ##\vec{r_{1}}=(x_1,x_{2},x_{3})## and ##x_j=x_j(q_1,...,q_n)##. Awkward inex usage but the point is just that the each variable is contained in exactly 1 vector. Is it correct to in...

Hello, I solved this exercise, but I probably did mistake in simplification... f(x)=${\left(-2{x}^{2}+3\right)}^{4}$${\left(9{x}^{2}+7\right)}^{12}$ They asked to find derivative, so here is what I did...

So i have an equation problem that i need to find the 2nd derivative of, but my understanding of the chain rule is not the best. I tried working it out but i don't know if i did it correctly. i was given the equation y=4(x2+5x)3 So to take the first derivative, i started off by using the chain...

I need help in understanding the proof of the Chain Rule for differentiation, as presented in Theorem 5.1.6 in Manfred Stoll's book: Introduction to Real Analysis. Theorem 5.6.1 in Stoll (page 173) reads as follows:In the above proof we read the following: " ... ... By identity (3) and then...

Hi. I was looking for a chain rule in vector calculus for taking the gradient of a function such as f(A), where A is a vector and f is a scalar function. I found the following expression on wikipedia, but I don't understand it. It's taking the gradient of f, and applying that to A, and then...

I'm currently reviewing my knowledge of calculus and trying to include rigourous (ish) proofs in my personal notes as I don't like accepting things in maths on face value. I've constructed a proof for the chain rule and was wondering if people wouldn't mind checking it and letting me know if it...

After completing calculus 2 with an A I now realize I know nothing of mathematics. We used stewart calculus and I did not really like it, due to a lot of hand waiving. I got an older edition of thomas calculus with analytic geometry 3rd ed, and so far I'm having a blast learning proofs from...

by chain rule or by homegenious function idk how to start with chain rule or uing homoginus function

Homework Statement Derivative question f=f(x) and x=x(t) then in one book I find \frac{d}{dx}\frac{df}{dt}=\frac{d}{dx}(\frac{df}{dx}\frac{dx}{dt}) =\frac{dx}{dt} \frac{d^2 f}{dx^2} Homework EquationsThe Attempt at a Solution Not sure why this is correct? \frac{dx}{dt} can depend of f for...

View image: IMG 20141102 00094 i know chain rule but it more complicated i can't go far with it please any help ??

In one physics problem if $$r^2= \lambda^2(1+\frac{m}{2\lambda})^2$$ what is ##dr^2 ?## Should I find ##dr## starting from ##r= \lambda(1+\frac{m}{2\lambda})## first and then square or find ##dr^2## starting from r^2? I know this is a basic question in differentiation using chain rule but it...

Mentor's note: These posts were split off from a thread in the textbooks forum. Most of them are about calculus, even though they start off with a non-calculus question. I was too lazy to split them further into two threads ----------------------------------- I don't know what books to...

Homework Statement Let g(x) = f(sin(2x) f(cos x)), where f(0) = 2, f'(0) = 3, f(-1) = -1/3 , and f'(-1) = -1. Find the equation of the tangent line to the curve of y = g(x) at x = pi. 2. The attempt at a solution Point of Tangent: (pi, 2) g(pi) = f(sin(2pi) f(cos pi)) = f(0 * f(-1)) = f(0) =...

Not homework, just having fun. Every reference I find illustrates the chain rule for composite functions of two variables in this way: \begin{align*} B &= f(x,y) \\ x &= g(w,z) \\ y &= h(w,z) \\ \frac{\partial B}{\partial w} &= \left( \frac{\partial B}{\partial x} \cdot \frac{\partial...

This is a problem that has been bugging me for ages. I just can't wrap my head around this weird result. I know I went wrong somewhere [as a matter of fact, that was the answer I was hoping for], but most sources, (including, but not limited to, wikipedia), suggest otherwise. I will cut to the...

Homework Statement How does one get the r" equation from r'? Homework Equations r = distance v = r' = ds/dt a = r'' = dv/dt chain rule, dy/dt = dy/dx * dx/dt The Attempt at a Solution I can easily get to r' from r using the chain rule but how do you derive r" from r'? How do you apply...

Definition of 'Limit of function (f) at x=p' Let E be domain of f and p be a limit point of E. Let Y be the range of f. If there exists q∈E such that for all ε>0 there exists δ>0 such that for all t∈E for which d(t,p)<δ implies d(f(t),q)<ε. Then we say that f(t)->q as t->p. 1) Suppose f...

For this function y=\sqrt{2ln(x)+1} if I use the chain rule properly, should I be getting this answer? \frac{dy}{dx}=\frac{2}{x} \times \frac{1}{2} \times \frac{1}{\sqrt{2ln(x)+1}} My aim of doing this is to verify that \frac{dy}{dx}=\frac{1}{xy}

I was playing around with some simple differential equations earlier and I discovered something cool (at least for me). Suppose you have y=sin(x^2) \Rightarrow \frac{dy}{dx}=2xcos(x^2) What if, instead of taking the derivative with respect to x, I want to take the derivative with respect to...

[SIZE="4"]Definition/Summary The chain rule is an elementary rule of calculus, but it can be understood without any knowledge of calculus: If a depends on b, and b depends on c, then the rate at which a changes with respect to b times the rate at which b changes with respect to c equals...

The question: This is the solution that was given by my teacher Attempt: I understand how the work is done until the 3-4 line. Where did the 1-cos2x disappear to in the 4th line? I know you can use the outside inside method but try as I might, I can't seem to understand how the final...

Need to find the Derivative using the chain rule y = x2sin4(x) + xcos-2(x) I am not sure where to start. answer in book is 2xsin4(x) + 4x2sin3(x)cos(x) + cos-2(x) +2xcos-3(x) xsin(x)

Hello, I have a tricky chain rule question, I think understanding it is more difficult than solving. For the function z=f(x,y) it is given that: f_{y}(0,-3)=-2 and \[f_{x}(0,-3)=3\] so for the function \[g(x,y)=f(2\cdot ln(x+y),x^{4}-3y^{2})\] choose the correct answer: (1)...

hey pf! suppose i have a function ##f( x , y)##. i make a change of variables such that ##z(x,y)## in such a way that now ##f( z , y)##. how do i find $$\frac{\partial f}{\partial y}$$ $$\frac{\partial f}{\partial x}$$ $$\frac{\partial^2 f}{\partial y^2}$$ $$\frac{\partial^2 f}{\partial x}$$...

Homework Statement if z = x2 + 2y2 , x = r cos θ , y = r sin θ , find the partial derivative [SIZE="4"]\left(\frac{\partial z}{\partial \theta}\right)_{x} Homework Equations z = x2 + 2y2 x = r cos θ y = r sin θ The Attempt at a Solution The textbook says that the equation...

Suppose we have a function V(x,y)=x^2 + axy + y^2 how do we write \frac{dV}{dt} For instance if V(x,y)=x^2 + y^2, then \frac{dV}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} So, is the solution \frac{dV}{dt} = 2x \frac{dx}{dt} + ay\frac{dx}{dt} + ax\frac{dy}{dt} + 2y \frac{dy}{dt}

Homework Statement Problem: Given C is the graph of the equation 2radical3 * sinpi(x)/3 =y^5+5y-3 Homework Equations (1) Prove that as a set C= {(x,y) Exists at all Real Numbers Squared | 2radical3 * sinpi(x)/3 =y^5+5y-3 is the graph of a function differentiable on all real...