Chain rule Definition and 511 Threads

Hello, I'm trying to follow Goldstein textbook to show that the Lagrangian is invariant under coordinate transformation. I got confused by the step below So ## L = L(q_{i}(s_{j},\dot s_{j},t),\dot q_{i}(s_{j},\dot s_{j},t),t)## The book shows that ##\dot q_{i} = \frac {\partial...

find y' $$y=\sqrt{7x+\sqrt{7x+\sqrt{7x}}}$$ ok this was on mml but they gave an very long process to solve it don't see any way to expand it except recycle it via chain rule any suggest...

Homework Statement Using the chain rule, find a, b, c, and d: $$\frac{\partial}{\partial x'} = a\frac{\partial}{\partial x} + b\frac{\partial}{\partial t}$$ $$\frac{\partial}{\partial t'} = c\frac{\partial}{\partial x} + d\frac{\partial}{\partial t}$$ Homework Equations Chain rule...

Homework Statement Find the derivative of ##y=cos^3(πx)## *Must be in Leibniz notation Homework EquationsThe Attempt at a Solution (i) $$Let~ w=y^3 , y=cos(u), u=πx$$ (ii) $$\frac{dw}{dy} = 3y^2,~ \frac{dy}{du} = -sin(u),~ \frac{du}{dx}=π$$ (iii) By the Chain Rule, $$\frac{dw}{dx} =...

Homework Statement This isn't really a homework problem, as the entire solution is laid out in the text. My question is in regards to a possible typo, which I have highlighted in blue in the given picture. Usually I don't like to second guess the text, but this one has been absolutely plagued...

I am confused when I should use the ∂ notation and the d notation. For example, on http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx, in Case 1, the author wrote dz/dt while in Case 2, the author wrote ∂z/∂t. Could anyone please explain to me when I should use the ∂ notation and the...

Homework Statement I am facing problem in applying the chain rule. The question which I am trying to solve is, " Find the second derivative of " Homework Equations The Attempt at a Solution So, differentiated it the first time, [BY CHAIN RULE] And now to find the second derivative I...

1. The problem statement, all variables, and given/known data Given is a second order partial differential equation $$u_{xx} + 2u_{xy} + u_{yy}=0$$ which should be solved with change of variables, namely ##t = x## and ##z = x-y##. Homework Equations Chain rule $$\frac{dz}{dx} = \frac{dz}{dy}...

Homework Statement Homework EquationsThe Attempt at a Solution I am trying to repair my rusty calculus. I don't see how du = dx*dt/dt, I know its chain rule, but I got (du/dx)*(dx/dt) instead of dxdt/dt, if I recall correctly, you cannot treat dt or dx as a variable, so they don't cancel...

Greg Bernhardt submitted a new PF Insights post Demystifying the Chain Rule in Calculus Continue reading the Original PF Insights Post.

Homework Statement I am going through a derivation of the thermal energy equation for a fluid and am stumped on one of the steps. Specifically, the text I am using converts the term: P/ρ*(Dρ/Dt) to: ρ*D/Dt(P/ρ) - DP/Dt where: ρ = density P = pressure D/Dt = material derivative The text...

I'm trying to understand why $$\left(\frac{\partial P}{\partial T}\right)_V = -\left(\frac{\partial P}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_P$$ where does the minus sign come from?

<Moderator's note: Moved from a technical forum and thus no template.> Question: A certain amount of oil on the sea surface can be considered as circular form and the same thickness throughout its surface. At a certain time, the following are noted Data: Oil is supplied to the spot at 5m^3/min...

Homework Statement This is a chain rule problem that I can't seem to get right no matter what I do. It wants me to find the derivative of y=sqrt(x+sqrt(x+sqrt(x))) Homework Equations dy/dx=(dy/du)*(du/dx) d/dx sqrtx=1/(2sqrtx) d/dx x=1 (f(x)+g(x))'=f'(x)+g'(x) The Attempt at a Solution My...

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 6: Differentiation ... I need help in fully understanding an aspect of the proof of Theorem 6.1.6 ...Theorem 6.1.6 and its proof ... ... reads as follows: In the...

I am having difficulty trying to figure the following . What is \frac{\mathrm{d} }{\mathrm{d} x}f(x,y) where x is a function of s and t. Here is my calculation \frac{\mathrm{d} }{\mathrm{d} x}f(x(s,t),y) = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial...

In my quest to understand the Euler-Lagrange equation, I've realized I have to understand the chain rule first. So, here's the issue: We have g(\epsilon) = f(t) + \epsilon h(t). We have to compute \frac{\partial F(g(\epsilon))}{\partial \epsilon}. This is supposed to be equal to \frac{\partial...

Homework Statement How to obtain the famous formula of velocity transformation using a chain rule. I know that there is a straightforward way by dividing ##dx## as a function of ##dx`## and ##dt`## on ##dt## which is also a function of them. But I would rather try using the chain rule. Homework...

Homework Statement Evaluate the derivative of the following function: f(w)= cos(sin^(-1)2w) Homework Equations Chain Rule The Attempt at a Solution I did just as the chain rule says where F'(w)= -[2sin(sin^(-1)2w)]/[sqrt(1-4w^(2)) but the book gave the answer as F'(w)=(-4w)/sqrt(1-4w^(2))...

Homework Statement To show that ##K=V^uK_u## is conserved along an affinely parameterised geodesic with ##V^u## the tangent vector to some affinely parameterised geodesic and ##K_u## a killing vector field satisfying ##\nabla_a K_b+\nabla_b K_a=0## Homework Equations see above The Attempt at...

Homework Statement Use the top line to get 1) and 2) Homework Equations above The Attempt at a Solution So for 2) split the log up using ##log (AB)=log (A) + log (B) ## and this is simple enough I think I may be doing something stupid with 1) though. I have ##\frac{\partial}{\partial...

Hi, I have a probably very stupid question: Suppose that there is an expression of the form $$\frac{d}{da}ln(f(ax))$$ with domain in the positive reals and real parameter a. Now subtract a fraction ##\alpha## of f(ax) in an interval within the interval ##[ x_1, x_2 ]##, i.e. $$f(ax)...

Homework Statement Question has been attached to topic. Homework Equations Chain rule. The Attempt at a Solution $$\frac {dy}{dt} . \frac{dt}{dx} = \sqrt{t^2+1}.cos(π.t)$$ $$\frac{d^2y}{dt^2}.(\frac{dt}{dx})^2 = 2 $$ $$\frac{d^2y}{dt^2}.(t^2+1).cos^2(π.t)= 2 $$ and for the t=3/4...

(Sorry for the mistakes first thread using hand held device) Hello, I was working on Harold T. Davis Introduction to Nonlinear Differential and Integral Equations I saw this following equations 1-So equation 4 came as a result of chain rule applies on equation. 3 ? 2- how did equation 5...

Homework Statement (a) Light waves satisfy the wave equation ##u_{tt}-c^2u_{xx}## where ##c## is the speed of light. Consider change of coordinates $$x'=x-Vt$$ $$t'=t$$ where V is a constant. Use the chain rule to show that ##u_x=u_{x'}## and ##u_{tt}=-Vu_{x'}+u_{t'}## Find ##u_{xx},u_{tt},##...

I understand that when you use the chain rule you multiply the exponent by the number in front and then reduce the power by 1. So the derivative of 2x^3 = 6x^2 I'm confused now however on how you would solve something like e^-3x, the answer turns out to be -3e^-3x Am I missing a rule? Why...

I'm reading a textbook that says: "The directional derivative in direction ##u## is the derivative of the function ##f( \mathbf x + \alpha \mathbf u)## with respect to ##\alpha##, evaluated at ##\alpha=0##. Using the chain rule, we can see that ##\frac {\partial}{\partial \alpha} f( \mathbf x...

We start with: d2y/dx2 And we want to consider x as function of y instead of y as function of x. I understand this equality: dy/dx = 1/ (dx/dy) But for the second order this equality is provided: d2y/dx2 =- d2x/dy2 / (dx/dy)3 Does anybody understand where is it coming from? The cubic...

let df=∂f/∂x dx+∂f/∂y dy and ∂f/∂x=p,∂f/∂y=q So we get df=p dx+q dy d(f−qy)=p dx−y dqand now, define g. g=f−q y dg = p dx - y dq and then I faced problem. ∂g/∂x=p←←←←←←←←←←←←←←← book said like this because we can see g is a function of x and p so that chain rule makes ∂g/∂x=p but I wrote...

Hello, Here is the question: I can not figure out how we are to apply chain rule to the second order derivative. May somebody clarify that?

I want to evaluate $$ \frac{d}{dt}\int_{0}^{^{\eta(t)}}\rho(p,t)dz $$ where p itself is $$ p=p(z,t) $$ I have the feeling I have to use Leibniz rule and/or chain rule, but I'm not sure how... Thanks.

Homework Statement -here is the problem statement -here is a bit of their answer Homework Equations Chain rule, partial derivative in spherical coord. The Attempt at a Solution I tried dragging out the constant and partial derivate with respect to t but still I can't reach their df/dt and...

Hello! (Wave) Suppose that $u(t,x)$ is a solution of the heat equation $u_t-\Delta u=0$ in $(0,+\infty) \times \mathbb{R}^n$. I want to show that $u_k \equiv u(k^2 t, kx)$ is also a solution of the heat equation in $(0,+\infty) \times \mathbb{R}^n, \forall x \in \mathbb{R}^n$. If we have a...

I was thinking if the known methods of integration are enough to integrate any given function. In differentiation, we've evaluated the derivatives of all the basic functions by first principles and then we have the chain rule and product rule to differentiate any possible combination (product or...

I have ## \int_{t = 0}^{t = 1} \frac{1}{x} \frac{dx}{dt} dt = \int_{t = 0}^{t = 1} (1-y) dt ## [1] The LHS evaluates to ## ln \frac{(x(t_0+T))}{x(t_0)} ##, where ##t_{1}=t_{0}+T## My issue is that, asked to write out the intermediatary step, I could not. I am unsure how you do this when the...

Homework Statement a. Given u=F(x,y,z) and z=f(x,y) find { f }_{ xx } in terms of the partial derivatives of of F. b. Given { z }^{ 3 }+xyz=8 find { f }_{ x }(0,1)\quad { f }_{ y }(0,1)\quad { f }_{ xx }(0,1) Homework Equations Implicit function theorem, chain rule diagrams, Clairaut's...

Homework Statement Suppose $$z=x^2 sin(y), x=5t^2-5s^2, y=4st$$ Use the chain rule to find $$\frac{\partial z}{\partial s} \text{ and } \frac{\partial z}{\partial t}$$ Homework Equations $$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} +...

Homework Statement F(x) = (integral from 1 to x^3) (t^2 - 10)/(t + 1) dt Evaluate F'(x) Homework Equations Using the chain rule The Attempt at a Solution Let u = x^3 Then: [((x^3)^2 - 10) / (x^3 + 1)] ⋅ 3x^2 *step cancelling powers of x from fraction* = (x^3 - 10)(3x^2) = 3x^5 - 30x^2 I am...

While solving an equation, the lecturer was using substitution in this video: x=au was subbed in for Psi at timestamp 39:27 d/dx = (1/a)(d/du). I get that. But then the second derivative is stated as being d2/dx2 = (1/a2)(d2/du2) How is it (1/a2) if we do not know if there is an "a" in the...

$\tiny\text{Whitman 8.7.18 chain rule} $ $$\displaystyle I=\int { \left({t}^{3/2}+47\right)^3 \sqrt{t} } \ d{t} ={ \left({t}^{3/2}+{47}^{}\right)^4/6 } + C$$ $$\begin{align} \displaystyle u& = {t}^{3/2}+47& du&=\frac{3}{2}{t}^{1/2} \ d{t}& \\ \end{align}$$...

Consider a surface defined by the equation ##g(x, y, z)=0##. The intersection between this surface and the plane ##z=c## produces a curve that can be plotted on an x-y plane. Find the gradient of this curve. By chain rule, ##\frac{\partial y}{\partial x}=\frac{\partial y}{\partial...

Homework Statement To show that ##\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0## where ##p=p(z)##, ##p'=p(z+dz)##, ##s'=s(z+dz)##, ##s=s(z)## Homework Equations I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions...

Is the chain rule below wrong? What I propose is as follows: Given that ##x_i=x_i(u_1, u_2, ..., u_m)##. If we define the function ##g## such that ##g(u_1, u_2, ..., u_m)=f(x_1, x_2, ..., x_n)##, then ##\frac{\partial g}{\partial u_j}=\sum_{i=1}^n\frac{\partial f}{\partial x_i}\frac{\partial...

$$\tiny\text{Whitman 8.7.15 Chain Rule} $$ $$\displaystyle I=\int \frac{\sec^2\left({t}\right)}{\left(1+\tan\left({t}\right)\right)^2}\ d{t} =\frac{-1}{2\left(1+\tan\left({t}\right)\right)} + C$$ $\begin{align}\displaystyle u& = \tan\left({t}\right)& du&= \sec^2 \left({t}\right)\ d{t} \\...

Suppose you have a parameterized muli-varied function of the from ##F[x(t),y(t),\dot{x}(t),\dot{y}(t)]## and asked to find ##\frac{dF}{dt}##, is this the correct expression according to chain rule? I am confused because of the derivative terms involved. ##\frac{dF}{dt}=\frac{\partial...

Differentiate the following two problems. 1. x divided by the square root of x squared+ 1 2. The square root of x + 2 divided by the square root of x - 1 Thank you.

Homework Statement Use the Chain Rule to find the 1. order partial derivatives of g(s,t)=f(s,u(s,t),v(s,t)) where u(s,t) = st & v(s,t)=s+t The answer should be expressed in terms of s & t only. I find the partial derivatives difficult enough and now there is no numbers in the problem, which...

while solving differential equations, I got a bit confused with chain rule problem. The solution says below yprime = z then y double prime = z (dz/dy) = z prime but I don't understand why the differentiation of z is in that form. Please help...

Homework Statement Suppose ω = g(u,v) is a differentiable function of u = x/y and v = z/y. Using the chain rule evaluate $$x \frac{\partial ω}{\partial x} + y \frac {\partial ω}{\partial y} + z \frac {\partial ω}{\partial z}$$ Homework EquationsThe Attempt at a Solution u = f(x,y) v = h(y,z)...

(df/dx) + (df/dy)* (dy/dx) = df(x,y)/dx My book mentions the chain rule to obtain the right side of the equation, but I don't see how. The chain rule has no mention of addition. The furthest I got was applying the chain rule to the right operant resulting in: df/dx + df/dx = 2(df/dx)