Chain rule Definition and 511 Threads

If I have u = u(x,y) and let (r, t) be polar coordinates, then expressing u_x and u_y in terms of u_r and u_t could be done with a system of linear equations in u_x and u_y? I get: u_r = u_x * x_r + u_y * y_r u_t = u_x * x_t + u_y * y_t is this the right direction? Because by...

Homework Statement n=y*sqrt((V)/(v*x) and Q=sqrt(v*V*x)*f(n) so i have V=-dQ/dx=(dQ/dn)*(dn/dx) and the final answer is V=(1/2)*sqrt((v*V)/x)(n*df/dn-f) Homework Equations The Attempt at a Solution i have tried diff. by hand and also by maple and cannot get the answer. What am i...

I have a function F(u,v) that I need to get first and second order partial derivatives for (Gradient and Hessian). F(u,v) is very ugly, so I'm thinking of it like F(x,y,z) where I have another function [x,y,z]=G(u,v). So, I got my first orders, e.g.: dF/du = dF/dx*dx/du + dF/dy*dy/du +...

Okay so I'm doing chain rule work to go over the stuff from calc 1 before I take a departmental exam and I've run into this problem: Homework Statement Take the derivative of: f(x) = \frac{sin(x^2)}{ln sinx} Homework Equations Here's the formula I used (and always do) for the...

In a simple electric circuit, Ohm's law states that V = IR, where V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms. Assume that, as the battery wears out, the voltage decreases at 0.03 volts per second and, as the resistor heats up, the resistance is...

Homework Statement f(x) = ((x^2+2)^2)/(x+2)^1/2 Use the chain rule to find the derivative Homework Equations None The Attempt at a Solution ((x^2+2)^2)(x+2)^-1/2 PS: Answer in the book is 3x((x^2+2)^1/2) I have no idea how they get it there, would like some help, thx!

Hello hello. In class we went over the ''mini-chain rule'' once, and haven't gone over the real chain rule yet. I really want to understand how to go about solving this equation, and to really understand what is happening here. x=u3-3uv2 y=3u2v-v3 z=u2-v2 Define z as a function of x and...

Homework Statement z=f(x,y) x=escos(t) y=essin(t) show d2z/dx2+d2z/dy2 = e-2s[d2z/ds2+ d2/dt2 Homework Equations dz/dt=dz/dz(dx/dt)+(dz/dy)dy/dr The product rule The Attempt at a Solution I found d2x/dt2=2e2ssin(t)cos(t)d2z/dydx + e2scos2(t)dz/dy2 But, now I'm...

I stumbled upon this document that discusses the single variable chain rule: http://math.rice.edu/~cjd/chainrule.pdf At the bottom, there is an incorrect proof of the validity of the chain rule, but the author does not cite why the proof is wrong. I'm wondering if the problem is...

I am trying to find the second derivative of the function C:[0,1]^{2} \rightarrow [0,1] ,\quad \mbox{defined by }C=C(u,v) evaluated at u=F(x)=1-\exp(-\lambda_{1} x),\quad \lambda_{1} \geq 0 and v=G(x)=1-\exp(-\lambda_{2} x),\quad \lambda_{2} \geq 0 First I work out the first...

Hi there, I'm a new user to the forums (and Calculus) and I 'm hoping you can give me your opinion on my chain rule form below. When learning the chain rule, I was taught two forms. This form: \frac{d}{dx}f(g(x))=f'(g(x))g'(x) As well as the Leibniz form...

I'm trying to find the derivative of 0 = 3xcosƟ with respect to time. I know I should use the product rule for x and cosƟ. But I don't know what I should do with the constant 3. would it be like this? 0 = 3x(-sinƟ)(dƟ/dt) + 3(dx/dt)(cosƟ)

I'm having some trouble following this equation: \frac {d \Phi_B} {dt} = (-) \frac {d}{dx_C} \left[ \int_0^{\ell}dy \ \int_{x_C-w/2}^{x_C+w/2} dx B(x)\right] \frac {dx_C}{dt} = (-) v\ell [ B(x_C+w/2) - B(x_C-w/2)] \ Shouldn't the differentiation of the bracketed terms "killed" the...

Homework Statement 2 straight roads intersect at right angles. Car A, moving on one of the roads, approaches the intersection at 60km/h and car B moving on the other road, approaches the intersection at 80km/h. At what rate is the distance between the cars changing when A is 0.5km from the...

Use the chain rule to compute the partials of F(z,w) = f(g_1(z,w),g_2(z,w),z,w) where f(x,y,z,w)=x^2 +y^2 +z^2 −w^2 and g_1(z,w) = wcosz , g_2(z,w) = wsinz Evaluate the partials at z = 0, w = 1. Confirm your result by writing out F explicitly as a function of z and w, computing its...

Homework Statement A function f is called homogeneous of degree s if it satisfies the equation f(x1, x2, x3,... xn)=t^s*f(x1, x2, x3,... xn) for all t Prove that the \sum from i=1 to n of xi * df/dxi (x1, x2, x3,... xn) = sf(x1, x2, x3,... xn). Homework Equations The Attempt at a Solution...

If z = f(x,y) and x = r \cos{v}, y = r\sin{v} the object is to show that d = \partial since it's easier to do on computer Show that: \frac{d^2 z}{dr^2} + \frac{1}{r} \frac{dz}{dr} + \frac{1}{r^2} \frac{d^2 z}{dv^2} = \frac{d^2 z}{dx^2} + \frac{d^2 z}{dy^2} It's from Adams calculus, will...

Homework Statement Skethch the greaph of x^3/(x^3+1). Identify all extrema and points of inflection, asymptote equations, and easily found intercepts Homework Equations If a/b=0, a must be 0.(thats how I got critical points from first derivative) And chain rule: F'(x) = f '(g(x)) g '(x) And...

Could someone explain this to me please where n=y/squareroot(4vt) ∂C/∂t=(dC/dn)(∂n/∂t)=-(1/2)(n/t)(dC/dn) When i take the derivative of 1/t^1/2 i get -(1/2)t^(-3/2) so where does the (-3/2) go to in the final answer of -(1/2)(n/t)(dC/dn). Thank you very much!

D e^(b*t*ln(t)) + ln(x) respect to t my answer: t^(b*t)*(ln(t)*b)+b + 1/x

Chain Rule - intuitive "Proof" Suppose y = f(u), and u = g(x), then dy/dx = dy/du * du/dx. In an intuitive "proof" of the chain rule, it has this step: dy/dx = \lim_{\Delta x \to 0} \frac {\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac {\Delta y}{\Delta u} * \frac {\Delta u}{\Delta x}...

Homework Statement Differentiate f(x)=(3x^{2}+4)^{3}(5-3x)^{4} Homework Equations N/A The Attempt at a Solution I can see that this derivative is a product, yet also involves using chain rule. With this being said, am i just supposed to evaluate these separately using chain...

We have f(x(y,z),t(y,z)). This is more of a study question. I don't know how to expand out d^{2}f/dz^2 I know df/dz = df/dx*dx/dz + df/dt*dt/dz, but I don't know how to expand this to the 2nd derivative. I think the product rule comes into play? Not really sure. Thanks for your help.

Homework Statement Let T= g(x,y) be the temperature at the point (x,y) on the ellipse x=2sqrt2 cos(t) and y= sqrt2 sin(t), t is from 0 to 2pi. suppose that partial derivative of T with respect to x is equal to y and partial derivative of T with respect to y is equal to x. Locate the max and...

Homework Statement http://img21.imageshack.us/img21/6784/probi.jpg Homework Equations The Attempt at a Solution i have no idea where to begin and my textbook doesn't have any examples that look like this question.. can someone give me hints? whats the equation that l=10m after...

Homework Statement Use the chain rule to find (d/dx)(xx) by using the function f(y,z)=yz. Homework Equations Chain rule: \frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt} The Attempt at a Solution I honestly have no clue on how to use...

Homework Statement The length l, width w, and height h of a box change with time. At a certain instant the dimensions are l = 7 m and w = h = 9 m, and l and w are increasing at a rate of 6 m/s while h is decreasing at a rate of 3 m/s. At that instant find the rates at which the following...

Homework Statement I need to find the derivative of: y=\left(4x+3\right)^{4}\cdot\left(x+1\right)^{-3} Homework Equations Chain Rule Quotient or Product Rule The Attempt at a Solution So I tried to use quotient rule because...

There is a theorem in partial derivative If x= x(t) , y= y(t), z= z(t) are differentiable at t_{0}, and if w= f(x,y,z) is differentiable at the point (x,y,z)=(x(t),y(t),z(t)),then w=f(x(t),y(t),z(t)) is differentiable at t and \frac{dw}{dt}=\frac{\partial w}{\partial x}\frac{dx}{dt} +...

Homework Statement Suppose that w=f(x,y), x=r*cos(θ), y=r*sin(θ). Show that: (\frac{\partial w}{\partial x})^2 + (\frac{\partial w}{\partial y})^2 =(\frac{\partial w}{\partial r})^2 + \frac{1}{r^2} (\frac{\partial w}{\partial \theta})^2 Homework Equations the multivariable chain rule...

I'm having a bit of a hiccup understanding the differentiation that I am doing... I'd like to be clear on the concept rather than just knowing 'apply chain rule'. So I have a particle with equation: y=a(1+cos\theta) now the derivative with respect to time (the velocity in y) is...

Find the value of (f o g)' at the given value of x. f(u) = u5 + 1 u = g(x) = sqrt(x) x = 1 Ok so the section is based on the chain rule and came right out of my calculus book. I seem to be doing the problem right, i check my attempt over a few times and cannot seem to find the problem (the...

Homework Statement Prove that if z(x,y)=e^y f(ye^{\frac{x^2}{2y^2}}) is differentiable, then (x^2-y^2) \frac{\partial z}{\partial x} + xy\frac{\partial z}{\partial y} = xyz Homework Equations Chain Rule. The Attempt at a Solution A similar question is solved like this: Have this: z(x,y) =...

Okay, I know how to differentiate regular functions. But when it comes to fractions, I'm hopeless. This may be an extremely simple one to some, here is the function; "1/4x-7" I have to differentiate that using the chain rule. I think that u=4x-7, but I am not sure. As i said, I am horrible...

Hello all, I am stuck on what seems like a rather simple problem: Let f:\mathbb{R}^3 \rightarrow \mathbb{R} and g:\mathbb{R}^2\rightarrow \mathbb{R} be differentiable. Let F:\mathbb{R}^2 \rightarrow \mathbb{R} be defined by the equation F(x,y)=f(x,y,g(x,y)). Find DF in terms of the...

Find the derivative of y = [x + (x + (sin(x)2))5]3 I know that power and chain rule combined uses the equation n[g(x)]n-1 * g'(x) I don't even really know where to start with so many layers in the equation. I can only find examples with only one power. with my attempt I got...

In fluid mechanics velocity is given in the form \textbf{V}=u\textbf{i}+v\textbf{j}+w\textbf{k} Homework Statement A two-dimensional velocity field is given by \textbf{V}=(x^2-y^2+x)\textbf{i}+(-2xy-y)\textbf{j} At (x_o,y_o) compute the accelerations a_x\text{ and }a_y I am...

Hello everyone, I was looking at the proof of chain rule as posted here: http://web.mit.edu/wwmath/calculus/differentiation/chain-proof.html" I am having trouble understanding why delta(u) tends to 0 as delta(x) tends to 0. Can someone point out to me why that is so? Many thanks, Luca

Homework Statement x2+y2=1 I want to differentiate this equation. I know that the answer is 2x+2y*y'=0. Homework Equations The chain rule. The Attempt at a Solution I don't understand how you get 2y*y' from y2. Shouldn't it just be 2x+2y=0?

Homework Statement Hi all. I have an expression given by V(x,y) = ay+x2y2, where a is a constant. I wish to find the time-derivative of V(x,y), and this is what I have done: \frac{dV}{dt} = a\dot y + \frac{d}{dt}x^2y^2, where the dot over y represents differentiation w.r.t. time. My...

Homework Statement Homework Equations The Attempt at a Solution My dought is about if ,dw/dy*dy/dt = (-2)t^(-3)*dy/dt

Hi, I'm new to these forums so not exactly sure where to place this question, although calculus seems a good bet, so here goes: I'm currently taking a mechanics course at my university (current subject is work/energy), and I'll just post this snippit from our textbook (Physics for Scientists...

Homework Statement Differentiate y = \left(\frac{x+2}{\sqrt[3]{x}}\right)3 Homework Equations -Chain Rule -Quotient Rule -Power Rule -Product Rule? The Attempt at a Solution First I got rid of the fraction by taking the negative of x^3, and then used the chain rule to differentiate...

Homework Statement First problem: Let f(x,y) = x-y and u = vi+wj. In which direction does the function decrease and increase the most? And what u (all of them) satisfies Duf = 0 Second problem: Let z = f(x,y), where x = 2s+3t and y = 3s-2t. Determine \partial{z^2}/\partial{s^2}...

Homework Statement It is given that, \left(e^{-t^2}y\right)'=e^{-t^2}\left(y'-2ty\right), which I am trying to work out. Homework Equations f'(t)=h'(g(t))g'(t) (u\cdot v)'=u'v+uv'The Attempt at a Solution f(t)=e^{-t^2}y=h(g(t)) \text{let}\;g(t)=u=t^2\;\text{and}\;h(u)=e^{-u}y...

Hello! I got one question for you. How come that (f \circ g)'(x) = f'(g(x)) g'(x) ? Since (f \circ g)'(x)=f(g(x))' , f'(g(x))=f'(g(x)) g'(x). And now we can rewrite the equation like 1=g'(x) I don't understand that part. Also I don't understand why the flawed proof of the chain rule...

Hello. Let g(x,y) be a function that has second order partial derivatives. Transform the differential equation \frac{\delta ^{2}g}{\delta x^{2}}-\frac{\delta ^{2}g}{\delta y^{2}}=xyg by chaning to the new variables u=x^2-y^2 and v=xy The equation doesn't have to be solved. Okay, so this is...

Homework Statement Z is defined implicitly as a function of x,y by equation (z^2)x + 3xy^2 + e^((y^2)z) = 4. Find dz/dx Homework Equations dz/dx = -Fx/FzThe Attempt at a Solution Fx= z^2 + 3y^2 Fz=2zx+(y^2)e^((y^2)z) dz/dx= (z^2+3y^2)/[2zx+(y^2)e^((y^2)z)] I'm not sure if I used the partial...

This is stuff I do in order to understand analytical mechanics better, I encounter the followin thing: \frac{\partial L}{\partial \dot{\phi}} = \text{?} Where \dot{\phi} = \frac{\partial \phi}{\partial q} \frac{dq}{dt} = \frac{\partial \phi}{\partial q} \dot{q} I should know this! It is...

Homework Statement 2x^2+5xy-y^2=1 Homework Equations d/dx(f(u)x))=df/du * du/dx The Attempt at a Solution i got (2y-4x)/5x but I'm almost certain that its wrong...can anyone help me?