Differentiation Definition and 1000 Threads

Hi, Just a question on an example in a maths textbook. See attached image for question below. So, I understand that if you set u=sin(x) and v=e^-cos(x) f'(x)=u'.v + u.v' But I'm stuck looking at e^-cos(x), could it also be classified e^(w)? Also, the second step in differentiating...

Folks, I am stuck on an example which is partial differenting a functional with indicial notation The functional ##\displaystyle I(c_1,c_2,...c_N)=\frac{1}{2} \int_0^1 \left [ \left (\sum\limits_{j=1}^N c_j \frac{d \phi_j}{dx}\right )^2-\left(\sum\limits_{j=1}^N c_j \phi_j\right)^2+2x^2...

I've got some maths homework to do over the summer before I go back to uni and there's this stupid question on there which is one of those 'so basic I don't know it' kind of questions, so here goes. Homework Statement What is LimΔx→0\frac{y(x+Δx) - y(x)}{Δx} ? The Attempt at a Solution...

Hi there! If one would want to prove that the indefined integral : \int[f(x)+g(x)]dx = \int f(x)dx + \int g(x)dx. Would this be apropriate: A(x) = \int[f(x)+g(x)]dx; B(x) = \int f(x)dx; C(x) = \int g(x)dx. And since the primitive of a fuction is another fuction whose derivative...

what exactly does Δy/Δx mean. for instance i know that when y=x2 Δy/Δx=2x but what does Δx/Δy equal? also why is the derivative always Δx/Δy? also what does Δx by itself mean for instance if y=x2 what is Δx i appreciate any and all answers thanks:smile:

Homework Statement Find the velocity and acceleration of a particle with the given position function: r(t)=<2cos t, 3t, 2sin t/t+1> The Attempt at a Solution v(t)=r'(t) dt= <-2sin t, 3, (2cos t/t+1) - (2sin t/(t+1)2)> a(t)=v'(t) dt = <-2cos t, 0, (4sint t/(t+1)3-(2sint t/(t+1)-(4...

Homework Statement This question is about entropy of magnetic salts. I got up to the point of finding H1, the final applied field. The Attempt at a Solution But instead of doing integration I used this: dS = (∂S/∂H)*dH = (M0/4α)(ln 4)2 I removed the negative...

Homework Statement I'm try to implicitly differentiate the function: xlny+√y=lnx The Attempt at a Solution And I got to the stage where I have: dy/dx = (1/x-lny)/(x/y+1/(2*√y)) which is where I have no idea on how to clean this up. Could someone please explain to me how to simplify a...

Hi everyone, I need help with a derivation I'm working on, it is the differentiation of the norm of the gradient of function F(x,y,z): \frac{∂}{∂F}(|∇F|^{α}) The part of \frac{∂}{∂F}(\frac{∂F}{∂x}) is bit confusing. (The answer with α=1 is div(\frac{∇F}{|∇F|}), where div stands for...

f(x)=x^2-3x-3x^-2+5x^-3 I need help finding the 1st derivative of this function using the power rule.If you can help can you explain how you got the answer.I tried like five times but the differentiation calculator says I am getting the wrong answer.heres how o attempted it. f(x)'=(2)(x)^2-1 -...

1. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/hr and ship B is sailing north at 25 km/hr. How fast is the distance between the ships changing at 4:00 pm? 2. None 3. I have the distance as 150 km. I have the variables \frac{dx}{dt} = 35 and...

Something about this worked problem looks off. Is this example correctly solved using logarithmic differentiation? The original problem is y = (2-x)^(sqrt x). If anyone who is rather confident with this could double check this example it would really help me out. Thanks. I attached the...

Homework Statement y = [2x + 1]^5 * [(x^4) - 3]^6Homework Equations I take the derivative of the natural log of both sides: (y' / y) = [(10 ln(2x + 1)^4) / (2x + 1)] + [(24x^3 ln(x^4 - 3)^5) / (x^4 - 3)] then I multiply both sides by the original function: y' = [((10 ln(2x + 1)^4) / (2x +...

Homework Statement Find an equation of the tangent line to x^3 + y^3 - 6xy = 0 at the point ((4/3), (8/3)) Homework Equations I got y = -(8/5)x + (24/5). Is this correct? The Attempt at a Solution Lots of algebra involved. Sorry. but I'd rather not type it. I take the derivative of...

Homework Statement Find dy / dx for sqrt(xy) = x - 2y. Homework Equations I don't know how to simplify [(xy' + y) / 2sqrt(xy)] = (1 - 2y') to y' = [- y + 2sqrt(xy)] / [x + 4sqrt(xy)]. The Attempt at a Solution I do everything Wolfram Alpha does here...

I am not very used to jugglery of tensors...I am learning it all now-a-days...I am trying to read a paper...and stuck at a point..:( ...It will be of great help if someone could help me get at eqn (34) from eqn (32) (cf. attached.) d/d\tau=u^\alpha\partial_\alpha (I think) and semi-colon is for...

if first derivative is the slop of the given functions, then what is the physical meaning of exponential function remaining the same function after differentiation?? does it mean its vertical tangency make it indifferentiable? please clarify me the concept... regards

To get the value of g, the period(T), length of pendulum (l) and radius of pendulum bob (a) were measured. Well, my question is actually to find which accuracy of one of these measurements need to be improved? The formulae given to find the probable error are ε _gT = (partial differentiation...

Hello everyone, I have a question that I have spent many nights pondering and hours on my whiteboard considering. I apologize in advance if this question seems a bit elementary, but to me it is something that I believe is all important before I can understand all of calculus. How is...

Hi everyone, I know that if z = f(x,y) = x^2y + xy^2 then \frac{\partial z}{\partial x}=2xy+y^2 and \frac{\partial z}{\partial y}=x^2+2xy Please correct me if I am wrong. In the physics, can anyone please tell me what is the meaning of below formula? \frac{\partial V}{\partial t} Where...

Homework Statement Need to find the tangent to the curve at: e^(xy) + x^2*y - (y-x)^2 + 3 I just implicitly differentiate the expression to find the gradient and then use the points given to find the equation, right? Or does this involve partial differentiation? Homework Equations...

Homework Statement Image of the problem: http://prntscr.com/addkf Homework Equations My question is how I can solve the equation I gave above. Should I use logarithmic differentiation? Because I think that the logarithmic differentiation is used when y = (the equation) but my problem is f...

Homework Statement R(x) := ∫ exp ( -y^2 - x^2/y^2 ) dy The Attempt at a Solution I move the derivative operator inside the integral and differentiate with respect to x R'(x) = ∫ [ - 2x/y^2 ] exp ( -x^2/y^2 - y^2 ) dy Then I let: t = x/y and dy = - x/t^2 dt R'(x) = 2 ∫ [ - x ] [ t^2 /...

Homework Statement An angular displacement θ radians in time t seconds is given by the equation θ = sin 3t. Find a:) angular velocity when t = 1 second b:) the smallest positive value of t for which the angular velocity is 2rad/s c:) the angular acceleration when t = 0.5 seconds d:) the...

I am reading about this topic, and I came across this sentence "Remember, every time we want to differentiate a function of y with respect to x, we differentiate with respect to y and then multiply by dy/dx." What exactly does this mean?

I heard that the formula below can be used to evaluate some kinds of integrals but I can't find what kinds and how to do it.Could someone name those kinds and also the procedure? \frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x,b(x)) b'(x) - f(x,a(x)) a'(x) + \int_{a(x)}^{b(x)}...

Homework Statement A transverse wave on a cord is given by D(x, t) = 0.19sin(2.9x - 35t), where D and x are in m and t is in s. 1) At t = 8.6*10^-2 s, what is the displacement of the point on the cord where x = 0.62 m? 2) At t = 8.6*10^-2 s, what is the velocity of the point on the cord...

Homework Statement u = x^u + u^y Find the partial derivatives of ##u## w.r.t. ##x, y##.Homework Equations Only the one. The Attempt at a Solution I've attempted reducing the problem using logs, but the resulting equations seem no more tenable to me. I'm sure there is a nice trick... it...

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)...

differentiation of a functional Where \phi = \phi(x) and the functional F=F(\phi(x)) = \int d^d x [\frac{1}{2}K^2(\bigtriangledown\phi)^2+ V (\phi)] , the author says the derivative with respect to phi gives \frac {\partial F} {\partial \phi(x)} = -K^2\bigtriangledown^2\phi + V'(\phi)...

This is from my text, "Linear Algebra" by Serge Lang, pg 11: -The two functions et, e2t are linearly independent. To prove this, suppose that there are numbers a, b such that: aet + be2t=0 (for all values of t). Differentiate this relation. We obtain aet + 2be2t = 0. Subtract...

I have a dataset in two columns X and Y, sorted in ascending values of X. I'm trying to find its numerical derivative, however, the "noise" (it's very hard to see any noise in the dataset itself when plotted), but the noise gets massively amplified to the point where the numerical derivative...

Differentiate y = ln(x^{2} + sin x)exp^{4xcos(x)} (2x + 1)^{8} Thanks in Advance for Your Help

Hey there guys, So I've been doing some Thermodynamics revision particularly involving the equation pV^{\gamma}=constant , which is the adiabatic equation of state. Now in my notes it says: "we can differentiate this to obtain a relation between changes in volume and pressure...

Homework Statement for a.) f(x) =1/ ( (1+x)^2 ) what is the radius of convergence? b.) Use part a.) to find a power series for f(x)=1/ ( (1+x)^3) c.) Use part b.) to find a power series for f(x) =x^2 /( (1+x)^3) Homework Equations I want to check my work. I used properties of functions...

Homework Statement a) if f(x)= ln(x/√(a-x^2)) show that f'(x) = a^2/x(a^2-x^2) [b] ∫1/x(25-x^2) dx The Attempt at a Solution for a) i tried differentiating the top (ans. = 1) then the bottom.. obviously the bottom's where hte prob is at lol.. i kno d/dx ln[f(x)] --> 1/(f(x) χ f...

Homework Statement Give an example of a continuous function f:R^2→R having partial derivatives at (0,0) with f_1 (0,0)≠0,f_2 (0,0)≠0 But the vector (f_1 (0,0),f_2 (0,0)) does not point in the direction of maximal change, even though there is such a direction. (If this is too difficult...

Homework Statement Suppose that a and b are real numbers. Find all values of a and b (if any) such that the functions f and g, given by a) f(x)={ax+b if x<0 and sin(x) if x≥0} b) g(x)={ax+b if x<0 and e2x if x≥0} are (i) continuous at 0 and (ii) differentiable at 0...

Homework Statement Find dy/dx in terms of x and y if.. x2-√(xy)+y2=6 Homework Equations The Attempt at a Solution so I started by.. x2-√(xy)+y2=6 deriving the LHS 2x+2y(dy/dx)-1/2(xy)-1/2(1(y)+x(dy/dx)) Simplifying the last term...

Hi, for a presentation I am requested to give some examples of the Real world applications of Parametric Differentiation. Now i know its to do with a differentiation of 3 variables that are connected, but for the love of god i cannot think of any examples of its practical uses. any help...

Homework Statement Prove or disprove: Suppose f:[a,b]->R is continuous. If f is diff on interval (a,b) and f'(x) has a limit at b, then f is diff at b. Homework Equations We say that f is differentiable at x0 to mean that there exists a number A such that: f(x)=f(x0)+A(x-x0)+REM...

Homework Statement (x,y) = x√(xy) The answer says: fx=3/2*√(xy) fy=(x√x) / (2√y) fxx= (3√y) / (4√x) fxy= (3√x) / (4√y) fyx =(3√x) / (4√y) fyy = -(x√x) / (4y√ I don't get from the beginning. shouldnt fx be equal to (3/2)x^2 * (x^3 * y)^-(3/2)?? When I do second derivative fxx from fx, it...

Homework Statement use implicit differentiation to find an equation of the tangent line to the curve a the given point. y^2(y^2-4) = x^2(x^2-5) at (0,-2) Homework Equations y^2(y^2-4) = x^2(x^2-5) The Attempt at a Solution I got dy/dx to be (3x^2-10x)/(4y^3-8y) but...

Homework Statement differentiate the following y= x(x+2)(x+3) Homework Equations dy/dx The Attempt at a Solution The answer I'm given is dy/dx = 2x+5 Would this not be for (x+2)(x+3) = x2 +5x +6 dy/dx = 2x +5 +0 My problem is the x at the beginning of the brackets. Please...

I have x=x(t) and y=y(t) and I'm working in polar co-ordinates so $$x=rcos{\theta}$$ and $$y=rsin{\theta}$$. I want to find ${\theta}'(t)$ so by the chain rule I want $${\theta}'(x)*x'(t)+{\theta}'(y)*y'(t)$$. I know $${\theta}=arctan(y/x)$$ but how do I partially differentiate theta w.r.t x and y?

Homework Statement Differentiate sin(ax), cos(ax) and tan(ax) from first principles. Homework Equations The Attempt at a Solution I have used first principles to differentiate the three expressions and have been successful until I encountered limits of some expressions in the...

I am trying to differentiate the functions xn, eax and ln(ax) from first principles. I have successful in all three, but here's my problem. In finding the limit in each problem, you need to first Taylor expand to remove Δx from the denominator. But the very process of Taylor expansion uses...

Hello. I know how to do implicit differentiation taught in calculus 1, but I'm confused by something regarding it. Take the example: y3+y2-5y-x2=4 If we do implicit differentation we get: 3y2(dy/dx)+2y(dy/dx)-5(dy/dx)-2x=0 dy/dx=2x/(3y2+2y-5) Now, it makes sense how to...

why do you differentiate twice? In some of the questions in my homework, we have been asked to differentiate an equation twice. I understand that when you differentiate once, you are finding the gradient. When you intergrate, your finding the area. However what is the reason for differentiating...

Homework Statement Consider the following implicit scheme: y_{n+1}=y_{n}+\frac{\Delta t}{2}\left [f(y_{n+1})+f(y_{n})] By linearization one can obtain an explicit scheme which is an approximation to this - with approximation error O(\Delta t^{3}) Homework Equations The solution is...