Differentiating Definition and 270 Threads

For this problem, The solution is, My solution is, Where I solved these equations to find where the function is ##f(x) > 0## and ##f(x) < 0##. Using ##x^2(1 - x) \geq 0## and ##x^2(1 - x) < 0## First equation: ##x^2 \geq 0 \implies x \geq 0## and ## 1 \geq x## Second equation: ##x^2(1...

Why when we differentiate ## E = \frac {1}{2}mv^2 + \frac {1}{2}kx^2 ## with respect to time the answer is ## \frac {dE}{dt} = mva + kxv ##? I though it would be ##\frac {dE}{dt} = ma + kv ##. Many thanks!

I differentiated both sides of Euler's formula with respect to x : e^ix = sin x + i cos x => ie^ix = cos x - i sin x Then for comparison I multiplied both sides of Euler's formula by i: e^ix = sin x + i cos x => ie^ix = i sin x - cos x Each of these two procedures seems to yield the...

What is in our DNA that separates us from other primates?

I was wondering if anyone could write out Maxwell's relations for partial derivatives with respect to particle count ##N_i##. Starting from the fundamental thermodynamic relation, $$dU(S,V,N_i)=TdS-PdV+\sum_{i}\mu _idN_i$$ $$dU(S,V,N_i)=\left ( \frac{\partial U}{\partial S} \right )_{V,\left...

Hi, I have recently learned the technique of integration using differentiation under the integral sign, which Feynman mentioned in his “Surely You’re Joking, Mr. Feynman”. So, I decided to try it on the Gaussian Integral (I do know the standard method of computing it by squaring it and changing...

Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to...

From the question, $$f(F,\theta)=F \cos \theta$$ 1. If I use: $$df=dF \cos{\theta} -F \sin {\theta} d\theta$$ and using radian, $$df=dF \cos{\theta} -F \sin {\theta} d\theta \frac {\pi}{180^\circ}=5.28 N$$ 2. But, if I take logarithm to both side: $$ln f=ln F+ln \cos{\theta}$$ differentiate...

Hello, I found this link very useful: https://www.quora.com/Why-is-cosine-used-in-dot-products-and-sine-used-in-cross-products I understand all of Anders Kaseorg's discussion except for ONE PONT. At the very end, he writes: "[the evil twin of the dot product] is not differentiable at parallel...

I tried doing it a few times and this is all I get: c(˙f1+˙f2)=a˙a2+a2˙a−3ca2+˙ha2+2ha+2˙af′2+2a˙f′2c(f1˙+f2˙)=aa˙2+a2a˙−3ca2+h˙a2+2ha+2a˙f2′+2af2′˙ Please let me know where I'm going wrong. Thanks

<mentor: change title> In thermodynamics, there is a function which, for the three variables x, y, and z, can be given as ##G = xG_x+yG_y+zG_z + N[x\ln(x) + y\ln(y)+ z\ln(z)]+E(x,y,z)## where G_x, G_y, G_z, and N are some constants and E is some arbitrarily complicated term. There is a...

i'll be grateful for any advice I've already tried using but it just gets massive and makes me get lost.

T = (x+\frac{1}{\alpha}) sinh(\alpha t) X = (x+\frac{1}{\alpha}) cosh(\alpha t) - \frac{1}{\alpha} Objective is to show that ds^2 = -(1 +\alpha x)^2 dt^2 + dx^2 via finding dT and dX and inserting them into ds^2 = -dT^2 + dX^2 Incorrect attempt #1: dT= (dx+\frac{1}{\alpha})...

My Question : Shouldn't differentiating ##-log B## give ##\frac{-\delta B}{B}##? (Note : A, B and Z are variables not constants) By extension for ##Z=A^a \,B^b\, C^c## where ##c## is negative, should ##\frac{\Delta Z}Z=|a|\frac{\Delta A}A+|b|\frac{\Delta B}B-|c|\frac{\Delta C}C##?

Homework Statement Find the first and second derivatives of ##\displaystyle f(x)=\frac {1} {x^2+6}## Homework EquationsThe Attempt at a Solution [/B] ##\displaystyle f(x)=\frac {1} {x^2+6}## ##\displaystyle f(x)=(x^2+6)^{-1}## ##\displaystyle f'(x)=-1(2x)(x^2+6)^{-2}## ##\displaystyle...

Homework Statement Differentiate ##F(x)=4^{3x}+e^{2x}## Homework EquationsThe Attempt at a Solution I have an exam coming up and need some help with this problem. ##F(x)=4^{3x}+e^{2x}## ##f(x)=4^{3x}## ##G(x)=e^{2x}## First I need to find f'(x) and g'(x), which I thought I did correctly...

Hello, I have this potential: ## V(\chi) = \frac { F ’’ (\chi) [ 2 F(\chi) - \chi F’ (\chi) ]}{ (F’(\chi))^3} ## How to get ## \frac{ d V(\chi)}{d \chi} = \frac{ \chi F’’ + F’ - F’ }{ F’^2} - 2 \frac{ \chi F’ -F }{ F’^3} F’’ ~~~~~~(*)## My trail, ## V( \chi) = 2 F F’’ F’^{-3} - \chi F’’...

I am looking at a solution to an integral using differentiation under the integral sign. So let ##\displaystyle f(t) = \frac{\log (tx+1)}{x^2+1}##. Then, through calculation, ##\displaystyle f'(t) = \frac{\pi t + 2 \log (2) - 4 \log (t+1)}{4(1+t^2)}##. The solution immediately goes to say that...

Hi. If I have a vector v , say for velocity for example then v.v = v2 and I differentiate wrt t v.v I get 2v.dv/dt but if I differentiate v2 I get 2v dv/dt but v.dv.dt is not the same as v dv/dt so what am I doing wrong ? Thanks

Homework Statement [/B] Differentiate the power series for ##\frac 1 {1-x}## to find the power series for ##\frac 1 {(1-x)^2}## (Note the summation index starts at n = 1) 2. The attempt at a solution ##\sum_{n=1}^\infty n*x^{n-1}##

Homework Statement Find the derivative of the following functions. h(x)=8x2√x2+1 Homework EquationsThe Attempt at a Solution I just had a lesson on friday but it was a bit confusing to me. I am able to solve some problems but ones that contain multiple rules confuse me. If someone could...

What is the justification for differentiating some integrals with respect to constants in order to obtain result, i.e. ∂/∂a[∫e^(− ax^2).dx] =∫-x^2.e^(-ax^2) dx?I mean what if we say "a" was 3 then differentiating wrt 3 would have no significance?How can we treat it like a multivariable function :/

Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...

Homework Statement : [/B]find the dy/dx of xy=a constantHomework Equations : basic differentiation formulae[/B]The Attempt at a Solution :[/B] I know we can use logarithmic differentiation for differentiating x y..But can we differentiate it using chain rule and get answer as yxy-1.dy/dx =0. ?

I'm working through the problems in Mary Boas's Mathematical Methods text. Here's how she began solving one problem... "We take differentials of the equation 1/i + 1/o = 1/f (f=constant) to get -di/i2 - do/o2 = 0." So on the left side the first term was differentiated with respect to i and...

Hello! I have the equation FM = qvBsinθ . As the end result, I am trying to figure out what B I need to change θ even a little bit. To do that, I was planning to find the minimum B by differentiating B=(μe/4π)(qv x R / R3) in terms of R and setting it equal to zero. . I am assuming that this...

I've taken up to Calc. III (vector/multi-variable calculus) but have not had any classes that used intensive calculus for a few semesters. However, I'm now in a quantum physics class (its a glorified title really--it's more or less "modern physics") and am looking to see if I am correct in my...

I need to prove that for $-1 < x < 1$ $$\frac{1}{(1 - x)^2} = 1 + 2x + 3x^2 + 4x^3 ...$$ So, according to the textbook, the geometric series has a radius of convergence $R = 1$ (I'm not sure how this is true). In any case we can compare it to: $$\frac{1}{1 - x} =\sum_{n = 0}^{\infty} x^n$$...

can someone go through the steps how to differentiate (1+sin²x)³

Homework Statement Find the expression for the slope on the lower half of the circle y^2 + x^2 = 25. 2. Attempt at a solution. The text says you get 2x + 2y(dy/dx) = 0. I got this and then solved for dy/dx to get dy/dx = -2y - 2x. Then, I substituted for y the x value-expression for the...

Homework Statement I was reading about the Hermite integration scheme for N-body simulations, as seen here: http://www.artcompsci.org/kali/vol/two_body_problem_2/ch11.html#rdocsect76 This scheme uses jerk, the time rate of change of acceleration. The problem is that I don't know how to...

So I was just trying to differentiate (for no good reason) the equation : x=x0sin(wt) (w= angular frequency, x0= maximum displacement, t=time) to obtain the expression : a= -w2x I differentiated twice with respect to time the initial expression for x and got: a= -w2x0sin(wt) I must have...

Let ##k:\mathcal{O}\times\mathbb{R}^n\to\mathbb{R}##, with ##\mathcal{O}\subset\mathbb{R}^m## open, be such that ##\forall x\in\mathcal{O}\quad k(x,\cdot)\in L^1(\mathbb{R}^n) ##, i.e. the function ##y\mapsto k(x,y)## is Lebesgue summable on ##\mathbb{R}^n##, according to the usual...

Hello, folks. I'm trying to figure out how to take the partial derivative of something with a complex exponential, like \frac{\partial}{\partial x} e^{i(\alpha x + \beta t)} But I'm not really sure how to do so. I get that since I'm taking the partial w.r.t. x, I can treat t as a constant term...

I(α) = 0∞∫e-(x2+α/x2) dx Differentiating under the integral sign leads to: I(α) = 0∞∫-e-(x2+α/x2)/x2 dx Here I am supposed to let u = sqrt(a)/x, but the -x2 doesn't cancel out, Wolfram-Alpha tells me the answer is: e(-2 sqrt(α) sqrt(π))/(2 sqrt(α)). I understand where the sqrt(π))/(2)sqrt(α)...

Homework Statement Find all f(x) satisfying: ∫f dx ∫1/f dx = -1 Homework EquationsThe Attempt at a Solution I solved for ∫1/f dx and differentiated both sides (using the quotient rule for the right side): ∫1/f dx = -1 / ∫f dx 1/f = f / (∫f dx)2 (∫f dx)2 = f2 ∫fdx = ±f f = ±f' Solving the...

How do you differenciate tan2x and what are the steps?

Hello, I am having trouble finding the proper justification for being able to pass the derivative through the integral in the following: ## u(x,y) = \frac{\partial}{\partial y} \int_0^\infty\int_{-\infty}^\infty f(x') K_0( \sqrt{ (x - x')^2 + (y-y')^2 } \, dx' dy' ## ##K_0## is the Modified...

Homework Statement Differentiate Homework EquationsThe Attempt at a Solution du/dx = cos(x) dv/du=cos(u) dg/dv=cos(v) dg/dx = dg/dv.dv/du.du/dx = cosx.cos(sinx).cos(sin(sinx)) I know the answer is correct but my issue is in the understanding of the...

Homework Statement Given that ##Q' = \int d^{3}x \bigg[ \rho \big( \frac{vx}{c^{2}},x,y,z \big) - \frac{v}{c^{2}} j^{x} \big( \frac{vx}{c^{2}},x,y,z \big) \bigg]##, show the following: ##\frac{dQ'}{dv} = \frac{1}{c^{2}} \int d^{3}x \bigg[ x \big( \frac{\partial \rho}{\partial t} -...

I'm trying to understand the definition of maps between vector spaces (in normed vector spaces) listed in the following link: http://ocw.mit.edu/courses/mathematics/18-965-geometry-of-manifolds-fall-2004/lecture-notes/lecture3.pdf On the surface, this seems similar to what I expected from the...

Homework Statement Hi all, I'm trying to learn how to manipulate tensors and in particular to differentiate expressions. I was looking at a Lagrangian density and trying to apply the Euler-Lagrange equations to it. Homework Equations Lagrangian density: \mathcal{L} = -\frac{1}{2}...

I'm having some trouble with the terminology used in calculus. My book states: "Fortunately we don't need to solve an equation for Y in terms of X in order to find the derivative of Y. Instead we can use the method of implicit differentiation. This consists of differentiating both sides of the...

I know how to derive the expression Vout = -CR*d(Vin)/dt for a circuit differentiator that consists of a capacitor and a resistor, how I can understand the attached circuit, which is part of practice exercise for my class. Why do we need to add R2 and R3? When I run the simulation Vout =...

Can experiments differentiate like vs. opposite charges? Two electrons repel, whereas an electron and positron attract. But for macroscopic observers, in the absence of annihilation, could anyone tell whether paths deflected due to attractions or repulsion? Or, is there always annihilation...

Hi PF! I'm reading my math text and am looking at the heat eq ##u_t = u_{xx}##, where we are are given non-homogenous boundary conditions. We are solving using the method of eigenfunction expansion. Evidently we begin by finding the eigenfunction ##\phi (x)## related to the homogenous...

I keep getting the wrong answer when i try to differentiate cotx.. this is what i get: cotx = 1/tanx =cosx/sinx=cosx ⋅ sin^-1 so by the product and chain rule we have: sinx⋅(sin x)^-1+cos⋅(-1sin^2 x)^-1 ⋅(cosx)^-1 = sinx/sinx - cosx/cosx ⋅ sin^2x =1-1/sin^2 x where as the correct answer...

So, I have $$\d{x}{t} = 20 sex^2∂ \d{∂}{t}$$ And the text goes to: $$ \d{∂}{t} = \frac{1}{20}cos^2 ∂ \d{x}{t}$$ I don't understand where the cos comes from? Is it a trigonometric identity? If so I can't find it.

I do not fully understand how non local correlation functions are derived from the differentiation of local ones. Specifically how in the following paper equation 23 is derived from 21 since the paper implies H and dpdq are invariant under time...