Differentiation Definition and 1000 Threads

What is ##\frac{d}{dx}(\frac{x}{y^2})##? Please tell me is it correct or not: ##\frac{d}{dx}(\frac{x}{y^2}) = \frac{[\frac{d}{dx}(x)] ⋅ (y^2) - (x) ⋅ [\frac{d}{dx} (y^2)]}{(y^2)^2}## ## = \frac{(x) ⋅ (y^2) - (x) ⋅ (\frac{d}{dy} (y^2)) ⋅ \frac{dy}{dx}}{y^4}## ##= \frac{xy^2 -...

Is Differentiation exact or just an approximation? I am wonder whether this question is meaningful or not. Slope is expressed as "it is approaching to a value as x is approaching 0" so it is inappropriate to ask such question. But when I deal with uniform circular motion, it is very confusing...

I believe there is a mistake in the second equation of (5.139). The equation is obtained from (5.138) using the Euler-Lagrange equation ##\frac{d}{dt}\frac{\partial L}{\partial\dot{\theta}}=\frac{\partial L}{\partial\theta}.## LHS##\,\,=\frac{d}{dt}\frac{\partial...

Homework Statement in the notes , 'by applying chain rule to LHS of the above equation ' , which equation is the author referring to ? it's given that f /x + (f/z)(z/x) = 0 , As we can see , the function contain variable x , y and z Homework EquationsThe Attempt at a Solution why not f /x +...

I'm having trouble evaluating the following expression (LATEX): ##\nabla_{i}\nabla_{j}T^{k}= \nabla_{i} \frac{\delta T^{k}}{\delta z^{j}} + \Gamma^{k}_{i m} \frac{\delta T^{m}}{\delta z^{i}} + \Gamma^{k}_{i m} \Gamma^{m}_{i l} T^{l}## What are the next steps to complete the covariant...

I have a basic understanding of the reason why we look for derivative or integration in Physics, based on the water flow example, where integration is the process of accumulating the varying water flow rate "2x" , while we reverse to the water flow rate by differentiating " x squared " the...

I have a derivative of a function with respect to ##\log \left(r\right)##: \begin{equation*} \frac{dN\left(r\right)}{d \log\left(r\right)} = \frac{N}{\sqrt{2\pi} \log\left(\sigma\right)} \exp\left\{-\frac{\left[\log \left(r\right) - \log\left(r_M\right)\right]^2}{2...

Homework Statement A particle moves along a straight line so that its acceleration t seconds after passing a fixed point O on the line is (2 - 2t) cm/s2. Three seconds after passing O, the particle has a velocity of 5 cm/s. Find and expression, in terms of t, for the velocity of the particle...

a particle of mass m is projected towards a point O with initial speed √5/3 m/s from a point P where [OP]=3 metres. the particle is repelled from O by a force of magnitude 4m/x³ where x is its distance from O. (i) show that the equation of motion is v dv=4x^-3 dx (ii) find how close the particle...

a car starts from rest. when it is at a distance s from its starting point its speed is v and its acceleration is 25v+v³. show that dv=(25+v²)ds and find (correct to 2 decimal places) its speed when s=.01 this is how i went about it dv/dt= acceleration dv/dt=dv/ds×ds/dt=v×dv/ds v dv/ds=25v+v³...

Hi, I have a doubt on the topic of bases and alkalis. I have learned that a alkali is a soluble base so does that mean Sodium Oxide(Solid) is a alkali and Lithium Hydroxide(Solid) is a alkali. Or are they considered alkali when they are dissolved? For example, Sodium Oxide becomes sodium...

Homework Statement a) Differentiate the following equation with respect to: 1) θ 2) Φ 3) ψ (Ua - Ub)' * C * r where: C is a 3 x 3 rotation matrix: [ cos θ cos ψ, -cos Φ sin ψ + sin Φ sin θ cos ψ, sin Φ sin ψ + cos Φ sin θ cos ψ] [ cos θ sin ψ, cos Φ cos ψ + sin Φ sin θ sin...

Explain to me: Why the 2πf came in front? I lost touch and sort of forgot.

I'm revising differentiation for tomorrow, I am hopelessly stuck on some things. I have some examples with answers, but I have no idea which rules are being used. y=2sin3x dy/dx=6cos3x I see that sin goes to cos but why is the two multiplied by the 3 (if that's what's happening)...

Does methane gravitationally differentiate in air, or does fluid dynamics mess things up & make it mix in with the air?

Homework Statement An inverted cone has a depth of 10cm and a base radius of 5cm. Water is poured into it at the rate of 1.5 cm^3/ min. Find the rate at which level of water in the cone is rising, when the depth of water is 4cm. Homework EquationsThe Attempt at a Solution [/B] I know that...

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.

Hi, in the above why is the left-hand side simple differentiation, i.e V is only function of t but in the right it is function of t, x, y, and z. It is very strange that one side is different than the other. Would you like to explain it? Thank you.

Since we have this relationship between x and y, as the two sides are equal, so are their derivatives. We just have to remember that as y is a function of x, any function of y is also a function of x, with the inner function "y" composed inside whatever is being told to do to the y. So to...

I know that ##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu} \phi \big) = \partial_{\mu} \phi##. Now, I need to prove this to myself. So, here goes nothing. ##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu}...

Homework Statement \int_1^2 \frac {e^x}{x}\,dx Through the use of differentiation under the integral sign. 2. The attempt at a solution Tried inputting a several times, each one resulting in another function without an elementary anti-derivative (example given below) I(a)=\int_1^2 \frac...

To perform implicit differentiation we must make use of the chain rule. Basically if you have a function composed in another function, its derivative is the product of the inner function's derivative and the outer function's derivative. All other rules (such as the sum rule, the product rule...

I'm a bit iffy with the whole of the 'related rates' topic of my calculus course. I've tried coming up with a question of my own to see if I can solve it. The question is as follows: The distance between a point on the ground and the bottom of a pole is 26m. The angle of inclination from that...

Homework Statement Hi,I saw a statement in my physics notes like this(Anyway it is a maths problem): where L is a general differential operator.G is a green's function(I guess it is irrelevant) My question is related to the red line: Suppose we have this: ∂/∂x ∫ f(x-y)g(y) ∂y is it...

Hello, friends! 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...

So to find the x values of the stationary points on the curve: f(x)=x3+3x2 you make f '(x)=0 so: 3x2+6x=0 x=0 or x=-2 Then to find which of these points are maximum or minimum you do f ''(0) and f ''(-2) so: 6(0)+6=6 6(-2)+6=-6 so the maximum has an x value of -2 and the minimum has an x value...

I am trying to estimate the second derivative of ##sin(x)## at ##0.4## for ##h=10^{-k}, \ \ k=1, 2, ..., 20## using: $$\frac{f(x+h)-2f(x)+f(x-h)}{h^2}, \ \ (1)$$ I know the exact value has to be ##f''(0.4)=-sin(0.4)= -0.389418342308651.## I also want to plot the error as a function ##h## in...

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...

Hi I have a pretty specific question. It is in regards to tissues in multicellular organisms. Is there any information on how different cell grouping arose in multicellular organisms? I have some ideas from what I've so far read and learned: - Would this have happened because two different...

Hello all, Here's something I've been trying to wrap my head around: In general, it seems that integration is 'harder' than differentiation. At least analytically. Numerically it may be the other way round. For one thing, it's often easy to differentiate implicit functions. For example...

I'm watching Susskind's Classical Mech. YouTube lecture series and am really confused about something he's doing where otherwise I've followed everything up until this point without a problem. In Lecture 3 he's dealing with the Euler-Lagrange equation applied to minimizing the distance between...

Plz give me an easy explanation I do know about the differentiation and second differentiation. I just don't get how that negetive sign comes in front of the exponent in the second differentiation

Hello! Can someone help me with the process of solving \sqrt{x}+\sqrt{y}=5 on point (4,9)? With implicit, I differntiated both sides and ended up with 1/2x^-1/2+1/2y^-1/2\d{y}{x}=0 and I tried to isolate the dy/dx, but how do I get rid of the others? And with explicit, I isolated y to one side...

So it has been quite a few years since I learned about implicit differentiation so the content is a bit rusty in my mind. x=rcos(θ) How do you find dx/dt? I know the answer but I am trying to figure out why. I mean dx/dt can be written as (dx/dθ)*(dθ/dt) so why is the answer not just...

hi guys, i have a question. i saw this picture, and i don't really understand how they derived with the formula. The aim is basically to find the formula for the surface area of a spherical cap. why do you differentiate the x=sqrt(rˆ2-yˆ2)? how does that help to find the surface? and then...

Hi! I recently came upon this problem : the height of a right angled triangle is increasing at a rate of 5cm/min while the area is constant. How fast must the base be decreasing at the moment when the height is 5 times the base? I drew a picture of the triangle, labelled the height (h) and...

If $y^2+\cos(x+y) = 1$ find $\frac{dy}{dx}$. How do I differentiate $\cos(x+y)$ bit?

Mod note: Moved from the Homework section 1. Homework Statement This might seem like a stupid question but I'm unsure what z= ƒ(x/y) means? I'm not sure how I would find ∂z/∂x , ∂z/∂y just from this statement either. Thank you Homework EquationsThe Attempt at a Solution

The working out states that v = (m/r)sin(theta) but surely this should be (-m/r)sin(theta) ? I mean there is a negative in front of the ∂Ψ/dx ? Also, if v=0 then 0 = (m/r)sin(theta), therefore sin(theta) = 0? And from this we know cos(theta) = 1 or -1 which would mean u = Uinfinity + (m/r) or...

Hi Community, I have this question. When I graph it I get: When I answer the first sets of questions a, Intercepts: x=(-6), x=0, x=6 Stationary points: (0,0) , (-3\sqrt{2}, -324) , (3\sqrt{2}, -324) Points of inflection: x=0 This is where I was wrong. (It's a turning point isn't...

Homework Statement Homework Equations The Attempt at a Solution Note: by real solution I mean the correct implicit derivative, not an actual real solution... Please help![/B]

1. Given the function ##xy+cos y+6xy^2=0## , it follows that ## dy/dx=-y/x-siny+12xy##2. My problem is how do we integrate this derivative ## dy/dx=-y/x-siny+12xy## to get back the original function3.## ∫dy/dx dx=y ##

Homework Statement *according to my teacher, the problem can be solved either through differentials or quick algebra, but she prefers differentials as the answer in order for us to get used to Calculus, so that's the route I'd like to take it A pendulum made from a string and small metal ball...

Folks, Differentiate implicitly \phi(x,y)=0 I get: wrt to x \phi_x+\phi_y \frac{dy}{dx} and wrt to y \phi_y+\phi_x \frac{dx}{dy} however I don't know how this is derived \phi_x dx+\phi_y dy=0

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Hi Folks, It is been given that differentiation of \phi(x,y)=0 is \phi_{x} dx+ \phi_{y} dy=0 however I arrive at \phi_{x} dx/dy+ \phi_{y} dy/dx=0 via the chain rule. Where \phi_{x}=d \phi/dx etc What am I doing wrong? Thanks

Hello Is what I have done calculated here correct? Please correct me if I have done something wrong. Thanks in advance.

Hi, I have an mathematics assignment to do, and I wonder if the topic I have chosen is doable for me. I want to minimize the surface area of a cobbler cocktail shaker, and until now my plan was to get the curve equation for the side of it, and get the area equation from surface of revolution...

Homework Statement http://postimage.org/][/PLAIN] click image upload Homework Equations http://postimage.org/][/PLAIN] free image upload The Attempt at a Solution So I wanted to differentiate B(t) by saying B(t) = P(t)(1.05)20-t ln(1.05) Apparently this is the wrong answer. I'm...

Homework Statement Differentiate Homework Equations Chain Rule: dg/dx = du/dx . dv/du . dg/dv The Attempt at a Solution My answer(wrong): Correct answer provided to us(not mine): I understand the correct solution that was provided to us, but what I don't understand is why my method...