Differentiation Definition and 1000 Threads

When the textbook differentiated with respect to time, I see that the middle term is R(dI/dt). Why can't it be I(dR/dt)? When I differentiate, how do I know which letter to differentiate? 3. The Attempt at a Solution 2. Homework Equations 1. Homework Statement

Homework Statement I need to proof that Rodrigues’ formula satisfies Laguerre differential equation Homework Equations Rodrigues’ formula of Laguerre Laguerre differential equation The Attempt at a Solution first,I have to calculate = I tried to sum both terms and this is what I got...

Homework Statement Find the general solution of the following equation: u(t): u' = u/t + 2t Homework Equations y' + p(x)y = Q(x)....(1) yeI = ∫ dx eIQ(x) + constant.....(2) The Attempt at a Solution I rearranged the equation to give: u' - u/t = 2t Then I considered the following...

Hey, In my class we just learned about logarithmic differentiation. I can see this being useful when taking the derivative of a complex function since it could be messy. But, I tried it on simpler equations as well. Everything I tried it on it seemed to work. Are there ever instances that it...

I have an equation: r^2 = x^2 So I found out dr/dx = x/r. But when I try to find the second derivative, I get d2r/dx2 = -x^2/r^3 when the text says it should be (r^2 - x^2)/r^3. Can anyone help? My working out: r^2 - x^2 = 0 r^2 = x^2. Assume r is a function of x. rr' = x (first derivative...

I have the equation: x = 2*L*sin(θ/2) and in my lecture notes they have converted it to: ϑx = L*cos(θ/2)*ϑθ Is it correct to do the following to get this answer? x = 2*L*sin(θ/2) x = 2*L*sin(θ/2)*(ϑ(θ/2)/ϑx) x*ϑx = 2*L*sin(θ/2)*ϑ(θ/2) 1*ϑx = (1/2)*2*L*cos(θ/2)*ϑθ ϑx = L*cos(θ/2)*ϑθ My problem...

$$ \frac{d^{2n}}{dx^{2n}}\left(x^2-1\right)^n = (2n)! $$

Homework Statement Uploaded on the bottom Homework Equations None The Attempt at a Solution I took dt of both sides. The part that confuses me is when you take the derivative of 1 over t . The one becomes a 0 and the t becomes a dt. Just want to make sure I did that right.

Hey all, I am reading Goldstein and I am at a point where I can't follow along. He has started with D'Alembert's Principle and he is showing that Lagrange's equation can be derived from it. He states the chain rule for partial differentiation: \frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial...

Homework Statement 3(√x2+xy)-2x2+10=0The Attempt at a Solution There are x and y on the same side. If I expand it directly, it is so complicated! Can anyone show me the full step? Thanks!

Homework Statement Given the potential energy function V(x,y)=V(ax-by) where a,b is an arbitrary constants differentiate with respect to x and y. Homework Equations Multivariavle differentiation The Attempt at a Solution The answer yields (d/dt)p1=-aV'(ax-by) (d/dt)p2=+bV'(ax-by). The right...

Homework Statement Homework EquationsThe Attempt at a Solution 1. The centripetal acceleration of a particle moving in a circle is ##a = \frac{v^2}{r}##, where v is the velocity and r is the radius of the circle. Approximate the maximum percentage error in the calculation of the acceleration if...

Homework Statement Homework EquationsThe Attempt at a Solution 1. The combined electrical resistance R of two resistors connected in parallel is ## R = \frac{R1R2}{R1+R2}##, where R, R1 and R2 are measured in ohm. R1 and R2 are increasing at rates of 1 and 1,5 ohm per second respectively. Find...

Homework Statement Homework EquationsThe Attempt at a Solution 1. If z=x+sin(##x^2##y) + ln y find ##\frac{\partial ^2z}{\partial x^2}## and ##\frac{\partial ^2z}{\partial y^2}## 2. Second order partial differentiation. 3. ##\frac {\partial z}{\partial x}## = 1 + ##cos(x^2y)## . (2x) =...

Homework Statement with answers given: Homework Equations use implicit differentiation The Attempt at a Solution I always get this answer but not the second one PLs explain the second answer for I am very desperate. Thank You

Hi there Getting stuck on this equation. e^(-x) / ln(x) Solving it by quotient rule, however answer has extra x in numerator. using the dy/dx = (v(du/dx)-u(dv/dx))/v^2 dy/dx = (ln(x)*-e^(-x) - (-e^(-x)*1/x)/ln(x)^2 Answer = (-e^(-x) (x ln(x)-1))/x(ln(x))^2 Would appreciate help with...

Homework Statement for a projectile show that d^2 (v^2) / dt^2 = 2g^2 2. The attempt at a solution =d/dt (d(v^2)/dt) =d/dt (2v)

Given f(x) = \arctan\left({\frac{\sqrt{1+x}}{\sqrt{1-x}}}\right) I differentiated and this was my answer. \d{y}{x} = \frac{1}{2\sqrt{1+x}\sqrt{1-x}{(1-x)}^{2}} I used implicit differentiation on the elliptic curve {x}^{2}+4{y}^{2} = 36 and it wants two horizontal tangents through (12,3)...

Homework Statement a match box consists of an outer cover, open at both ends, into which slides a rectangular box without a top. The length of the box is one and a half times its breadth, the thickness of the material is negligible, and the volume of the box is 25cm^3 . If the breadth of the...

Homework Statement A man wishes to fence in a rectangular enclosure of area 128m^2.One side of the enclosure is formed by part of a brick wall already in position. What is the least possible length of fencing required for the other three sides? 2. Homework Equations The Attempt at a Solution...

$$6x-\sqrt{2xy}+xy^3 ={y}^{2}$$ $$6-?+3x{y}^{2}{y'}^{}+{y}^{3}=2y{y'}^{}$$ Got stumped on this one answer was complicated...

Homework Statement Let f(x) = 1 - x2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem? Homework EquationsThe Attempt at a Solution f(x) = 1 - x2/3. f(-1) = 1 - 1 = 0 f(1) = 1 - 1 = 0 f' = 2/3 x -1/3. I don't...

the variables x and y are positive and related by x^a.y^b=(x+y)^(a+b) where a and b are positive constants. By taking logarithms of both sides, show that dy/dx=y/x. provided that bx not equal to ay.

If I have a function ##f(u,u^*) = \int u^* \hat{O} u d^3\mathbf{r}## both ##u## and ##u^*## are functions of ##\mathbf{r}## where ##\mathbf{r}## position vector, ##\hat{O}## some operation which involves ##\mathbf{r}## (e.g. differentiation), and the star sign denotes complex conjugate. Now I...

Homework Statement Suppose we have an equation, ex + xy + x2 = 5 Find dy/dx Homework Equations Now I know all the linear differentiation stuff like product rule, chain rule etc. Also I know partial differentiation is differentiating one variable and keeping other one constant. The Attempt at...

Homework Statement Given f(x,y)=ycosx,x(t)=t^2,y(t)=sint Calculate \frac{df((x(t),y(t))}{dt} t∈ℝ Homework Equations For parametric equations I know \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}} The Attempt at a Solution So far I have found \frac{∂f}{∂x}=-ysinx \frac{∂f}{∂y}=cosx...

I want to prove that differentiation with respec to covariant component gives a contravariant vector operator. I'm following Jackson's Classical Electrodynamics. In the first place he shows that differentiation with respecto to a contravariant component of the coordinate vector transforms as the...

Hi - looking at 'discretizing elliptical PDEs'. I understand the normal lattice approach, but this approach uses the variational principle. I have a couple of questions please. The text says: $ \: Given\: E=\int_{0}^{1} \,dx\int_{0}^{1} \,dy\left[\frac{1}{2}\left(\nabla \phi\right)^{2} -...

What has done here in the second line of the proof for product rule?, from Mathematical methods for physicists from Riley, Hobson they defined f(x)=u(x)v(x) and these steps are given, I have no idea how to proceed further please help me.

Hi. A very long problem brought me to a derivative in the form \frac{\mathrm{d}f(x)}{\mathrm{d}g(x)} I'm assuming that \mathrm{d}g(x)=\left(\frac{\mathrm{d}g(x)}{\mathrm{d}x}\right)\mathrm{d}x So, is it correct to say that...

Homework Statement Given: z = f(x,y) = x^2-y^2 To take the partial derivative of f with respect to x hold y constant then take the derivative of x. ∂f/∂x = 2x What I don't understand is why such would equal 2x, when the y is still there it just isn't variable and is ignored. Wouldn't it be...

I can't convince myself whether the following functional derivative is trivial or not: ##\frac \delta {\delta \psi(x)} \big[ \partial_x \psi(x)\big],## where ##\partial_x## is a standard derivative with respect to ##x##. One could argue that ## \partial_x \psi(x) = \int dx' [\partial_{x'}...

Question: I have a function of time. Its expression has a constant 'b' in it. I am asked to ascertain how changing 'b' affects the function. Specifically, I have velocity as a function of time which accounts for drag forces; 'b' is the drag coefficient. I am asked to ascertain how changing 'b'...

Simple question really, I'm not sure why the constant pulled out of the derivative becomes negative (-w2). I've tried looking for answers by googling but can't come up with anything. I feel like its because the first term (1,1) is negative but I want to be sure. Thanks

Homework Statement f(x) = 1/ln (10-x) -- I would assume it to be a fairly simple equation, but I am screwing it upHomework Equations What is f'(x)?The Attempt at a Solution f'(x) = (ln (10-x))^-1 = -(ln (10-x))^-2 * -1 * 1/(10-x) -- 2 negatives cancel out = 1/(10-x) (ln(10-x))2 --...

For a research project, I have to take multiple derivatives of a Yukawa potential, e.g. ## \partial_i \partial_j ( \frac{e^{-m r}}{r} ) ## or another example is ## \partial_i \partial_j \partial_k \partial_\ell ( e^{-mr} ) ## I know that, at least in the first example above, there will be a...

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'm in Calc 1 and the Chain Rule is giving me one hell of a rough time. I've spent about 10-12 hours over the last few days just on the homework problems in this one section (only getting about 15-20 problems done) and still feel like I barely understand it. Does anyone have any tips, tricks...

Homework Statement Please see the attached file. I am trying to understand the sensitivity of two related variables - Y and K - to an independent variable M. a. Is my differentiation of equation 2 correct? b. I can see that, based on eq. 4, K is more senstive to M than Y is, however I am not...

If ## f\in L_{p}^{\rm loc}(\mathbb{R}^{n}) ## and ## 1\leq p<\infty ##, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}=|f(x)|$$ for almost all ##...

Homework Statement Find \frac{\partial}{\partial x} if: f(x,y) = \begin{cases}x^2\frac{\sin y}{y}, & y\neq 0\\0, &y=0 \end{cases} Homework EquationsThe Attempt at a Solution If y\neq 0 , then it's simple, but I get confused about the second part. How can I exactly utilize the limit definition...

I need help solving all three parts to this question, never seen a question regarding applications of differentiation that is this hard before! All help is much appreciated.

This might seem like a naive question to ask, but a full explanation of why these two concepts are different would be welcome. I am confused because parametric equations are ##y = 8t^2## and ##x = 5t##, but at the same time, these two equations can describe the ##x## and ##y## components of a...

For the problem of differentiating ##y = x^5(3x-1)^3## using logarithmic differentiation, the solution provides the first step as rewriting the functions as ##\left |y \right | = \left | x \right |^5 \cdot \left | 3x-1 \right |^3##. This confuses me. First, how are we, mathematically, able to...

I have a question about logarithmic differentiation, especially concerning the absolute value involved. For example, if we have the function ##y = 2^x \sin x##, the domain is all real numbers. So what happens when we take ##\ln## of both sides of the equation? The antilogarithm must be greater...

To express the derivative of a particular function, I have recently come across a "new" notation. For the function x2-3x for example, can you write the derivative operator like this? x2-3x dx . I heard this is called the Euler notation, is it valid?

I am reading several books on multivariable analysis/calculus and am trying to get a precise and rigorous theoretical understanding of implicit differentiation, including the Implicit Function Theorem ... in particular I am reading: Vector Calculus (Second Edition) by Susan Colley and...

I unfortunately keep on getting the wrong answer to this problem. I am supposed to find: dw/dy(1/(w^2+x^2)+1/(w^2+y^2)) I attached a picture of how I tried to solve it. Help would be much appreciated.

relation between integration and differentiation ? how is instantaneous slope(differentiation) related to area under the curve(integration) ? thank you!

Hi everybody, I'm having a little difficulty understanding the differentiation of x with respect to x. When a function, f(x) is differentiated, each term is differentiated with respect to x, correct? So, when differentiating y=x, we would have d(y)/dx = d(x)/dx. To my (very limited) knowledge...