Differentiation Definition and 1000 Threads

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...

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...

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...

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...

I need help in understanding the proof of the Chain Rule for differentiation, as presented in Theorem 5.1.6 in Manfred Stoll's book: Introduction to Real Analysis. Theorem 5.6.1 in Stoll (page 173) reads as follows:In the above proof we read the following: " ... ... By identity (3) and then...

I have J - matrix x and y - vector d [ J(x) y(x)] / dx I can multiply the matrix and vector together and then differentiate but I think for my application it would be better to find an identity like {d [ J(x) y(x)] / dx } = J(x) d y(x) / dx + d J (x) / dx y(x) I am not sure if this identity...

I saw this in http://en.wikipedia.org/wiki/Momentum_operator From equation 4 to 5, it seems that a function is canceled out from the partial derivatives, is this possible?

If I understand what's going on (quite possibly I don't), I think my book is using bad (confusing) notation. Homework Statement As written: "Calculate ##\frac{\delta H[f]}{\delta f(z)} \ \text{where} \ H=\int G(x,y)f(y)dy##" and ##\frac{\delta H[f]}{\delta f(z)}## is the functional derivative...

Homework Statement Differentiate the following functions with respect to x Homework Equations (√2x-1)/ (lnx)? The Attempt at a Solution ln x (2x-10)^-1/2 -√(2x-1)(1/x)/(ln x)^2 ..........{This is a denominator} thank you [/B]