Differentiation Definition and 1000 Threads

Good afternoon, This is not actually a homework question; it's for self-study. I'm reading a Calculus book, and one of its exercises asks the following: If xnym = (x+y)n+m, show that xDxy = y (where Dxy is the derivative of y with respect to x). The only way I could think of to get the correct...

Hello guys Let y = x^i x is real, i imaginary, then is it right to say: y' = ix^{i-1} and must y be real?

All I know about Leibnitz is \frac{d^{n}}{dx^{n}}f(x) = \sum^{n}_{i=0} \left(\stackrel{n}{i}\right) g^{n-i}(x)h^{i}(x) and I don't understand what the answer sheet says: Parts of it perhaps - I can see they take the 0th, 1st and 2nd derivative, but why not more? I mean why stop at 2? I know...

I've been studying Turbulence, and there's a lot of averaging of differential equations involved. The books I've seen remark offhandedly that differentiation and averaging commute for eg. < \frac{df}{dt} > = \frac{d<f>}{dt} Here < > is temporal averaging. If...

Homework Statement d(x(t)2)/dt Homework Equations The Attempt at a Solution I guess that this should be: 2x(dx/dt) but I'm not sure how to justify it: u= x d(u2)/du = 2u (d(u2)/du)(du/dt) = d(u2)/dt So 2u(du/dt) = 2x(dx/dt) Is this right?

Hi, can someone help me solve this double derivative please: \frac{1}{c}\frac{d^2}{dt^2}\frac{1}{c^2t^2-x^2-y^2-z^2} I am assuming you get use the quotient rule but I am getting so many terms I am getting confused... Its in my notes and I am not quite getting it... Thanks

Question: y6 + 6 (x^2+4)6 = 9 6y5 .dy/dx . 6(x2 + 4)5 . (2x) = 0 6y5 .dy/dx = -6(x2+4)5 .(2x) dy/dx = 6y5 / -6(x2 + 4)5 .(2x) dy/dx = 6y5 / 12x(x2 +4)5 Although the answer is ment to have y5 as the numerator, not 6y5? ------------- Another Q. [Simplifying result from...

Homework Statement So here's a question from my textbook 'Calculus: Concepts and Contexts' 2nd ed. by James Stewart. This is section 3.6 # 54 We have Cartesian coordinates set up with an ellipse at x^2 + 4y^2 = 5 To the right of the ellipse a lamppost (in 2D!) stands at x=3 with...

I'm a self-learner (so far) when it comes to Calculus. I have completed College Algebra (with a grade of B) and PreCalculus (with a grade of A) and am a later-in-life than usual student (age 50). I plan on returning to College in January of 2012 and will start on a degree in either...

Homework Statement derive (s2t3) / (rs2t3) with respect to s The Attempt at a Solution equation becomes s2t3*(rs2t3)-1 which becomes s2t3r-1s-2t-3 then just differantiate like a polynomial? i tried this on an online partial differentiation calculator and it gave me an...

Homework Statement Find y' using logarithmic differentiation if y=x^{sin(x)} Homework Equations The Attempt at a Solution lny=sin(x)ln(x) \frac{1}{y}(y')=sin(x)(\frac{1}{x})+ln(x)\cdot cos(x) (y')=[sin(x)(\frac{1}{x})+ln(x)\cdot cos(x)]y Did I do this one properly? Thanks in advance.

let be the integrals \int_{0}^{\infty}dxx^{2} (x-a)^{1/2}=I1 and \int_{0}^{\infty}dxx^{2} (x-a)^{-1/2}=I2 is then correct that I2= 2\frac{dI1}{da} whenever applying a regularization scheme , is it correct to differentiate with respct to external parameters ??

Homework Statement Find the first derivate of \frac{2}{x+1}. Formula: \frac{f(x+h)-f(x)}{h} Homework Equations Formula: \frac{f(x+h)-f(x)}{h} The Attempt at a Solution \frac{\frac{2}{x+h+1}-\frac{2}{x+1}}{h} h(\frac{2}{x+h+1}-\frac{2}{x+1}) h(\frac{2(x+1)-2(x+h+1)}{(x+1)(x+h+1)}...

Greetings, We know that the two operations (integration and differentiation) are inverse. I mean, if you integrate a function and differentiate the result of the first operation you go back to the first function. But I'm trying to understand WHY. Of course there are a lot of mathematical...

I want to verify simple integration and differentiation of a vector and verify that the direction of the derivative and integral of a vector is not the same direction of the original vector. Let: \vec A = \hat x A_x + \hat y A_y + \hat z A_z 1) Differentiation: \frac {d \vec A}{dx} =...

Homework Statement Given two functions F and G, I will use the following notation to indicate partial differentiation: Fx means dF/dx Gz means dG/dz (for example) I would like to develop the following two expressions. I don't want them grouped into brackets as they're now, but I have...

Homework Statement "Find dy/dx at the given point by using implicit differentiation" x2y + y2x = -2 at (2, -1) and (x+y)3 = x3 + y3 Homework Equations The Attempt at a Solution 1) x2(dy/dx) + y(2x) + y2(1) + 2y(dy/dx)(x) = -2 x2(dy/dx) + 2xy + y2 + 2xy(dy/dx) = -2...

At time t = 0 and position s = 0 a plane starts its descent into an airfield. From this point, the distance s in km as a function of time t in hours is given by; s = 300 + 400t - 200t^3 The inital velocity I have calculated to be 300 km/hr and the acceleration after 1/2 hr is 475 km/hr2. I...

Homework Statement The diagram attached shows a solid engineering component which consists of a solid cylinder of length L and radius r, together with a right circular cone with semi-vertical angle = 11 degrees. The total surface area of the component is 481. The component is to be...

Homework Statement A "big screen" AB of height L is placed at a height h above a point C on the ground as shown (in the attachment). A person wishes to sit on the ground at a point D in order to watch a video on the big screen. She wishes to sit at a horizontal distance x away from C which...

Hi, I have this derivation, and I am not sure how to derive it: \frac{d}{dp(x)}\int_{-\infty}^{\infty}p(x)\,log(p(x))\,dx I mean, what to do with the integral? Another thing, how to integrate: \int_{-\infty}^{\infty}e^{x^2}\,dx Thanks in advance

Homework Statement Hi all, I'm trying to understand why my second way of solving this problem doesn't work: A 3.00kg object is moving in a plane, with its x and y coordinates given by x= 5t^2–1 and y=3t^3 + 2, where x and y are in meters and t is in seconds. Find the magnitude of the net force...

Please go easy on me, 2 days ago I didnt even know what implicit differentiation was. Homework Statement If x tan y − y tan x = 1, use implicit differentiation to determine dy/dx, expressing your answer in the form dy/dx = f(x, y), The Attempt at a Solution Differentiate first...

Homework Statement Use differentiation to verify that the following integrals are correct (where a is not = 0 is a constant and c is an arbitrary constant (a) integrate xsinax dx= ( −x/a ) (cosax) +(1/a2) sinax+c (b) integrate tanax dx=(−1/a) ln(cosax)+c Homework Equations Composite rule...

Homework Statement Let f (x) =int(x,0) x sin(t^2)dt. Show that f''(x)= 2 sin(x^2) + 2x2 cos(x^2) Homework Equations The Attempt at a Solution I can't get f''(x)= 2 sin(x^2) + 2x2 cos(x^2), i can only get f''(x)= sin(x^2) + 2x2 cos(x^2). Because f'(x)=xsin(x^2). can anyone see...

Differentiate and simplify: y=(x+1)(2x-3)^{4} I got: 8(x+1)(2x-3)^{3} + (2x-3)^{4}. But the answers in the answer booklet say: 5(2x+1)(2x-3)^{3} I put both answers in Wolfram Alpha and found they were both equal. So this is just a matter of simplifying/rearranging. Could someone please...

Could someone help me with this simple differentiation? u= 5root(1/x^8) --> =(1/x^8)^1/5 = (1/5)/(x^8/5) du/dx= ? I don't know what to do after here, as I keep getting the wrong answer. The answer is du/dx = (-8/5)x^-13/5 :( Thanks!

Homework Statement (A) \int{\frac{(v^2+2v+4)dv}{v^3+v^2+2v+4}} (B) \frac{\partial{M}}{\partial{y}}=(1-xy)^{-2} \frac{\partial{N}}{\partial{x}}=y^2+x^2(1-xy)^{-2} Homework Equations (A) How can I integrate this? (B)After getting the partial derivatives, are they equal? The Attempt...

Homework Statement Show that 4 = \sum from n = 1 to \infty (-2)^{n+1} (n+2)/n! by considering d/dx(x^{2}e^{-x}). Homework Equations Power series for e^{x} = \sum x^{n}/n! from 0 to \infty. The Attempt at a Solution So I started with the power series for e^{-x} = \sum -x^{n}/n...

Suppose the function f has the following four properties: 1. f is continuous for x >=0; 2. f'(x) exists for x > 0; 3. f(0) = 0; 4. f'is monotonically increasing. I'm just looking for functions that have these 4 properties to better understand what f represents. So far, I came...

I'm learning about group theory and wanted to apply it to something other than + and *, and I'm having trouble understanding how it can be applied to differentiation. I can see that integration would be the inverse, but is it binary of unary? Since you would write it as d(f), it appears unary...

Homework Statement Determine the equation of the line that is tangent to y=8^x at the point on the curve x=1/2. Homework Equations Differentiation Rules m=y2-y1/x2-x1 The Attempt at a Solution y=8^x dy/dx=8^x(ln8) =8^0.5(ln8) =5.9 y=8^x =8^0.5 =2.8 5.9=y-2.8/x-0.5...

I've tried looking online, but I haven't found the answer. For instance, when can you say (dFx/dt)=(dF/dt)x, where subscript x indicates partial differentiation with respect to x. I know that partial differentiation is pretty much always interchangeable, but what about in this case? I have a...

What conditions does the real scalar function f(x,y) (on the particular range of integration) have to satisfy in order to put \frac{d}{dx}\int{f(x,y)dy}=\int{\frac{\partial}{\partial x}f(x,y)dy} ?

Homework Statement Using the product rule, differentiate the following function: Homework Equations y = etsintcost The Attempt at a Solution The three term product rule says: d/dx (uvw) = u'vw + uv'w + uvw' I find u = et, u' = et, v = sint, v' = cost, w = cost and w' = -sint...

Homework Statement Let X,Y be vector fields and x(t) be a curve satisfying \dot x(t) = X(x(t)) + u(t) Y(x(t)), u(t) \in \mathbb R [/itex] and assume there exists p(t) an adjoint curve satisfying \dot p(t) = -p(t) \left( \frac{\partial X}{\partial x}(x(t)) + u(t) \frac{\partial Y}{\partial x}...

Optimization under differentiation! Homework Statement OK I have a upside down looking curve structure (½ ellipse). It has the following specifications: The building has a rectangular base 150m long and 72m wide. The max height of the structure should not exceed 75% of its width or be less...

in partial differentiation why we have to use the jacobian?what does signifies?how does it differ from normal partial derivative? thanks

Homework Statement Consider a smooth manifold M, and smooth functions H_i: T^*M \to \mathbb R, i=0,1 on the cotangent bundle. Further, let z:[t_1,t_2] \subset \mathbb R \to M define a trajectory on T^*M by \frac{dz}{dt} = \vec H_0(z(t)) + u(t) \vec H_1(z(t)) where \vec H_i is the...

Homework Statement I need to know why: \frac{d^{2}y}{dx^{2}}=\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}} y=y(t) , x=x(t) The Attempt at a Solution I know that: \frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt} From here: \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}...

Hi everyone - I'm in my first year at uni, and I was given a derivative that i don't quite get: y(x) is a function of x then y` = (dy/dx) d/dx (ln y(x))=1/y(x)*(dy/dx) =(y`/y) This is given by the chain rule. I don't understand this step... Any help? Thanks in advance, Owen.

Homework Statement Calculate the length of the graph/equation: x^(2/3) + y^(2/3) = a^(2/3) The graph is formed as an s.c. asteroid, almost like a diamond. It seems to be some sort of modified unit circle. Homework Equations The length of the graph between x1 and x2 can be described as L=∫...

Homework Statement dM/dT = .5m-24 I know this is a linear differential equation, with an integrating factor of .5 I get my final answer to be M = 48+ke^(-.5t) Next, when t = 0, M = 7e^(.5t)... Giving me k = 7e^(.5t) - 48 so to solve for M.. M = 48 + (7e^(.5t)-48)e^(-.5t)...

Homework Statement Using logarithmic differentiation calculate the derivative of y=e^(x^x) The Attempt at a Solution y=e^x^x LNy=LNe^x^x LNy=x^xLNe ... Stuck! This seems to be the only way you can do it, but once I get to that part I'm not sure what else there is to do. I...

Homework Statement Use logarithmic differentiation to find the derivative of the function. y=xln(x) also, if anyone could help me with this... i have (for a diff problem) 1/(x2-1)1/2 * 1/2(x2-1)-1/2*2x i know that is right, but i don't know how to get from that, to x/(x2-1) i keep...

Homework Statement Hi again Practicing log differentiation now :P Sorry for all the questions, i have a midterm on thursday and i need to get this stuff by then 1. ex2ln(2x+1) 2.ln(2x/(2x+1)) 3. (3x-2)x (use logarithmic differentiation even though unnecessary The Attempt at a Solution 1...

Homework Statement Ok so I found out last time that the derivative of the function they asked me to find was: \frac {dF}{d\theta} = \frac { - \mu^{2}Wcos(\theta)+\mu W sin(\theta)} { (\mu sin(\theta) + cos (\theta))^{2}} and now they are asking me when is this 0, specifically they...

Homework Statement suppose f is a differentiable function such that f(g(x)) = x and f'(x) = 1 + [f(x)]^2. Show that g'(x) = 1/(1+ x^2) Homework Equations The Attempt at a Solution since f(g(x)) = x, i think that f is the inverse of g. so f = g{inverse} f'(x) = f'(g(x)) *...

Homework Statement An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle θ with the plane, then the magnitude of the force is given by the following equation, where μ is a constant called the coefficient...

Homework Statement Given 5y^2 = 4x - 3/4x + 3 Homework Equations is it permissable to say this is equal to y^2 = 4x -3 /5(4x + 3) and then 2y(dy)/(dx) = what the right side equals thru using the quotient rule? I know the answer is dy/dx = 12/5y(4x + 3)^2 but I don't know how to...