Differentiation Definition and 1000 Threads

Homework Statement Find a function of the form f(x) = a + b \cos cx that is tangent to the line y = 1 at the point (0,1), and tangent to the line y = x + 3/2 - \pi /4 at the point (\pi /4 , 3/2). Homework Equations The Attempt at a Solution f(0) = a + b = 1, so a = 1 - b...

Homework Statement Use implicit differentiation to find dy/dx if y - sin(xy) = x^2. What I've got is dy/dx y - cos(xy)(y+x dy/dx) = 2x I don't know what I did and I don't know where to go from here.

Homework Statement The vector a depends on a parameter t, i.e. a=a(t)=a_x(t)i +a_y(t)j +a_z(t)k.. it satisfies the equation da/dt= j \times a show that d^2a_x/dt^2 =-a_x , da_y/dt=0 and d^2a_z/dt^2 =-a_z. For the vector a, find its value for t=pi if at t=0 a(0)=i+j and da/dt(0)=0k...

Homework Statement Hey >.< another simple problem; find \frac{\partial u}{\partial y} , \frac{\partial v}{\partial y}, \frac{\partial w}{\partial y} given that u - v + 2w = x + 2z 2u + v - 2w = 2x - 2z u - v + w = z - y The Attempt at a Solution I don't know, this question is really...

Homework Statement Given y = x^2 + 6 / x find x if dy/dx equals 0 zero Homework Equations dy/dx = v du/dx - u dv/dx / v^2 The Attempt at a Solution I have got as far as dy/dx = x^2 - 6 / x^2 However then zero must equal Sqrt 6, which 2.44... -...

Homework Statement Find \frac{\partial z}{\partial x} \frac{\partial z}{\partial y} where z=\left( [x+y]^3-4y^2 \right)^{\frac{1}{2}}Homework Equations -The Attempt at a Solution I know that \frac{\partial z}{\partial y}=\frac{3(x+y)^2-8y}{2\sqrt{(x+y)^3-4y^2}} but I am unsure whether...

Homework Statement find \frac{\mathrm{d}y}{\mathrm{d}x} where y is defined implicitly as a function of x Homework Equations x\sin(xy)=x The Attempt at a Solution x(\cos(xy)(x\frac{\mathrm{d}y}{\mathrm{d}x}+y))+\sin(xy)=1...

Homework Statement Prove that if F is a twice differentiable function with F(0) = 0 and F(1) = 1 and F'(0) = F'(1) = 0, then |F''(x)| >= 4 for x in (0,1). Hint: Prove that either F"(x) >= 4 for some x in (0,1/2) or else F"(x) <= -4 for some x in (1/2,1) Then, show that we...

Homework Statement Differentiate the following cos^2 x The Attempt at a Solution \frac{d(cos^2 x)}{dx} = 2 (-sin^{2-1} x).\frac{dx}{dx} = 2 (-sin x).1 = -2 sin x But the answer given is- -2 cos x sin x Can you please tell me what's wrong !

Homework Statement How do i differentiate a sum in MATLAB? So for an example, if I have: y = x^2 + \sum^{n}_{j=1}3x_{j}^4 I want to differentiate it with respect to x. It should return: 2x + \sum^{n}_{j=1}12x_{j}^3 Homework Equations The Attempt at a Solution If I want...

Homework Statement If x = yz and y = 2sin(y+z), find dx/dy and d^2x/dy^2 Homework Equations The Attempt at a Solution I am beyond confused at how to even start this one, the problem is not like any of the examples in my book. I know x = ydz + zdy but don't know how to deal...

The velocity of a particle moving in a plane in polar coordinates is {\bf{v}} = v_r {\bf{\hat r}} + r\omega \hat \theta where v_r = \frac{{dr}}{{dt}} and \omega = \frac{{d\theta }}{{dt}}. By differentiating w.r.t. time, show that the acceleration of the particle is {\bf{a}} =...

Homework Statement http://img94.imageshack.us/img94/3853/physicse.jpg The Attempt at a Solution I kept y fixed, and so I ended up with the following equation: Integ[dU/U] = Integ[x] And we end up with: U(x,y) = e^x * g(y) To solve g(y), we sub the solution into the 2nd PDE provided to...

Hello, I`m studying 1st year physics and I would really like to improve my basic maths. The thing is I don`t want to sit and first make up a random expression and then calculate the derivative, I would like to be able to print out hundreds and hundreds of small expressions with varying...

Prove that if P(a,b) is a point on the rotated ellipse C (whose equation is x^2 -xy +y^2=4 ), then so is Q(-a,-b), and that the tangent lines to C at P and Q are parallel. The equation of the line joining P and Q is y - b = m(x - a), where m= \frac{b-(-b)}{a-(-a)} = \frac{b}{a}, then the...

Homework Statement A swimming pool is 40 feet long, 20 feet wide, 4 feet deep at the shallow end, and 9 feet deep at the deep end. Water is being pumped into the pool at 10 cubic feet per minute. a. When the water is 3 feet deep at the shallow end, at what rate is the water level rising? b...

I was working through this problem, I understand the method, but got stuck trying to differentiate the arc tan terms... In the table of standard derivatives, \frac{d}{dx} \arctan{x} = \frac{1}{1+x^2} and for PDE's, you treat y as constant when differentiating w.r.t. x --- Any...

Homework Statement "An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other. One plane is 120 miles from the point and is moving at 400 miles per hour. The other plane is 160 miles from the point and has a speed of...

find derivative of y=(sqrt(8x^4-5))/ (x-1) ok...after working out the tricky calculations i get for my final answer: (32x^4sqrt(8x^4-5)-64x^3sqrt(8x^4-5)) / (16x^6-32x^5+ 16x^4-10x^2+20x-10) I don't know if you want to do the math...im just wondering if i can simplify it anymore. thanks

A Norman window has the shape of a square surmounted by a semicircle. The base of the window is measured as having 80cm with a possible error in measurement of 0.1cm. Use differentials to estimate the maximum possible error in computing the area of the window. So what i did was...

Homework Statement I'm having trouble deriving the following equation \frac {\partial^2 {\theta}}{\partial {x'^2}} = -y^2*exp(\theta) and y = x/x' my main problem is the exponent Homework Equations The Attempt at a Solution Normally i would use the equation (x')'' + k^2*x' = 0 x' = c1...

Homework Statement If f'x = 0 \forall x \in (a,b), show that f is constant there. I've gotten a final result, but I'm not entirely sure if I actually showed what I intended to show... Homework Equations The Attempt at a Solution Fix a point c \in (a,b) and choose a point x...

The tangent to the curve y=ax^3 at the point (-1,b) is perpendicular to the line y = 2x+3. Find the values of a and b. Could someone show me how to answer that?

Homework Statement Differentiate f(x) if f(x)=xxln(3x-6) Homework Equations None The Attempt at a Solution The problem in f(x) is the xx so what I did was let a=xx and differentiated a with respect to x. I ended up with da/dx= (ln(x)+1)xx. Afterwards, I modified f(x) by...

Okay, so my problem lies within taking the second derivative of a change of variable equation. w = f(x,y); x = u + v, y = u - v so far I have the first derivative: dw/dx = (dw/dv)(dv/dx) + (dw/du)(du/dx) = (d/dv + d/du)w Now I'm having problems in finding my second derivative...

Homework Statement Find all the points on the curve x^{2}y^{2}+xy=2 where the slope of the tangent line is -1. The Attempt at a Solution I differentiated both sides of the equation and got: \frac{dy}{dx}=\frac{-2xy^{2}-y}{x^{2}2y+x} I know that \frac{dy}{dx}=-1, but if I substitute...

Homework Statement i have been told to calculate DI/Dx i calculated I as being = m(L^2)/16 how do i differentiate this? I am confussed with what happens to the m is it a constant an just removed? same with the 1/16 i assume that is removed as a constant? just need some clarification...

Homework Statement Given the equation y= f(x) , at a certain point the slope of the curve is 1/2 and the x-coordinate decreases at 3 units/s. At that point, how fast is the y-coordinate of the object changing? The Attempt at a Solution Dy/dx = f ' (x) dx/dt Would that be...

Homework Statement f(x) = (2x-1)(3x-2)(5x+1) d/dx = ? Note that I am letting (2x-1)(3x-2) = f and 5x+1 = g Homework Equations d/dx (fg) = f d/dx g + g d/dx f The Attempt at a Solution d/dx y = (2x-1)(3x-2)d/dx(5x+1)+(5x+1)d/dx((2x-1)(3x-2)) = (2x-1)(3x-2)(5)+...

Homework Statement (x^(3)+2x) / (x^(2)-5) d/dx = ? Homework Equations d/dx (f/g) = (g d/dx (f) - f d/dx (g)) / g^2 The Attempt at a Solution d/dx y = { (x^(2)-5) d/dx (x^(3)+2x) - (x^(3)+2x) d/dx (x^(2)-5) } / (x^(2)-5)^2 This is where it gets complicated for me. {...

Homework Statement Find the slope of the curve y=\frac{x}{3x+2} at the point x = -2. Homework Equations \lim_{h \rightarrow 0}\frac{f(x_{0} + h) - f(x_{0})}{h} = m The Attempt at a Solution If x = -2, then y = 1/2. I'm not sure what to do from here. This is the first step, but I don't get...

Homework Statement Consider: x^3+y^3+2xy=4, y=1 when x=1 a.) Find the equation of the tangent line to the curve when x=1. b.) Find y'' at x=1. c.) Is the graph of y=f(x) concave up or concave down near x=1? Homework Equations Any derivative rules... The Attempt at a Solution For Part a...

Homework Statement y = x^(18x^-4) Homework Equations chain rule dy/dx a^x = a^x ln a The Attempt at a Solution first i used the second equation from above to get x^(18/x^4)*ln(x) then i use the chain rule to get x^(18/x^4)*ln(x)*(-72x^-5) the computer program i am using is...

Alright, I'm not technically stuck on this one, but I consistently get a result that disagrees with what Wolfram Alpha shows when I enter the problem to check my answer. Sorry 'bout the lack of LaTeX, but it should be simple enough to read. Here goes: Problem: Differentiate y=sin-1[x/(1+x)]...

Homework Statement Suppose f(3)=2 , f'(3)=5 , and f''(3)= -2 . Then d²/dx² (f²(x)) at x=3 is equal to ____? A. -20 B. 20 C. 38 D. 42 E. 10 The Attempt at a Solution I am confused about how to find the function to get the derivative from that function. Any Ideas? Thanks.

Find d^2/dx^2(3y^2+8y=3x) I managed to get dy/dx = 3 / (6y + 8) but I have no clue where to go from here. According to WolfRamAlpha, the answer is -27/(4(16 + 9x)(4 + 3y)), but since dy/dx doesn't have any x value in it, I don't see how the derivative of it would. I've played around...

Homework Statement Find y´(x) for x + y = 1/x^2 + 8/y^2 Homework Equations The Attempt at a Solution Rewrite the eq. as x + y = x^-2 + 8y^-2 differentiate. 1 + y´(x) = (-2x^-3) + (-16y^-3)(y´(x)) Rearrange 1 + (2x^-3) = (-16y^-3)(y´(x)) - (y´(x)) (1 + (2x^-3))/(16y^-3) =...

1. Suppose the curve f(x)=(x^4)+a(x^3)+b(x^2)+cx+d has tangent line when x=0 with equation y=2x+1, and a tangent line when x=1 with equation y=2-3x. Find a,b,c,d 2. d/dx f(x)g(x)=g*f`+f*g` 3. f`(x)=4x^3+3ax^2+2bx+c x=0 y=2x+1 y=1 (0,1) is a point on f(x) x=1 y=2x-3 y=-1 (1,-1) is a point on...

Homework Statement If g(x) + x sin g(x) = x^2 find g'(0) Homework Equations The Attempt at a Solution At this point I have tried a few things but hit deadends. Any help would be appreciated.

Homework Statement Having trouble understanding this the example I saw was; Solve \int^{\infty}_{0} x^3 e^{-9x} dx using integration by parametric differentiation.The Attempt at a Solution well, i do know how to do this, so i set out my integral; \int^{\infty}_{0} e^{-\alpha x} dx =...

Homework Statement Find y'' by implicit differentiation. Homework Equations The Attempt at a Solution I get to this point in the problem, which is I solved for y'. But then when I attempt to take y'', in other words take the derivative of my answer for y', I...

2xy=3x-y^2 find dy/dx and d2y/dx2 I just want to make sure my answer is right and simplified I got dy/dx= (3-2y)/ (2x+2y) Now d2y/dx2 took some time but this is what i got: (-12x+2x2y+9+4yx-14y-4y^2) / ((x+y)(2x+2y)^2)

I guess the following identity isn't right. Can anyone tell me what the LHS is really equal to? \frac{\partial}{\partial x}\int_{-\infty}^\infty\int_0^x f(s,t) dsdt = \int_{-\infty}^\infty f(x,t)dt

1. use implicit differentiation to evaluate y(prime) for x^3+y^3=4xy+1 at the point (2,1)

so to solve this i differentiated each part and got 6dy/dx + 8 = cos(xy^2)(y^2*x2ydy/dx) then i divided both sides by cos(xy^2) then serpatated the 6dy/dx + 8 and put them both over cos(xy^2) then i took out a yprime and ended up with -8/(6-2y^3xcos(xy^2)) as an answer but its wrong...

Homework Statement The function f is defined as f(x)=x^4, then; f'(3x^3)=? d[f(3x^3)]/dx=? f'(xy^3)=? Homework Equations The Attempt at a Solution My problem here is not so much doing the differentiation itself but understanding the notation. 1. for f'(3x^3) i want...

some help with this differentiation question thanks Question : Find the indicated partial derivative . frss , frst f(r,s,t) = r ln (rs^2t^3) differentiating with respect to r gave s^2t^3 1* ---------- rs^2 t^3 but this is not correct something is...

Hi all, I was reading a paper in which implicit differentiation was used as follows x \in R, \lambda \in R Given G(x,\lambda) = 0 \frac{\partial G(x,\lambda)}{\partial x} \frac{\partial x}{\partial \lambda} + \frac{\partial G(x,\lambda)}{\partial \lambda} = 0 My doubt is...

Hi, So, I am reviewing Cal III, and I have come across something that I do not understand regarding implicit differentiation with partial derivatives: x^3 + y^3 + z^3 + 6xyz = 1 implicit differentiation of z with respect to x: 3x^2 + 3z^2(dz/dx) + 6yz + 6xy(dz/dx) = 0 *notive the...

Homework Statement Prove if f(x) is differentiable at x=a then f'(a)=lim_{h\rightarrow0}\frac{f(a+h)-f(a-h)}{2h}Homework Equations I know that the derivative is defined as f'(a)=lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}The Attempt at a Solution Starting from the definition I used a known...