Derivatives Definition and 1000 Threads

Homework Statement Demonstrate that C_{Y,N}=\left ( \frac{ \partial H}{\partial T } \right ) _{Y,N} where H is the enthalpy and Y is an intensive variable. Homework Equations (1) C_{Y,N}=\frac{T}{N} \left ( \frac{ \partial S}{\partial T } \right ) _{Y,N} (2) T= \left ( \frac{ \partial...

Homework Statement Find the derivative of f(x): f(x)= 1/ ((ln(x)^2)) Homework Equations f(x)= ln(x) f'(x)= 1/((ln(x)) The Attempt at a Solution Dx(1/ln(x)^2) = Dx((ln(x))^-2)= -2*(ln(x)^-3) * Dx(ln(x)) = -2*(ln(x)^-3) * 1/x = -2/(x*ln(x)^3) Are these the correct...

I'm not sure if it's OK to post this question here or not, the Calculus and Beyond section doesn't really look very heavily proof oriented. I'm trying to prove that if continuous complex valued function f(z) is such that the directional derivatives(using numbers with unit length) preserve...

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I'm struggling to understand what the derivative of an attitude quaternion really is and how to use it. I need it to solve a problem relating to a rotating frame of reference relative an inertial frame. The information I have is a vector of Euler angular velocities (i.e for roll \phi, pitch...

Homework Statement the Celsius temperature in a region in space is given by T(x,y,z)=2x^2-xyz. a particle is moving in this region and its position at time t is given by x=2t^2, y=3t, z=-t^2, where time is measured in seconds and distance in meters a) how fast is the temperature experienced...

f(x,y)=\sqrt{|xy|} Do the partial derivatives of f exist at x=0, y=0?

There is already an article on physics forums that kind of addresses my issue: https://www.physicsforums.com/showthread.php?t=107516 and I'm not really satisfied with the wikipedia article. I am generally confused on what the derivative should be. I'm familiar with Jacobian matrices but am...

Can this problem be solved? I made up the problem myself so I am not sure a solution exists. It is known that: h(0) = 0 h'(0) < 0 h'' > 0 Prove that there exists a value c > 0 such that h(c)=0 It makes sense visually. I have tried applying the MVDT/IVT in various ways, but...

Homework Statement If h(x):=0 for x<0 and h(x):=1 for x≥0. Prove there dne f:ℝ→ℝ such that f'(x) = h(x) for all x in ℝ. Give examples of two functions not differing by a constant whose derivatives equal h(x) for all x ≠ 0 Homework Equations The Attempt at a Solution I don't know...

For a function such as w=5xy/z How would you find the partial derivative of w with respect to y or z? I've tried using basic logarithmic differentiation, but can't arrive at the correct answer. For reference, the correct answer is wy=5*(xy/z/z)*ln(x)

1. In the Khan academy video I watched on partial derivatives, I understand absolutely everything except for the last 20 seconds which confused me. http://www.youtube.com/watch?v=1CMDS4-PKKQ Using the formula: Z = x² + xy + y² @z/@x = 2x +y x=0.2, y=0.3 2(.2) + .3 = .7 What...

Homework Statement A bush-walker is climbing a mountain, of which the equation is h \left( x,y \right) =400-{\frac {1}{10000}}\,{x}^{2}-{\frac {1}{2500} }\,{y}^{2} The x-axis points East, and the y-axis points North. The bush-walker is at a point P, 1600 metres West, and 400 metres South of...

Quick question. This is kind of embarrassing actually. Suppose I have functions x(t,s) and y(t,s) (say they're parametric equations of a surface of something) and I want to know what dy/dx is. Specifically, I have x and y in terms of the parameters, which are kind of complicated functions, and I...

Homework Statement Find the approximate percentage changes in the given function y = f(x) that will result from an increase of 2% y = x2 Homework Equations The Attempt at a Solution dy/dt = dx/dt * d/dx * x2 dy/dt = dx/dt * 2x dy/dt = 2/100 * 2x dy/dt = 4x% ? I don't know if I...

Homework Statement Hi. I am attempting to get the Euler-Lagrange equations of motion for the following Lagrangian: L(ψ^{μ}) = -\frac{1}{2} ∂_{μ} ψ^{\nu} ∂^{μ} ψ_{\nu} + \frac{1}{2} ∂_{μ} ψ^{\mu} ∂_{\nu} ψ^{\nu} + \frac{m^{2}}{2} ψ_{\nu} ψ^{\nu} Homework Equations So, I want to get...

I had questions on 2 Problems in the Text: 1. The total cost C of producing x units of some item is a function of x. Economists use the term marginal cost for the rate of change of C with respect to x. Suppose that: C = 5x^2 + 15x + 200 What is the marginal cost when x = 15? Would this...

Homework Statement Use the polar graph to determine the signs (+,-,0) of each derivative at the point labeled A. Homework Equations dy/dx= dy/dtheta= dx/dtheta= dr/dtheta= The Attempt at a Solution Hi people, I need help with this question. See the picture of the graph...

I have a PDE in two variables, u and v, which takes the form \frac{\partial\psi}{\partial u\hspace{1pt}\partial v} + \frac{1}{r}\left(\frac{\partial r}{\partial u} \frac{\partial \psi}{\partial v} + \frac{\partial r}{\partial v}\frac{\partial\psi}{\partial u}\right) for an auxiliary...

Homework Statement f(x.y)=4x^2-y^2 Homework Equations Ʃ partial derivative components(?) The Attempt at a Solution The solution when θ=pi and f(1,-1) is -8. Does this mean that one of the coordinates of this function is (1,-1,-8)? What exactly is the directional derivative, and what does...

How can I use the directional derivative of a two variable function to show that the limit does not exist? For example, suppose I have a function f(x,y)=g(x)/f(y) and g(a)=f(b)=0 and the limit as x and y go to a and b is 0. How would I use the directional derivative to show that the limit at...

We have a function f:R^2->R and it has partial derivative of 2nd order. Show that f_{xy}=0 \forall (x,y)\in \mathbb{R}^2 \Leftrightarrow f(x,y)=g(x)+h(y) The <= is self explanatory, the => I am not sure I got the right reasoning. I mean we know that from the above we have: f_x=F(x) (it's...

Homework Statement The question says ti find the derivative of y with respect to the independent variable. The equation is: y=4 ln 2t. Homework Equations I know how to find derivatives using the product/quotient/chain/etc rules, but that isn't what the question is asking for. I...

Homework Statement hey Forum! I had a question here I'm struggling with and was wondering if someone could take a look. its Dealing with calculus, specifically derivatives and behaviors of the graph: http://i41.tinypic.com/mc6opj.jpg I just started and part a) already has me stumped D...

Homework Statement Hey forum, hope everyone is having a good day! If someone could check this question out for me that'd be great! Determine the first and second derivative: g(x) = (2x - 3)/(x + 4) The Attempt at a Solution g(x) = (2x – 3)(x + 4)^(-1) g’(x) = (2x – 3)’(x +...

Hi all, I've been fiddling around with this problem for a while. I intuitively understand that the parallel propagator is the path integral of the connection. I would like to be able to show the converse (connection is derivative of parallel propagator) mathematically, and I am having a...

Homework Statement Find the partial derivatives: f(x,y)= integral[x,y] cos(t^2)dt, find f_x(x,y) and f_y(x,y) Homework Equations I know from calculus that the derivative of an integral is the function. The Attempt at a Solution I found that the integral of [x to y]...

Homework Statement Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g differentiable. (a) What are the values of the derivatives f'(1), g'(1)? (b)...

Homework Statement a) d^2/(dx^2): ln(x+1) b) d^3/(dx^3): x^7 + 4x^6 - x^2 c) d^2/(dx^2): 1/(x + 1) d) d^4/(dx^4): cos(2x) Homework Equations none The Attempt at a Solution Can someone tell me if these are right? a)= -(1/(x + 1)^2) b)=210x^4 + 480x^3 c)= (2/(x + 1)^3) d)=...

Homework Statement The question is attached along with its solution. Homework Equations Partial differentiation and the implicit function theorem. The Attempt at a Solution My work is attached. I feel it's correct but is it incomplete? I have the following questions/confusions...

How do you solve (analytically or numerically) a differential equation of this form, $$\frac{\mathrm{d}y(x)/\mathrm{d}x}{\mathrm{d}z(x)/\mathrm{d}x} = a[1-y(x)-z(x)] + b$$ where a, b are constants. Also, $$y(0) = z(0) = 0$$

Homework Statement Assume f(1,1,1)=3 and f(1.1,1.2,1.1)=3.1 a) Which directional derivative Duf at (1,1,1) can be estimated from this information? Give vector u b) Estimate the directional derivative in part a Homework Equations Duf = del f (dot product) vector u del f =...

Homework Statement My question is how do I take the time derivative of (theta dot)^2? Homework Equations The Attempt at a Solution Is the answer just 2(theta double dot)^1 or do you use chain rule 2(theta dot)*(theta double dot)?

Homework Statement f(x)=X^2/(x^2-16) f(x)=1+x/1-X f(x)=X^3(X-2)^2 Ive done the first and second derivatives but they just don't seem right Homework Equations Quotient/Chain/Product Rule The Attempt at a Solution f(x)=X^2/(x^2-16) (X^2-16)(2X)-(X^2)(2X)/(X^2-16)^2...

Homework Statement Just trying to find the first and second derivatives. X^2/(X^2-16) 1+X/1-X X^3(X-2)^2 Homework Equations Quotient Rule/Power Rule/Chain Rule The Attempt at a Solution

Homework Statement Do the derivatives del and d/dt commute? Or in other words, is it true that: del(d/dt)X = (d/dt)del_X Homework Equations ? The Attempt at a Solution nm, I think I know how to show it now..

Homework Statement In the steps below, the ∂z/∂x does not seem to be obeying normal algebraic rules. I'm confused. This is not really a problem, I'm just trying to understand the steps. The Attempt at a Solution 1. 3z2∂z/∂x - y + y∂z/∂x = 0 2. ∂z/∂x = y/(y + 3z2) if ∂z/∂x were...

If you have a function f(x,y)=xy where y is a function of x, say y=x^2 then the partial derivative of f with respect to x is \frac{\partial f}{\partial x}=y However, if you substitute in y and express f as f(x)=x^3 then the partial derivative is \frac{\partial...

Homework Statement find the direction of P sub 0 in the direction of A see second post for attachment, I forgot to place it on this one. The Attempt at a Solution I'm only worried about the part that says gy(x,y,z) = -3zexsin yz. I also don't understand the conversion of gx and gz. 1. why...

Homework Statement ∂f/∂x (xy -1)2 = 2y(xy-1) The Attempt at a Solution I would think the answer would be 2(xy-1) I don't understand where the y comes from in 2y

Homework Statement I don't understand why ∂f/∂x = xy = y whereas ∂f/∂x = x2 + y2 = 2x Why does the y disappear in the second but not in the first?

Homework Statement Given that the acceleration vector is a(t) = (-16cos(-4t))i + (-16sin(-4t))j + -2tk, the initial velocity is v(0) = i + k and the initial position vector is r(0) = i + j + k, compute: The velocity vector v(t) = ___i + ____j + ____k The position vector r(t) = ___i +...

Are the following equalities between total and partial derivatives true if \frac{dy}{dx}=f(x,y)? \displaystyle \frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} f(x,y) \displaystyle \frac{d^2f}{dx^2} = \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} +...

Homework Statement Classify the equation and use the change of variables to change the equation to the form with no mixed second order derivative. u_{xx}+6u_{xy}+5u{yy}-4u{x}+2u=0 Homework Equations I know that it's of the hyperbollic form by equation a_{12}^2 - a_{11}*a_{22}, which...

Homework Statement Suppose the function f:R^2→R has 1st order partial derivatives and that δf(x,y)/δx = δf(x,y)/δy = 0 for all (x,y) in R^2. Prove that f is constant; there exists c such that f(x,y) = c for all (x,y) in R. There's a hint as well: First show that the restriction of...

How do CAS systems and programmable calculators evaluate the derivative of a function? Do they use matrix representation of linear transformations?

I was reading a section on vector fields and realized I am confused about how to take partials of vector quantities. If V(x,y)= yi -xj, I don't understand why the \partialx= y and the \partialy= -x. The problem is showing why the previous equation is not a gradient vector field (because the...

Can anyone give me an example of a continuous function that is NOT differentiable(other than the square root function)? I have to prove that not all continuous functions are differentiable. Thanks!

If a and b are constants, compute the expression KY'(K) + LY'(L) for Y = AK^a + BL^a Y'(K) means partial derivative with respect to K by the way. The answer in the book is KY'(K) + LY'(L) = aY I'm not sure what they did or what they're asking :/

The problem for this is a picture. Its basically a U shaped parabola such that it intersects the points (-3,0) (0,0) (3, 0) you have a graph such as: f(x) | | |------- y | | The problem I am having is this.. I know what it means when a point is unstable or stable we tended to draw arrows on...