For questions regarding partial derivatives. The partial derivative of a function of several variables is the derivative of the function with respect to one of those variables, with all others held constant.

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1answer
12 views

Taking the derivative of $\epsilon \cdot(\ln X + \ln \beta) - \ln(1 + X^{\frac{\alpha}{\alpha - 1}})$ with respect to $\ln X$

So I am taking the derivative of $$\epsilon \cdot(\ln X + \ln \beta) - \ln(1 + X^{\frac{\alpha}{\alpha - 1}})$$ with respect to $\ln X$, where $X$ is a variable, $\epsilon, \beta, \alpha$ are ...
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1answer
32 views

The region where the two variable function $xy/(x-y)$ is differentiable

I need to found the area where this function is differentiable $$ f(x,y) = \frac{xy}{x-y} $$ How do I need to proceed? For partial derivatives I got: $$ \frac{\partial f}{\partial x} = ...
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0answers
8 views

Extrema of 3D function in a plane slice

I have 3D surface described via RBF I need to compute maximal value (extrema) in a plane cut through the surface. Can it be done? For example (not RBF related image).. I have 3D parabola and cut ...
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0answers
9 views

Gradient Vector of Homogeneous Functions

I've been given the definition $f:\mathbb{R}^n \to \mathbb{R}^m$ is homogeneous of degree k if $f(\lambda x)=\lambda^kf(x)$ $\forall x\in\mathbb{R}^n, \lambda>0$ and asked to show $<\nabla ...
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1answer
31 views

Applications of the wave equation

I've recently started to take interest in PDEs and how to solve them, and I'm wondering a bit about real life applications of the wave equation. So far I haven't found anything about practical ...
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1answer
23 views

Partial derevatives of multi variables

Hi I don't understand this problem. $F$ is a function of $r$, right? I can find the partial derivatives $\frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}$, and $\frac{\partial ...
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0answers
29 views

Gradient of a vector [on hold]

Matrix $V$ is 200 by 785. Matrix $X$ is 785 by 1. Matrix $W$ is 10 by 201. Matrix $y$ is 10 by 1. First, I do: $ V * X$ Then, I apply $tanh()$ to every element of that resulting matrix. The result ...
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1answer
35 views

Derivative of log-likelihood cost function with respect to a matrix

Recently, I am learning derivative method to a function and thanks to @hans help, I can solve those which can be expressed by Frobenius product. But for the log-likelihood function, I do not how to ...
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1answer
43 views

Intersection of sphere and ellipsoid

Ellipsoid: $$ x^2+\frac{y^2}{4}+\frac{z^2}{2}=1 $$ Sphere: $$ x^2+(y-1)^2+(z-d)^2=1 $$ For what values of $d$, there is a common tangent plane to both curves? Part of my resolution: Consider ...
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0answers
24 views

Higher Order Derivative Tests in Multiple Dimensions

To evaluate the minima, maxima, and saddle points of a real function of 2 variables, we use the second derivative test after evaluating the critical points to identify the type of extrema they are. ...
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0answers
8 views

Permute partial derivatives, ODE

Show that in the one dimensional case, we have $ \frac{\partial \phi}{\partial x} (t,x)= exp(\int_{t_0}^t \frac{\partial f}{\partial x} (s, \phi(s,x))ds)$, where $\phi(t,x)$ is the solution of the ODE ...
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1answer
18 views

Differentiating a squared quantity

I was reading through my electromagnetism book where i came across this statement where when we differentiate wrt a squared quantity rather than a single quantity we multiply it by $\frac{1}{2}$. ...
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1answer
24 views

Local extrema of $x^3+y^2+6y$

I have to find local extrema of $x^3+y^2+6y$. I found out that the stationary points are $(0,-3)$. I also found the Hess matrix for this function and computed the determinant, which is $12x$. But now ...
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1answer
21 views

Calculatin a partial derivative

If we had: and we were to calculate How is it equal to w ?
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1answer
37 views

Cost minimisation calculus

(Note: I'm a beginner at calculus): I'm solving a calculus problem in microeconomics intermediate course. The question is: Determine the factor demand functions, 𝑥∗(𝑤 , 𝑤 , 𝑦), by using the cost ...
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0answers
39 views

Showing existence of a partial derivatives

How would one show that that $$f(x,y) = \frac{xy(x^2-y^2)}{x^2+y^2}$$ for $(x,y) \neq (0,0)$ and $f(x,y)=(0,0)$ if $(x,y)=(0,0)$ has second order partials but $f_{xy}(0,0) \neq f_{yx}(0,0)$. I was ...
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1answer
24 views

Partial derivative using irregular variables?

I'm trying to find the partial derivative with respect to $M$ for: $$\frac{d}{dM} \frac{4\pi r^{\frac{3}{2}}}{\sqrt{GM}}$$ I know how to solve for a partial derivative, but I'm having trouble because ...
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0answers
26 views

A question on polynomials.

Let a polynomial $f\in\mathbb{R}[x,y]$, and $f(x,y)=(x^2+y^2)p(x,y)^2-q(x,y)^2$ and $p,q$ are coprime to each other. When do, $f$ and $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial ...
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2answers
19 views

Sketching solutions to IVP

Consider the following initial value problem (IVP): $$u_t + \cos(t)u_x = −u, \, \, \, \, (x,t) ∈\mathbb R×(0,∞)$$ $u(x,0) = u_0(x)$, $x ∈\mathbb R$, where $u_0 : \mathbb R → \mathbb R$ is a prescribed ...
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4answers
54 views

Why does $f$ and $f′$ non-trivial factor? [closed]

Let a polynomial $f\in\mathbb{R}[x]$. Why do $f$ and its derivative $f′$ share a non-trivial common factor?
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0answers
47 views

Why does this equation has not any solution?

Suppose $P(x,y)$ and $Q(x,y)$ are polynomials, and $P,Q$ are coprime. Why does the equation $${\left( {\frac{{xP + {{\left( {\frac{Q}{P}} \right)}^2}{P_x}}}{{yP + {{\left( {\frac{Q}{P}} ...
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1answer
32 views

Prove partial derivatives exist, but not all directional derivatives exists.

During my analysis course my teacher explained the difference between partial derivatives and directional derivatives using the notion that a partial derivatives looks at the function as approaching a ...
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0answers
20 views

Finding Directional Derivatives with gradient

Find the derivative of the function at $P_0$ in the direction of $u$.$$f(x,\, y,\,z) = \tan^{-1}\left ( \frac{5x}{9y+2z} \right ),\,\,\, P_0(7,\,0,\,0),\,\,\, u = 12i - 3j+4k$$ I understand how to ...
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0answers
6 views

Third order partial derivatives in cylindrical coordinates

Do you know, where I can find formulas for third order partial derivatives in cylindrical coordinates? All I can find are second order partial derivatives. Thanks!
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1answer
26 views

Problem with a derivative inside an integral

I am having some issues solving for the derivative of $$\int_0^1 q^{\alpha} e^{tq} dq$$ with respect to $t$ when $t > 0$. I tried to perform a direct method of computing: $$\lim_{t \rightarrow t_0} ...
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0answers
21 views

How to find $\partial ^2 u/ \partial x^2$?

If we have $u=u(\xi , \eta )$ and $\xi=\xi (x,y)$, $\eta = \eta (x,y)$ and $$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial x} +\frac{\partial u}{\partial ...
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1answer
28 views

Let $f(x,y) = \sin{(x-y)}$, $x = r\cos{θ}$, $y = r\sin{θ}$ find $df/dθ$ and $d^2f/dr^2$

Let $f(x,y) = \sin{(x-y)}$, $x = r\cos{\theta}$, $y = r\sin{\theta}$ find $\frac{\text{d}f}{\text{d}θ}$ and $\frac{\text{d}^2f}{\text{d}r^2}$ Using chain rule I got: ...
0
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1answer
54 views

Solution for partial differential equation

How to solve this partial differential equation $$a(1-q)\frac{\partial}{\partial q}A[p,q]+(bp(q-1)+c(1-p))\frac{\partial}{\partial p}A[p,q]-(s+d(1-p))A[p,q]=0$$ where $a,b,c,d$ and $s$ are constants ...
0
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1answer
27 views

Partial differentiation, solving an equation

I have this question about partial differentiation which I try to solve but I seem to get wrong all the time. The question is: if $x=t\sin(s)$ and $y=t\cos(s)$, find $d^2f(x,y)/dsdt$. I'm assuming ...
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1answer
44 views

Find the volume of bullet shape solid.

Bullet function is given by $y = 16 - x^2 - z^2$ to the right of the $xz-$plane. I have set up the following integral but not sure whether it is true or not. $\int_{-4}^{4} \int_{0}^{2π} ...
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1answer
15 views

Are there identities which will make calculating $\Delta \frac{e^{ik|x|}}{|x|}$ more efficient and clearer?

Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate $$\Delta \frac{e^{ik|x|}}{|x|}$$ Doing this 'manually' be calculating second derivatives is really long and tedious ...
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1answer
20 views

Simplifying second order chain rule

In these notes: http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx We have $$\text{If } y = f(x(t)), x = g(t)$$ $$\text{Then } \dot y = \dot{f}(x(t)) =\dfrac{ \partial f}{\partial ...
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0answers
23 views

wave equation with discontinuous velocity and zero dirichlet boundary at both end

I am solving the following wave problem $$\begin{cases} u_{tt} - u_{xx} = 0, & \text{for } x \in \big( 0, \frac{1}{2} \big) \\ u_{tt} - 4u_{xx} = 0, & \text{for } x \in \big( \frac{1}{2}, 1 ...
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1answer
18 views

Max-Min values of $f(x,y) = x^3+y^3-6x^2-y-1$

I am asked to find the extrema of the function $$f(x,y) = x^3+y^3-6x^2-y-1$$ I understand that we have to equal the partial derivatives to zero, which means $$ f_x = 3x^2-12x = 0\\ f_y = 3y^2-1 = ...
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0answers
22 views

Find the critical points of the function $f(x,y)=(x^2+y^2)e^{y^2-x^2}$

Find the critical points of the function $f(x,y)=(x^2+y^2)e^{y^2-x^2}$. Now I have found the partial derivative of f(x,y) with respect to both x and y and got the following: ...
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2answers
79 views

can anybody please explain me the answer for Putnam Exam $2010 A-3$? [closed]

enter image description here How was $(x,y)$ transformed into $(au-bv,bu+av)$? and how did $∂g$ become $∂x$ and $∂y$?
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0answers
19 views

Show that the Neumann Problem Has no Solution

I am given the Partial BVP as follows: $$ \ u_{rr}+u_{\theta\theta} = 0 ,(r<1,-\pi\leq\theta<\pi) \\ \frac{\partial u}{\partial r}= sin^2\theta,(-\pi\leq\theta<\pi) $$ This has no solution, ...
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1answer
27 views

Finding extreme values using Lagrange multipliers given constraint

Find the extreme values of the function subject to the given constraint.$$f(x,\, y) = y^2 - x^2,\, x^2 + y^2 = 16$$ I understand how to to compute the extrema using Lagrange multipliers and lambda ...
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1answer
45 views

A question on partial-derivative

Let $f(x,y) = \sqrt {{x^2} + {y^2}} p(x,y) + q(x,y)$, and $p$ and $q$ are two polynomials(non zero) Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't ...
0
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1answer
62 views

How can we prove, by Bézout's theorem, that $L$ has finitely many singularities?

Bézout's theorem Bézout's theorem for curves states that, in general, two algebraic curves of degrees $m$ and $n$ intersect in m·n points and cannot meet in more than $m·n$ points unless they have a ...
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1answer
22 views

Second derivative of Kullback–Leibler divergence

The Kullback-Leibler divergence is defined here. I have to find the second derivative of $\textrm{KL}(p(s, \theta)||\mu(s) q(\theta, s))$ regarding $p(s, \theta)$, where $p(s, \theta)$ is a joint ...
2
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1answer
35 views

Second-order derivative with respect to a function of two variables.

I have a surface defined as a radius vector in spherical coordinates: $$r = r (\theta, \psi).$$ In Cartesian coordinates, the projections are calculated as follows: $$\begin{align} r_x &= r \sin ...
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1answer
22 views

Where am I going wrong in taking this partial derivative?

I am trying to use to known relations to derive the Gibbs-Helmholtz equation. $S = -\left( \frac{\partial F}{\partial T} \right)_V$ $F=E-TS$ Must result in $E = -T^2 \left( ...
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2answers
37 views

Is $f(z) = Arg(z)$ Holomorphic?

Using the Cauchy-Riemann equations, determine whether $f(z) = arg(z)$ is holomorphic or not. I understand that we can use the theorem $f(z)$ satisfies the CRE over $\mathbb{C}$ (ie. ...
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1answer
50 views

A question on polynomials

Let $f(x,y) = ({x^2} + {y^2})p{(x,y)^2} - q{(x,y)^2}$ and $p$ and $q$ are two polynomials. Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't have a ...
3
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0answers
21 views

Find the gradient of $f(x,y,x)$ given its maximum value at a point $P_0$ in the direction $(1,1,-1)$

I am asked to solve the following problem: The directional derivative of the function $f(x,y,z)$ has maximum value of $2\sqrt{3}$ at the point $P_0$ in the direction $(1,1,-1)$. What is the gradient ...
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2answers
46 views

Can we say that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't have a common factor?

Let $f(x,y)$ be a polynomial. Can we say that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't have a common factor?
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1answer
16 views

Inconsistency in partial derivatives in polar and Cartesian coordinates

We know that for polar $(r,\theta)$ and Cartesian $(x,y)$ coordinates: $r=\sqrt{x^2+y^2}$ (1) $x=r\cos\theta$ (2) I am trying to find $\dfrac{\partial r}{\partial x}$. I have tried two methods, ...
0
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0answers
18 views

Partial derivative of inverse function with two variables

For a function $$y = f(x,a)$$ we have by the Inverse Function Theorem that for $x = f^{-1}(y,a)$, $$ \frac{\partial f^{-1}}{\partial y} = \frac{1}{\frac{\partial f}{\partial x}} $$ Is there ...
1
vote
1answer
13 views

Differentiate $v(x,y)=-\int_0^x u_y(t,0)dt+\int_0^y u_x (x,t) dt$ w.r.t. $x,y$ to prove complex differentiability

The domain is an open unit box (if required) and $u$ is harmonic, $v$ is harmonic conjugate defined below. Prove Complex Differentiabilty Diffirentiate with respect to $x$ and $y$: ...