For questions which deal with taking partial derivatives.
10
votes
4answers
120 views
Closed form for $n$th derivative of exponential of $f$
What is the closed form for:
$$\frac{\partial^n}{\partial x^n}\exp(f(x))=\exp(f(x))\cdot[????]$$
2
votes
2answers
30 views
Are there real numbers a and b such that $f(x,y,t) = x^a t^b$ satisfies the heat equation?
The question is in the title. The heat equation is as follows:
$$
\frac{\partial f}{\partial t} = k \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right),\quad ...
3
votes
1answer
41 views
Closed form for $n$-th derivative of exponential
I need the closed-form for the $n$-th derivative ($n\geq0 $):
$$\frac{\partial^n}{\partial x^n}\exp\left(-\frac{\pi^2a^2}{x}\right)$$
Thanks!
By following the suggestion of Hermite polynomials:
...
1
vote
1answer
29 views
How to determine a function of 2 variables from its derivative?
Please even the slightest advice would help!
If I have a function $V$ made of 2 variables $x_1$ and $x_2$,
and its derivative $$\frac{dV}{dt} = \frac{dV}{dx_1}\frac{dx_1}{dt} + ...
2
votes
1answer
80 views
Gravitational fields
Could anyone help me with the following question?
Consider a right circular cone of constant mass density sigma, height h, and semi-vertical angle alpha. By dissecting the cone into discs, show that ...
2
votes
1answer
18 views
Finding partial derivatives for equations expressed in terms of $z$ where $z=f(x,y)$ to find tangent plane
I am having troubles finding partial derivatives.
If $f(x,y)=2x^2+y^2$ then,
$$f_x=4x$$
$$f_y=2y$$
That's simple enough. But when I see a $z$ in the equation, I get stumped. I know $z=f(x,y)$. I ...
7
votes
1answer
94 views
What is this limit called? Is it a different kind of derivative?
(first I should notice you this is not something I can look up in a textbook, because I'm learning partial derivatives, alike I do with most Maths, as a hobby. If something below is wrong, blame the ...
1
vote
1answer
43 views
Partial derivatives question?
I have to find $dw$ if $w=f(u,v,z)$ where $u=x^2+y^2,v=x^2-y^2,z=2xy$.
Now,I know that $dw= ( ∂w/ ∂z)*dz + (∂w/ ∂u)*du + (∂w/ ∂v)*dv$
The problem is that for example,if I want to find $∂w/ ∂z$ I ...
2
votes
1answer
64 views
Partial derivative chain rules.
$$x= \cosθ -r\sinθ$$
$$y= \sinθ + r\cosθ $$
Show that,
$$ \frac{\partial^2θ }{\partial x^2}= \frac{\cosθ }{ r^3} (\cosθ -2r\sinθ)$$
Please Help :)
I used the chain rule:
$$ \frac{\partialθ ...
1
vote
2answers
77 views
Partial derivatives.
Suppose
$$f(x+y, x^2 +xy + z^2) = 0.$$
Show that
$$x + y = 2z\left(\frac{\partial z}{\partial y}-\frac{\partial z}{\partial x}\right).$$
Please help I don't know where to start!
3
votes
2answers
33 views
partial derivative with Maxima
I have the following equation:
(v * w) / ( v * ((v * w) + (1 - v) * x) )
I want to find the partial derivative with respect to v. Using the quotient rule I get:
...
0
votes
1answer
70 views
Understanding some differential notations…
What this notation mean? (I know that is a partial derivative, but I don't understand the meaning of the evaluation bar at the right)
$$\frac{\partial g}{\partial T}\Big|_{SA,p}$$
Is this relation ...
3
votes
3answers
77 views
If $\frac{\partial \varphi}{\partial x}=f(x,y),\frac{\partial\varphi}{\partial y}=g(x,y)$, what is $\varphi$?
Suppose we have a real-valued function $\varphi(x,y)$ such that
$$
\frac{\partial \varphi}{\partial x}=f(x,y)\quad\text{and}\quad\frac{\partial\varphi}{\partial y}=g(x,y)
$$
for some functions $f$ and ...
0
votes
0answers
38 views
Combining two partial derivatives into one partial derivative
Can someone explain to me why the following is true (from derivation of conservation of mass general equation in fluid mechanics):
$\rho\dfrac{\partial u}{\partial x} + u\dfrac{\partial ...
1
vote
1answer
20 views
Let $z(t) = (3x+2y)\cdot exp(2x^2-y^2)$, $x = \cos{3t}$ and $y = \sin3t$. Evaluate $ z'(1)$.
Let $z(t) = (3x+2y)\cdot exp(2x^2-y^2)$, $x = \cos{3t}$ and $y = \sin3t$. Evaluate $ z'(1)$.
I know that $$\frac{dz}{dt} =\; \frac{\partial z}{\partial x}\,\frac{dx}{dt}\;+\,\frac{\partial ...
8
votes
0answers
77 views
Find all functions $F(x,y)$ such as $\frac{\sqrt{3}}{2}\frac{\partial{f}}{\partial{x}}+\frac{1}{2}\frac{\partial{f}}{\partial{y}}=0$
How to find all possible functions $f(x,y)$ such as:
$$ \frac{\sqrt{3}}{2}f_x+\frac{1}{2}f_y=0$$
(with $f_x = \frac{\partial{f}}{\partial{x}}$ )
Here's everything I tried:
1) I can guess the ...
-1
votes
0answers
44 views
$\ Find \ \frac{\partial r}{\partial x} \ of \ r^2 = x^2 + y^2\ $
$\large\ Find \ \frac{\partial r}{\partial x} \ of \ r^2 = x^2 + y^2\ $
I've tried it by doing $\ \large \frac{\partial r}{\partial x} \ of \ r = \sqrt{x^2 + y^2} \ to \ obtain \ \frac{x}{\sqrt{x^2 ...
1
vote
1answer
19 views
Question about partial derivatives
Let $f(x, y) = x^2 + g(y)$. Is the partial derivative subject to $x$ now $\frac{\partial f}{\partial x} = 2x$? Since $y$ is considered to be constant, I am guessing $g(y)$ will be always constant too. ...
0
votes
0answers
37 views
Interpretation of partial derivative
Let $u:D~(\subset\mathbb R^2)\to\mathbb R$ be a function. By the partial derivative of $u$ w.r.t. $x$ at $(x_0,y_0)\in D$ we mean the limit $$\lim_{h\to 0}\frac{u(x_0+h,y_0)-u(x_0,y_0)}{h}$$
I've ...
3
votes
1answer
40 views
Understanding fourier notation $F(\partial_x)$
Can somebody please help me understand some of the notion in the equations below, taken from a published paper on image de-blurring.
I have an energy $E(H)$ defined over an image $H$, a point-spread ...
1
vote
2answers
46 views
Chain Rule Definition $f=f(x,g(x,y))$
Here i'm asking again. This time is about Chain Rule:
Apply the chain rule to calculate: $\frac{\partial^2 f}{\partial x^2} $
$\ f=f(x,g(x,y)) $
I have a trouble in the the first member of F ...
2
votes
0answers
71 views
Derivation of Euler-Lagrange equation
Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation.
If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is
$\dfrac{\partial ...
1
vote
1answer
34 views
Partial derivatives question help?
I have to find $∂z/∂u$ if $$z=\arctan(x/y), x=4\sin u,y=e^v$$
Can I find it directly by replacing x and y in $$z=\arctan[(4\sin u)/(e^v)]$$ Now I have to find $$∂z/∂u={1/[1+ 4\sin^2u/e^{2v}]}[(4\sin ...
0
votes
1answer
34 views
Math question partial derivatives?
I have to find ∂z/∂x and dz/dx if z=ln(e^x+e^y) ,y=x^3
Awesome.Now, I write ∂z/∂x=(∂z/∂y)*(∂y/∂x) .I find ∂z/∂y=e^y/(e^x+e^y) ..but how do I find ∂y/∂x? what is its value?
0
votes
1answer
33 views
Subscript notation in partial derivatives
just want to be clear with partial derivative notation, and as to what respect we are taking the partial deriv of , for example; $$f_x, f_y, f_{xx}, f_{yy}, f_{xy}$$
so with what respect is each ...
0
votes
0answers
21 views
Using first order condition to derive comparative statics.
Suppose i have a profit function:
$ \pi = pf(X_1,X_2) - w_1X_1 - w_2X_2 - t(w_2X_2)$, t and f are functions.
How do i obtain the expression $\frac{dX_1}{dt} = w_2\frac{dX_2}{dw_1}$ ?
the first ...
2
votes
1answer
31 views
Differentiation under the Integral sign for the Lebesgue integral
I want to prove the following version of Liebniz's Rule:
Let $f:[a,b]\times [c,d]\to \mathbb{R}$ be integrable with respect to the first variable, $\phi,\psi:[c,d]\to [a,b]$ be differentiable and let ...
3
votes
1answer
47 views
How to find a partial derivative of an implicitly defined function at a point
Suppose that the relation $\frac{x^2}{2} + \frac{y^2}{2} + \frac{z^2}{2} + xy + xz =\frac{7}{2}$ defines $z$ as a function of $x, y$ around the point $(1, 1, 1)$. Find $\frac{dz}{dy}$ at $(1, 1, ...
0
votes
1answer
14 views
Vector derivative partial to w
What's the derivative of E (below equation) partial to w: ( consider that w is vector)
4
votes
2answers
53 views
derivatives transformation
I'm currently doing a calculation for the connection coefficients using the standard space-time coordinates, namely x[0],x[1],x[2],x[3]. The setup is a spherically symmetric problem.
In my ...
2
votes
2answers
60 views
Fairly complicated partial derivative
I have a stats assignment, that requires the use of non linear regression - this I am fine with in principle, however I can't get the initial $X$ matrix, because I don't understand partial ...
0
votes
1answer
44 views
Where is the PDE solution defined?
Given $\dfrac {\partial u}{\partial t} + \cos(t)\dfrac {\partial u}{\partial x}=0,\ t>0$ with $u(x,0)=f(x)\ 0<x<1$, I need to determine where this solution is defined.
I have found using ...
1
vote
1answer
64 views
Differential equation question
I have already asked for help on this question here, but I did not fully explain the question and therefore didn't get the answer I was looking for. I am struggling with part aii. Specifically where I ...
0
votes
0answers
40 views
Partial differentiate (matrix) by indexed variable ($\frac{\partial}{\partial{\theta_{a_i}}}$ …)
According to the literature I'm reading, this:
$$ \frac{\partial}{\partial{\theta_{a_i}}} \sum_{n=1}^N X_n\theta_{c_n} - ln(\sum_je^{X_n\theta_j}) $$
is
$$ \sum_{n=1}^N I(a == C_n)x_i - ...
1
vote
1answer
47 views
Function $T = x^2 + 2y^2 + 2z^2$ from $\mathbb{R}^3 \to \mathbb R$
The function $T : \mathbb{R}^3 \rightarrow \mathbb{R}$ gives the temperature (in degrees) at each point in space. Suppose a particle is at the point $p = (1, 1, 1)$.
$$T = x^2 + 2y^2 + 2z^2$$
In ...
2
votes
2answers
41 views
Calculating partial derivatives
Let f and g be functions of one real variable and define $F(x,y)=f[x+g(y)]$. Find formulas for all the partial derivatives of F of first and second order.
For the first order, I think we have:
...
0
votes
1answer
145 views
Show that this piecewise function is differentiable at $0$
I have shown (from first principles) that the Cauchy-Riemann equations for the following function are satisfied at $z=0$. But to properly prove differentiability at $z=0$, what should I do next? Do I ...
2
votes
4answers
65 views
confirm which one is correct?
Let $f(z)=-(x^2+y^2)^{1/2}$ and $\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$. Help to confirm which one is correct for $\Delta f$; this or ...
2
votes
2answers
68 views
$\delta\to \partial$: Is this argument valid?
Is the following reasoning valid?
Suppose $z=z(x,y)$, then
$$\delta z =\left({\partial z \over \partial x}\right)_y\,\,\,\delta x+\left({\partial z \over \partial y}\right)_x\,\,\,\delta y$$
Divide ...
0
votes
1answer
41 views
Partial Differentiation
I was just wondering whether anyone could help me with this question- I need to differentiate
$$
y_t = p - \left(\frac{a}{a_2 + c}\right)(m_0 +by_t + v_t) + \left(\frac{a}{a_2+c}\right) a_1 +u_t
$$
...
0
votes
1answer
66 views
Finding the x and y values such that the partial derivatives are zero simultaneously (part two)
$f(x,y)= \ln(x^2 + y^2 +1)$
The partial derivatives of $f(x,y)$ being:
$f_x(x,y) = \frac{2x}{x^2+y^2+1}$ and $f_y(x,y)=\frac{2y}{x^2+y^2+1}$
Setting each partial derivative equal to zero, adding ...
4
votes
3answers
203 views
Partial Derivative of $f(x,y) =\ln(x^{2} + y^{2}) + \sqrt{x^{2}\cdot y^{3}}$
$$f(x,y) = \ln(x^{2} + y^{2}) + \sqrt{x^{2}\cdot y^{3}}$$
What is the value of $f_{x}\left ( 0,1 \right )$ and $f_{y}\left ( 0,1 \right )$?
I tried but I found the denominator as zero.
-5
votes
1answer
46 views
$z=f(x+ay) + \phi(x-ay)$, where a is a constant. [closed]
$z=f(x+ay) + \phi(x-ay)$
Show that, $$\frac{\partial^{2} z}{\partial y^{2}}=a^2\frac{\partial^{2} z}{\partial x^{2}}$$
0
votes
1answer
73 views
Finding first and second partial derivatives of $u =(t^{-0.5})\exp\left(\frac{-x^2}{4k^2t}\right)$
$$u =(t^{-0.5})\exp\left(\frac{-x^2}{4k^2t}\right)$$
I'm trying to find $\frac{\partial u}{\partial t}$ and $\frac{\partial^2 u}{\partial x^2}$, but I'm finding this to be very annoying. Tia
0
votes
3answers
83 views
Show that $f$ is not totally differentiable at $x=(0,0)$ by proof by contradiction.
Show that $f$ is not totally differentiable at $x=(0,0)$ by proof by contradiction.
f(x,y) = \begin{cases}
\frac{x^3}{x^2+y^2},&(x,y)\neq (0,0)
\\
0,&(x,y) = (0,0)
\end{cases}
2
votes
1answer
79 views
Partial differentiation of integrals from u(x, y) to v(x, y)
I'm having trouble with this question.
If:
$$
w = \displaystyle\int_{xy}^{2x-3y}du/ln(u)\,du
$$
Find $$ \frac{\partial y}{\partial x} $$ at x = 3, y= 1.
I know that the general rule for ...
3
votes
2answers
120 views
The notation for partial derivatives
Today, in my lesson, I was introduced to partial derivatives. One of the things that confuses me is the notation. I hope that I am wrong and hope the community can contribute to my learning. In ...
0
votes
0answers
38 views
Partial derivate with sum of functions
I have Neural Net, forward feed. Now i need to calculate partial derivate of error between input and output.
My equations are as follows:
I need to calculate partial derivate X. Basicly, for every ...
3
votes
1answer
44 views
Transformation of domain in Evans
From Evans, Partial Differential Equations, Page 53.
Let $\Phi(x,s)=\frac{1}{{4\pi t}^{n/2}}e^{-\frac{|x|^{2}}{4t}}$. Evans used $E(x,t,r)$ to denote the region $$(y,s)\in \mathbb{R}^{n+1}|s\le t, ...
3
votes
1answer
54 views
Is this gradient an isomorphism on its range?
Consider the gradient (in the weak sense) as an operator $\nabla \colon H^1(\Omega)/\mathbb{R} \to [L^2(\Omega)]^d$, where $\Omega \subset \mathbb R^d$ is a domain with a smooth boundary and ...
