Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
9
votes
0answers
91 views
Ramanujan style nested differential Equation
So I was exploring some math the other day... and I came across the following neat identity:
Given $y$ is a function of $x$ ($y(x)$) and
$$
y = 1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + ...
5
votes
0answers
118 views
Partial derivative of a composite function $\mathbb{R}^n \to \mathbb{R}^n$
I am trying to understand a proof but I am stuck on this technical bit:
Apart from the small typo highlighted, I don't really see how to get the big formula for the partial derivative of $v_i$
...
5
votes
0answers
193 views
Closed form expression for constants
We have the constants $c_{k,n}$ defined by :
$$c_{k,n}=\frac{d^{k}}{ds^{k}}\left(\frac{e^{\frac{1}{n(ns-1)}}e^{\psi\left(\frac{s-1}{s} \right )}}{s} \right )$$
Where $\psi(s)\;$ is the Digamma ...
5
votes
0answers
194 views
Functions whose derivative is the inverse of that function
Everyone knows that there are at least three functions whose derivative is the function itself, namely $e^x, \ 0$ and $-e^{x}$. ( are there more?)
I was drawing some polynomials and their ...
4
votes
0answers
43 views
Inverse of a differentiable function equal to its derivative then f is analytic
I've found a nice problem concerning analytic functions. Here it is:
Let $f: (0, \infty) \rightarrow \mathbb{R}$ be a function differentiable on $(0, \infty)$ and such that $f^{-1} = f'$. Prove that ...
4
votes
0answers
33 views
If $D:A\to A$ is a derivation, what can be said about the range of $D$?
What can be said about the relation between the domain and range of a derivation as a function?
If $A$ is the domain, any space of functions, what does $D(A)$ look like, where $D$ is a derivation? ...
4
votes
0answers
141 views
“Simple” proof of Lebesgue's Differentiation Theorem for dimension 1?
Lebesgue Differentiation Theorem for $\mathbb{R}$: Let $f:[a,b]\to \mathbb{R}$ be intergable and $F(x)=\int_a^xf$. Then $F$ is differentiable almost everywhere in $[a,b]$ and $F'=f$ a.e.
Is there a ...
4
votes
0answers
35 views
Is $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in $R^n$?
Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors.
Then define
$H^{\alpha}_{ij,k}=e_k\langle ...
3
votes
0answers
72 views
Green's function for third order boundary value problems
How to find the Green's function $G(t,x)$ for the BVP consisting of the equation :
$$u'''(t)=0 , \quad t\in (0,1)$$
and BC :
$$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$
where $\frac12 < ...
3
votes
0answers
33 views
Trying to find the lyapunov function
I have the system that I want to show the global asymptotic stability of the origin
$$\dot{x_1} = x_2 \\
\dot{x_2} = -g(k_1 x_1 + k_2 x _2) $$ where k1 and k2 are positive numbers.
Also,
$$g(y)y ...
3
votes
0answers
101 views
Derivative chainrule on khanacadamy ignoring some terms
I watched the chain rule series on khanacademy.org and decided to do the "questions". One of the questions is:
Let $y = \sin(6x^2−4x−1+3x^{−1}−5x^{−2})$
$dy/dx=?$
The answer is ...
3
votes
0answers
77 views
Partial derivatives using variables after a transformation
I have a transformation $$(x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2))$$ and I wish to find $$\partial x'_1\over \partial x'_2$$ how might I evaluate this?
If it is difficult to find a general expression for ...
3
votes
0answers
188 views
Sign of derivative of a complicated function
EDIT (for bounty):
Consider the differential equation
$G(p;x,\lambda)p \left[1-\lambda-x(1+\lambda)\right] + x(1+\lambda)p + (1-x)(1-\lambda) \int_{p}^{1} z G'(z;x,\lambda) dz - (1-\lambda) = 0$,
...
3
votes
0answers
186 views
Taking derivative below an integral
I am trying to solve the following question:
If $t>0$, then
\begin{align*}
\int_{0}^{+\infty} e^{-tx} \; dx = \frac{1}{t}
\end{align*}
Moreover, if $t \geq a > 0$, then $e^{-tx} \leq ...
2
votes
0answers
29 views
Find values of C that satisfy the statement of theorem (Rolles/MVT)
I have the following function:
$x^3+x-1$ $[0, 2]$
And determined the following:
$f(0)=0^3+0-1=-1$
$f(2)=2^3+2-1=9$
And then this:
$f'(c)=(9-(-1))/(2-0)$
$f'(c)=3c^2+c-1=5$
And I'm stuck on ...
2
votes
0answers
69 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 ...
2
votes
0answers
39 views
Distributions - please check my solution
I have to find a derivative in a distributional sense of the following function (known as Cantor's singular function)
$$f(x)=\left\{
\begin{array}{l l l l l}
0, & \quad\text{$ x\leq 0 $}\\
1, ...
2
votes
0answers
125 views
Space of functions that are everywhere differentiable
Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with ...
2
votes
0answers
28 views
Looking for an analysis book which uses linear maps notation for multivariable differentiation
I'm taking an analysis course and I find it quite hard to follow what the professor is saying. So far we've been following elementary real analysis by bruckner^2 and Thompson but for the topic on ...
2
votes
0answers
266 views
Prove that sum is finite with the help of generating function
Please help me to prove that the following sum is finite
$$
\sum_{j=2l-2}^{\infty}j!\, a_j^{(l)},
$$
here the generating function of $\displaystyle{a_j^{(l)}}$ is ...
2
votes
0answers
80 views
Are there more types of differential in mathematics?
I am familiar with two types of differential
normal differential:
$$d^2x^a$$
covariant differential:
$${\mathcal D^2x^a}=d^2x^a+\Gamma^a_{bc}dx^bdx^c$$
(where the covariant differential is broken ...
2
votes
0answers
108 views
Problem with understanding first (and second) derivative of a two-sided infinite series
For the function $$f(x)=b^x-1 = x_1 \qquad g(x)=\log(1+x)/\log(b) $$ and its iterative notation $$ x_0=x \qquad x_h=f(x_{h-1})=g(x_{h+1}) \qquad x_{-1}=g(x_0) $$ with b from the interval $1 \lt b \lt ...
2
votes
0answers
190 views
Chain rule for matrix - i'm confused
I googled around and searched inside the forum but I'm still confused about a problem.
I have 2 matrix functions $f,g : \mathbb{R}^{n \times n} \times \mathbb{R}^{a \times b} \rightarrow ...
2
votes
0answers
34 views
Can I say something about the $f_{xy}$ of such function?
I have a function $f(x,y)$ defining on $x>0,y>0$ satisfying that
(1) $f>0$
(2) for $a>1, f(ax,ay)>af(x,y)$
(3) $f_x,f_y>0$
(4)$f_{xx},f_{yy}<0$
Can I say something about ...
2
votes
0answers
71 views
Lower bounds for the derivative of Laguerre polynomials
Let $ L_{d}^{(1)}(x)$ denote the generalized Laguerre polynomial of degree $d$ and order $\alpha=1$. Clearly, since all the roots $r_1,\dots,r_d$ of $L_{d}^{(1)}$ are simple, there exists a strictly ...
2
votes
0answers
114 views
rényi entropy as a derivative
Let $x=(x_i)$ be a probability measure on $\{1,\ldots,n\}$. Suppose $1<p<\infty$. The Rényi entropy of $x$ is
$$
H^p(x)=\frac{1}{1-p}\log \sum_{i} x_i^p.
$$
Does there exist a formula for ...
1
vote
0answers
37 views
Math question derivatives related?
So I have to find the constant c in order that the function $\displaystyle z= x^c e^{-y/x}$ proves the equation in the image
The problem is that I don't understand what $\displaystyle z_{yy}$ mean ...
1
vote
0answers
77 views
Taylor series expansion example
I was reading an article and there was a snippet with a taylor series expansion as shown below:
My question is, should (11) read as $F(xA+h)+(xΔA+Δh)\frac{\partial}{\partial x}F(xA+h)$ instead of ...
1
vote
0answers
35 views
Chain rule for function of two derivatives
Let $u = f(x, y)$ be a $C^2$-function. Let $x = rcos\theta$ and $y = rsin\theta$. Compute $\frac {d^2u}{dr^2}$ in terms of partial derivatives.
Answer: $\frac {d^2u}{dr^2}$ = $\frac ...
1
vote
0answers
55 views
Different Definitions on the Differentiability of Functions on a closed set.
I have encountered three different definitions on the differentiablity of functions on a closed set.
In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The ...
1
vote
0answers
46 views
gradient of an axis symmetric vector field in cylindical coordiantes
I am trying to calculate with a general approach the gradient of an axis symmetric vector field in cylindrical coordinates and then express it in cartesian coordinates.
First I write my vector ...
1
vote
0answers
33 views
Maximizing the value of an implicit function
I have $R$ as an implicit function of $N, K$ as defined in the following equation:
$$\left(1+K\frac{N}{12}\right)=\frac{R}{1+(1+R)^{-N}}$$
For those interested, this equation came about as a result ...
1
vote
0answers
115 views
Step in derivation of Euler-Lagrange equations of motion
From http://www.mathpages.com/home/kmath523/kmath523.htm
"Variations in x,y,z and X at constant t are independent of t (since each of these variables is strictly a function of t), so we have"
...
1
vote
0answers
23 views
Gâteaux derivate of the Tikhonov functional
Let $X,Y$ be Hilbertspaces, and let $A\colon X\to Y$ be a compact operator. The Tikhonov functional is given by
$$
F(x)=\lVert Ax-y\rVert_X^2+\alpha\lVert x\rVert_X^2.
$$
Calculate the Gâteaux ...
1
vote
0answers
41 views
Neglecting solutions and reforming the system of differential equations with reducing the order but to keep choosen solutions
Here I have one problem which should help me to understand how to transform the system of differential equations with the condition to neglect two of four solutions and to get the appropriate system ...
1
vote
0answers
96 views
Convergence of the reciprocal of a function whose derivative tends to infinity
If $$\lim_{x\to\infty} f'(x) = \infty$$ prove that $$\int_1^\infty \frac{1}{f(x)}\neq\infty$$
if $f'(x) \geq 1$ and $f(x) \geq1$ for all values of $x$. I'm thinking I can find a way to write it in big ...
1
vote
0answers
34 views
Velocity vectors on $S^{3}$
Consider $S^{3}$ as the unit sphere in $C^{2}$ under the usual identification $C^{2}\leftrightarrow R^{4}$. For each $z=(z^{1},z^{2})\in S^{3}$, define a curve $\gamma _{z}:R\rightarrow S^{3}$ by ...
1
vote
0answers
82 views
Why is the subdifferential of norm of a matrix ||A|| defined like this?
I read in a paper called "Characterization of the subdifferential of some matrix norms"
that it defines the subdifferential of the matrix norm like this:
$$\partial ||A||=\{G \in R^{m\times n} : ...
1
vote
0answers
55 views
Uniform convergence of complex exponent derivative
I'm trying to prove the following:
Let $\Re z > 0$. Then $$\lim_{\varepsilon \to 0} \frac{t^{z + \varepsilon} - t^z}{\varepsilon} = t^z \log t$$ uniformly in $t \in [0,1]$.
I've tried to ...
1
vote
0answers
76 views
Integrating derivative ($f$ of bounded variation)
Let $f$ be of bounded variation on $[a,b]$, and define $ν(x)=TV(f_{[a,x]}) ∀x∈[a,b]$,
I want to show: $\int_a^b|f'|= TV(f)$ iff $f$ is absolutely continuous on $[a,b]$.
My attempts:
I've shown ...
1
vote
0answers
71 views
Bounding derivative of a function
Consider $a(t)\in\mathbf{L}^{2}(\mathbb{R})$ and $a(t)>0$, is a low pass smooth function with $\hat{a}(f)=0, |f|>f_{max}$. Can we have a upper bound on the following,
...
1
vote
0answers
38 views
Implicit Differentiation???
Assume that $y$ is a function of $x$. Find $y' = \dfrac{dy}{dx}$ for $(x-y)^2 = x + y - 1$.
I've worked out the problem multiple times, but I continue to get a different answer than the correct ...
1
vote
0answers
68 views
Evaluation of function: derivative,integral, absolute continuity
I'm in trouble with this exercise tough I thought it was not that difficult at the begininng, now I definitely changed my mind.
So I hope someone could help me to solve it. Give the function:
$f:[0,1] ...
1
vote
0answers
67 views
Derivative of multivariate normal density wrt a scalar
Given a covariance matrix $$\left(\sigma^2 I + \lambda^{-1}K_{\theta}^{-1}\right)$$ where $\sigma,\lambda \in \mathbb{R}$ (both positive), $I$ is the $n \times n$ identity matrix and $K_{\theta}^{-1}$ ...
1
vote
0answers
28 views
variation of a final state due to changes in period (where the period is a parameter)
I have a simple ordinary differential equation
$\frac{dx}{dt}=f(x,t,p,T)$
$x(0) = x_0$, $x(T) = x_T$
where $p$ and $T$ are constant parameters. How do I compute $\frac{dx_T}{dT}$ ? Thanks!
NOTE: I ...
1
vote
0answers
66 views
Consider the correlation of two functions, what is the derivative of the result with respect to one of those functions?
I have a problem that comes up from time to time in signal processing applications.
Let $f(x)\geq0\, \forall x$ and $g(x)$ be real functions with finite range and support.
Let $I(f(x),g(x)) = ...
1
vote
0answers
43 views
$ g\left( z \right) = \int\limits_\gamma {\frac{{\varphi \left( u \right)}} {{u - z}}du} $ computing the derivate
Let $\gamma\colon[a,b]\to \mathbb{C}$ denote a piecewise differentiable path , and let $\varphi:$ Image $\gamma\colon \to \mathbb{C}$ be a continuous function.
Define $g: D = ...
1
vote
0answers
66 views
Partial derivative of matrix * constant vector wrt a vector
Is there an identity for simplifying partial derivatives of the form: $\frac{\partial A(\bf{x})\bf{b}}{\partial\bf{x}}$ ?
A is a square matrix that is a function of x, and b is a constant vector.
I ...
1
vote
0answers
39 views
Unable To Partially Differentiate A Magnetic Field Equation Due To A Steady Current
Given that $$H = \pm \frac{I}{2}\frac{\partial}{\partial x}(\frac{x}{\sqrt{r^2+x^2}})$$
I need to prove that $$H = \pm \frac{r^2I}{2\sqrt{(r^2+x^2)^3}}$$ where $H$ is the magnetic field vector due ...
1
vote
0answers
303 views
Implicit differentiation with multiple variables help
I need some help with some implicit differentiation. If this is a trivial question I can read about in a book then can you please recommend which book? I have looked up basic calculus books like ...
