Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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6 views

If $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$, then $f$ is a diffeomorphism

Suppose that $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ is a differentiable function such that $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$. (Note that $\mathbf{J}_f$ is ...
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1answer
36 views

How to differentiate a series?

I need help with this function. Can this function be differentiated? $$f(x) = (-a\sum_{i=0}^n x^b)$$ b>0 \frac{\partial}{\partial b}(-a\sum{x^b})$
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0answers
5 views

Matrix derivatives for the HJB and ARE relationship

How does one take the derivative of these matrix equations? (Backround:{My professor used them in the proof showing that the Hamilton-Jacobi-equation equivalently solves the free end-point ...
0
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1answer
31 views

differentiating the ln function

I tried it but I couldn't solve it. I have to differentiate w.r.t a and b: $$\ln(L)=n\ln(a) + n\ln(b) + (b-1) \Sigma \ln(x) + (-a^b\Sigma x^b)$$ Sorry for bad English.Thanks for helping.
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1answer
19 views

Behavier of function and its derivatives at infinty

If $\lim_{t\to \infty}(\phi(t))=0$ and $\lim_{t\to \infty}(\phi''(t))=0$ then can we say $$ \lim_{n\to \infty}(\phi'(t))=0$$ Can we have a $\phi(t)$ such that $\lim_{t\to \infty}(\phi(t))=0$ but ...
3
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1answer
17 views

Directly proving continuous differentiability

Let us say that we want to prove that a function $f: I \to \mathbb{R}$ defined on an open interval $I$ is continuously differentiable on $I$. One way to do this is to establish that $f'(x)$ exists at ...
2
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0answers
22 views

Derivatives 1, 2 and 3 and limits

A question for you. Show that if $\lim_{x\to+\infty} x\,f(x)=0$ and $\lim_{x\to+\infty} x\,f''(x)=0$ then $\lim_{x\to+\infty} x\,f'(x)=0$ Thanks ;)
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2answers
38 views

Can this inequality be solved with Mean value theorem

As my sub-assignment I have to solve inequality: $$ \ln\left(\frac{1}{x} + 1\right) -\frac{1}{x + 1} > 0 $$ If I understood MVT correctly, I should set $g(x)=\ln\left(\frac{1}{x} + 1\right) ...
2
votes
1answer
40 views

Does all function's domain stay the same\expands as we derivate them?

Lets define a funciton $f(x)$ with a domain of, lets say $a>x>b$. If I derivate this function, it's domain will always stay the same or expand? Or it can be "reduced"? Is that mean that $f'(x)$ ...
2
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0answers
75 views

How to find 50th derivative of $\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$?

I need to compute 50th derivative of $$\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$$ Of course I would not compute 50 derivatives. I want to find a certain regularity. And what I have: As ...
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1answer
35 views

How to show that $\lim_{x\to \infty}f'(x)=0$

Let $f$ be a real-valued, bounded, twice differentiable function defined on $(0,\infty)$ with $f'(x)\ge 0$ and $f''(x)\le 0$. Show that $$\lim_{x\to \infty}f'(x)=0$$ I understand $f: (0,\infty) ...
0
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1answer
20 views

Missing something about second derivative tests

I'm studying second derivative tests, concavity and inflection points in khan academy ...
0
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1answer
44 views

How to take the derivative of a matrix with respect to itself?

Could someone please explain how to take the derivative of matrix with respect to itself? $$\frac{\partial \textbf{X}}{\partial \textbf{X}}$$ where $\textbf{X}$ is an M x N matrix
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0answers
20 views

Do these statements prove this formula?

$$ \frac{d^n}{dx^n}[g(x)^{f(x)}] = g(x)^{f(x)} B_n(d_1,\cdots,d_n) $$ Calling $$ d_n = \frac{d^n}{dx^n}[ln(g(x))f(x)] $$ Since faa di bruno's formula states $$ \frac{d^n}{dx^n}[f(g(x))] = ...
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0answers
25 views

What is the point of reflection of this function

$$y = 3x(x+5)^{2/3}$$ Is there some kind of trick to simplify it?
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0answers
34 views

Proof Verification for $n2^{n-1} = \sum\limits_{k=1}^n k\binom{n}{k}$

$(1+x)^n = \sum\limits_{k=0}^n\binom{n}{k}x^k$ by binomial theorem $\frac{d}{dx}(1+x)^n =\frac{d}{dx}\sum\limits_{k=0}^n\binom{n}{k}x^k$ $n(1+x)^{n-1} = \sum\limits_{k=1}^n k\binom{n}{k}x^{k-1}$ ...
0
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1answer
16 views

Inverse of a function involving a Jacobian.

Why is it true that if the inverse of both $ \tilde{f} $ and $ f $ exists then: $$ \tilde{f}\left(\vec{x}\right) = [Df(x_{0})]^{-1} f(\vec{x}) $$ $$ \implies \tilde{f}^{-1}(\vec{x}) = ...
2
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0answers
28 views

Differentiability and Monotonic Functions

I just read proof from Royden of theorem: 'Every Monotonic functions are differentiable almost everywhere.' But proof use Vitali Covering Lemma. But Vitali Covering Lemma is based on fact if we assume ...
6
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4answers
93 views

Show that $\lim_{x \to +\infty}\left(f(x)+f'(x)\right)=0 \Rightarrow \lim_{x \to +\infty} f(x)=0$

How to show that $\lim_{x \to +\infty}(f(x)+f'(x))=0 $ implies $\lim_{x \to +\infty} f(x)=0$?
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0answers
17 views

Derivative of the detrminant map

Question : For $ v = (v_1, v_2) \in \mathbb R^2$ and $ w = (w_1, w_2) \in \mathbb R^2$, consider the determinant map $det : \mathbb R^2 \times \mathbb R^2 \rightarrow \mathbb R$ defined by $det ...
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1answer
20 views

prove integration formula relating to derivatives

Could any one help me solve this problem ? it is from Apostol's calculus volume 1
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0answers
32 views

Help differentiate long equation

I need to differentiate the following equation twice with respect to $\alpha$. It is a profile log likelihood equation, where I need the derivatives to get the information matrix. The equation is: ...
2
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2answers
25 views

Finite difference method

I wanted to ask something regarding the finite difference approximation. I used the finite difference to calculate the numerical derivatives of my function. The finite difference is given by the ...
3
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4answers
77 views

Derivation of the integral

Evaluate $$\large\frac{d}{dx}\int_{0}^{\large\int_0^{e^x}{\cos (s)\,\mathrm ds}}\sec(t^2)\,\mathrm dt$$ I got the answer to be $$e^x\cdot\sec(\sin^2(e^x))\cdot \cos(e^x)$$ but do not know if ...
0
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0answers
15 views

Maximum volume of an open box with a square base?

A box with a square base and an open top is to be made. You have 1200cm^2 of material to make it. What is the maximum volume the box could have? Here's what I did: 1200 = x^2+4xz; where x=length of ...
3
votes
1answer
73 views

If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$

Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints. I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} ...
0
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0answers
31 views

problem with deriving continuity equation

I am studying Aerodynamics, to be more precise, the fundamentals of Aerodynamics. The first law is the continuity equation, for which it is explained in the book that I am using. However, I wished to ...
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0answers
17 views

Parametric derivative of $x^2+y^2+sin(4x)+sin(4y)=4$.

I am trying to parametrize $x^2+y^2+sin(4x)+sin(4y)=4$. I need to find a way of taking the intersections between $x^2+y^2+\sin(4x)+\sin(4y)=4$, and $\tan(nx)$, as n increases from $0\le{n}\le{2\pi}$. ...
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0answers
14 views

Find a vector field $\mathbb{Y}$ satisfying $L_{\mathbb{X}}\mathbb{Y}=\mathbb{Z}$

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $\mathbb{X}=(1,y)$. Let $\mathbb{Z}$ be the vector field on $\mathbb{R}^2$ given by $\displaystyle \mathbb{Z}(x,y)= \bigg( ...
0
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1answer
17 views

$\frac{d}{dx} f(x)$ piecewise defined

my function is defined as follows: $\frac{1}{x} $if $x \not = 0$, $ 1 $ if $ x=0$. Does the $\frac{d}{dx} f(x)$ in $x=0$ exist?
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0answers
45 views

urgent help please solve this eqaution [on hold]

Solve the following equation: $$f' '(x) + 4 f(x)= 0$$ where $f(0)=1$ and $f'(0)=-1$
1
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4answers
55 views

Trig differentiation

Prove that there is a constant C such that $$ \arcsin{\frac{1-x}{1+x}} + 2\arctan (\sqrt{x}) = C $$ for all $x$ in a certain domain. What is the largest domain on which this identity is true? What ...
0
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0answers
20 views

Find the lie derivative of a particular integral

Let $\mathbb{X}=(1,y)$ be a vector field on $\mathbb{R}^2$. Let $\Phi_t$ be the flow of $\mathbb{X}$. The flow of $\mathbb{X}$ I have calculated to be $\Phi_t(x,y)=(x+t,ye^t)$ Given a function ...
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0answers
12 views

How to take derivative $||W^TWx-x||_2^2$ with respect to $W$? [on hold]

It bothers me for a long time... How to take derivative $||W^TWx-x||_2^2$ with respect to $W$? I could not get a very efficient representation for the gradient... Help.........
1
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1answer
29 views

Maple - Substitute into an expression involving derivative

I'm working with some matrices including derivatives like d/dt(x(t)). I need to replace this whole expression {d/dt(x(t))} with something like xdot. I have tried to use "subs", but it seems to refuse ...
0
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2answers
25 views

find the derivative of a function with more than one variable

I have a function $g(a)=f_i(x+a(y-x))$ where a$\in$$\Re$ and x,y$\in \Re^d$. How can I find the first and the second derivative of this function? The second part of the exercise is asking me to use ...
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0answers
18 views

Calculating mixed strategy Nash equilibria: using the derivative?

From roaming around and looking for ways to calculate the mixed strategy Nash equilibrium, I learned that a general way to do it is by determining the probability of choosing a strategy in such a ...
2
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0answers
51 views

Show that $f(x)$ is uniform continuity in $(0,1]$

Suppose that $f(x)$ is a continuously differentiable function in $(0,1]$,and $\lim\limits_{x\rightarrow0^{+}}\sqrt{x}f(x)$ exists. Show that $f(x)$ is uniformly continuous on $(0,1]$.
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0answers
37 views

Looking for differentiable function $f:\mathbb R \to \mathbb R$ whose derivative is nowhere continuous [duplicate]

Does there exist a differentiable function $f: \mathbb R \to \mathbb R$ such that its derivative $f'$ is nowhere continuous ?
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1answer
81 views

Taking the Derivative of $F(x)=\int_0^x f(t)\,dt$ [on hold]

Let $F(x)=\int_0^x f(t)\,dt$ What is the derivative of $F(x)$? I desperately need guidance!
0
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0answers
11 views

Tips on effectively representing this recurrence relation in a generalized form.

The recurrence relation is $$ y_n = d_1 y_{n-1}+\frac{d}{dx}[y_{n-1}] $$ a good thing to note as well is $$ \frac{d}{dx}[d_n] = d_{n+1} $$ This is terrible to expand out after a good while, The main ...
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1answer
18 views

The set of differentiability of an extension from half-plane to the plane

This question is related to: Differentiability: Partially Defined Functions Consider a real-valued function $f:\mathbb{H}^2\to\mathbb{R}$. Are there some that admit no extension differentiable in ...
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0answers
12 views

Polynomial Root Multiplicity Testing.

I would appreciate some help here. Either a reference or a proof or just a statement that helps me to conduct research of my own. Long ago when I was studying polynomials intently I read about a ...
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0answers
40 views

Does there exist a differentiable function $f:\mathbb R \to \mathbb R$ such that $f'$ is no-where differentiable on $\mathbb R$?

Does there exist a real valued differentiable function $f:\mathbb R \to \mathbb R$ such that $f'$ , the derivative of $f$ , is no-where differentiable on $\mathbb R$ ?
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0answers
13 views

$\phi:\mathbb{R}^2\to\mathbb{C}$,$\phi(x,y)=x+iy=z$,$F=\phi^{-1}f\phi$

$\phi:\mathbb{R}^2\to\mathbb{C}$ be a map $\phi(x,y)=x+iy=z$, let $f:\mathbb{C}\to\mathbb{C}$ be the function $f(z)=z^2$ and $F=\phi^{-1}f\phi$ then I need to say which of the following are correct. ...
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0answers
27 views

Proving a theorem about matrix derivations

Ok, so Im doing some research and I have to understand the following theorem. The theorem states: Let $h$ be a derivation on $Tn(R)$ with $h(e_{ij})=0,\,\, 1\le i \le j \le n$. Then $h=\bar\delta$ ...
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votes
3answers
130 views

What is the kernel? give a basis of the kernal [on hold]

Let $P_3$ denote the real vector space of polynomial functions of degree up to 3, i.e., $P_3 = \{ a_3x^3 + a_2x^2 + a_1x + a_0 \mid a_i \in R\}$. Consider the linear transformation $D : P_3 \to P_3$ ...
0
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1answer
41 views

Show that the $n$th derivative of $f(x)$ is zero for all $n \geq 0$. [duplicate]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=e^{-\frac{1}{x^2}}$ for $x \neq 0$ and $f(x)=0$ for $x=0$. I want to show that $f^{(n)}(0)=0$ for all $n \ge 0$. That is, the $n$th ...
2
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1answer
72 views

$f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and satisfies an inequality that involves its partials - show that f is a bijection.

Suppose that $f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and the partial derivatives of the components $f_1$, $f_2$ satisfy $$max(|\frac{\ df_1}{dx} -1|, |\frac{df_1}{d_y}|, ...
0
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0answers
24 views

Derivative with respect to a tensor in Mathematica

I am trying to differentiate a tensor with respect to another one in Mathematica but I cannot do it. Could anyone please help? The following is the code: ...