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

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0
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0answers
5 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
votes
0answers
18 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 ...
5
votes
4answers
67 views

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

How to show that $\lim_{x \to +\infty}(f(x)+f'(x))=0 $ implies $\lim_{x \to +\infty} f(x)=0$?
0
votes
0answers
15 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
2
votes
0answers
31 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
votes
2answers
22 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 ...
2
votes
4answers
57 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 ...
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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
68 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
25 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 $0\le{n}\le{2\pi}$. ...
0
votes
0answers
10 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
43 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
vote
4answers
54 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
18 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 ...
0
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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
vote
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 ...
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2answers
23 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 ...
0
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0answers
15 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
50 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
51 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 ...
0
votes
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
10 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 ...
-1
votes
0answers
39 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. ...
0
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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 $h=\bar\delta$ ...
-3
votes
3answers
106 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
votes
1answer
71 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: ...
1
vote
0answers
24 views

Nth Derivative of the function [duplicate]

Find the $n^{th}$ derivative of $$f(x) = e^x\cdot x^m$$ If i am not wrong i have following $1^{st}$ Derivative: $e^x\cdot m \cdot x^{m-1} + x^m\cdot e^x$ $2^\text{nd}$ Derivative: $e^x\cdot m \cdot ...
1
vote
0answers
29 views

Find an integrating factor such that $y'=\frac{1-x+y}{x-y}$ is exact

Yet another question of this sort, and hopefully the last. In the previous question I posted, we were lucky enough and the integrating factor was a function of only one variable, the ansatz $\mu_y=0$ ...
4
votes
2answers
28 views

Ordinary differential equations of the form $M(x,y)dx+N(x,y)dy=0$ question

An ODE of the form $M(x,y)dx+N(x,y)dy=0$ is called "good" if $\frac{\partial (M(x,y))}{\partial y}=\frac{\partial (N(x,y))}{\partial x}$ We are given the differential equation ...
1
vote
2answers
47 views

Nth Derivative of a function

Find the $n^{th}$ derivative of $$f(x) = e^x\cdot x^n$$ If i am not wrong i have following $1^{st}$ Derivative: $e^x\cdot n\cdot x^{n-1} + x^n\cdot e^x$ $2^\text{nd}$ Derivative: $e^x\cdot n\cdot ...
0
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0answers
16 views

Functional differentiation in Mehta's “Random Matrices”

I'm trying to understand a bit in this book about functional differentiation, which I don't know much about. According to Wikipedia, $\delta F=\int d^n\boldsymbol{r}\frac{\delta F}{\delta ...
4
votes
1answer
47 views

Nth Derivative of a fucntion

Find the $N^{th}$ derivative of $$f(x) = \sqrt{\frac {1-x}{1+x}}$$ I have got $1^{st}$ derivative as: $\frac{-1}{(1-x)^{1/2}(1+x)^{3/2}}$ and $2^{nd}$ derivative as: ...
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votes
2answers
23 views

How to find the rate of change of the following problem? [on hold]

Consider a right circular cylinder of height h and radius r. If the radius increases at the rate of 3 cm/sec and the height decreases at the rate of 5 cm/sec, find the rate of change of each of the ...
3
votes
1answer
76 views

If f ' = 0, then f is constant?

I'm a little confused. After finishing the online multi-variable calculus course from the MIT OCW offerings (I wanted to brush up on the subject more concretely, after my Analysis II course), I ...
-2
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0answers
30 views

Derivatives and second derivatives [on hold]

Create an equation with the following conditions : 4th degree polynomial, first and second derivative factorable, non zeros
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0answers
7 views

directional derivative of convex function always exist

How we can prove that directional derivative of any convex function is exist for any $x\in \text{dom}(f)$ where $\text{dom}(f)=\left\{ x | f(x)<+\infty\right \}$ and directional derivative of $f$ ...
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0answers
11 views

Holder continuity and gradient

I am trying to prove the implication of differentiability and constancy from Holder continuity. I have: $\frac{\left\lvert f(x)-f(y) \right\rvert}{x-y} \le M|x-y|^{\lambda} \implies \exists g:x ...
3
votes
1answer
37 views

How to calculate $\frac{d}{d y} y'$

Is there a way to evaluate the following expression? $$ \frac{d}{dy} y' $$ where $$ y' = \frac{dy}{dx}.$$
1
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2answers
44 views

If $f'(x)$ has a limit as $x\to x_0$, then the function $f$ is differentiable at $x_0$

I've got a question about mathematical analysis of one-variable functions. Assume that we have a function defined for $x \neq x_0$ as composition/sum/product of differentiable functions and also ...
0
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0answers
42 views

I am required to solve the boundary value problem $y'' = 4x^2y' + 2xy,\space y(1) = 4,\space y(2) = 2$ using the midpoint method.

I am required to solve the boundary value problem $$y'' = 4x^2y' + 2xy,\space y(1) = 4,\space y(2) = 2$$ using the midpoint method. In order to get two first order equations I have set $u_1=y\space ...
0
votes
1answer
17 views

Possible Points of Local Extrema

I know that we should check critical numbers (points where f'(x) is either zero or not defined) and endpoints (for a closed interval) as possible points of local extrema of f(x). Obviously, all these ...
1
vote
3answers
40 views

Inflection point of $\,f(t) = \frac{1}{1+e^{(-t)}}$

I am trying to calculate the inflection point of the logistic function $f(t) = \dfrac{1}{1+e^{(-t)}}$. According to the definition given in Wikipedia, "A differentiable function has an inflection ...