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

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51 views

total least squares derivation with matrices

Taken from a computer vision book: "to minimize the sum of the perpendicular distances between points and lines, we need to minimize $$ \sum_i (ax_i + by_i +c)^2$$ subject to $a^2 +b^2 =1$. Now using ...
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2answers
47 views

Why $\dfrac{d}{dt} \dfrac{dy}{dx} = \dfrac{d}{dx} [ \dfrac{dy}{dx} ] \quad \dfrac{dx}{dt} $ ? [Stewart P206 3.4.95, BDP P165 3.3.34]

If $y=f(x)$, and $x = u(t)$ is a new independent variable, where $f$ and $u$ are twice differentiable functions, what's $\dfrac{d^{2}y}{dt^{2}} $? By the chain rule, $\dfrac{dy}{dt} = \dfrac{dy}{dx} ...
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1answer
35 views

Formula for area under the curve

I don't know that the equation that I am going to explain below is correct or not, and this is why I am asking this question. So, I have found out that area under the curve could be found out by ...
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1answer
46 views

continuous partial derivative implies total differentiable (check)

Let $f: \mathbb{R^n}\rightarrow \mathbb{R}$ to have continuous partial derivatives, it suffice to show that $f$ is total differentiable at $(0,..,0)$ with $f_{x_i}(0,..,0) = 0$. Since each partial ...
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1answer
38 views

right derivative of a continuous function

Let $f:(a,b)\longrightarrow \mathbb{R}$ be continuous. Suppose $D_+f(x)=\lim_{h\to 0+}\frac{f(x+h)-f(x)}{h}\geq 0$ for any $x\in (a,b)$. Prove that $f(x_1)\geq f(x_0)$ whenever $x_1\geq x_0$. How to ...
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2answers
218 views

How prove that there exists $\xi\in(a,b)$ with $f'(\xi)=\frac{f(\xi)-f(a)}{b-a}$

Let $f(x)$ be continuous on $[a,b]$, differentiable on $(a,b)$, and with some $c\in(a,b)$ such that $f'(c)=0$. Show: There exists $\xi\in(a,b)$ such that $$ f'(\xi)=\dfrac{f(\xi)-f(a)}{b-a} $$ ...
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1answer
65 views

Derivative of $\frac{x}{(1+x^2}$ using the limit definition.

I just started learning Calculus on my own and understand where $\lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$ comes from but I'm having trouble with this one, I think my Algebra skills are letting ...
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3answers
109 views

Why is $\cos(x)$ the derivative of $\sin(x)$?

The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$. Is there a simple proof of this, preferably using pictures?
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1answer
25 views

Is there any trick you can use to derive f( h(x),x)

I was just wondering, is there a way to derive $ \frac{d}{dx} f( h(x),x)$ without knowing how the function looks? For example by some trick of using multivariable diferentiation of $f(h(x),y)$? Thank ...
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1answer
17 views

Rate of change, square area by time

The question describes a square with sides defined by $s = 2+t²$, where $t = time$. It asks to define the variation rate of the area when $t = 2$ My result: $ t = time $, $ s = side $, $ A = area $ ...
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1answer
32 views

$Df(x_0)$ is one-to-one. show $f$ is one-to-one on a neighborhood of $x_0$

Suppose $f:\mathbb{R}^n \to \mathbb{R}^m$ is $C^1$ and $Df(x_0)$ is one-to-one. Show $f$ is one-to-one on a neighborhood of $x_0$. I think it's about inverse function theorem. but i cannot prove ...
0
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1answer
27 views

Prove that function f has a local minima and maxima

$f:R->R, f(x) = (x^2+mx)e^-x$ Show that, for every m in R, the function f has a local ...
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0answers
38 views

Proof of pointwise convergence of derivative of power series

I proved: If $\sum_{n=0}^\infty a_n x^n$ converges for all $x \in (-R,R)$ then the differentiated series $\sum_{n=1}^\infty na_n x^{n-1}$ converges for all $x \in (-R,R)$. Please could somebody tell ...
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0answers
39 views

Meaning of this differentiation operators

I have been just reading this paper here: paper and was wondering how they carry out the differentiation in (4.9). In principle, this should be just the differentiation of 4.8 with the help of 4.7a. ...
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1answer
19 views

How to show where a funcion is differetiable?

I mean what is the basic technique on functions which are defined like $h(x)$= {$f(x)$ if $x$$\in$$[a,b]$ and $g(x)$ if $x$$\in$$[c,d]$}. An example:$f(x)$={$x$*$cos({1\over x})$ if $x$ is not $0$ ...
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1answer
65 views

Why can't I use the chain rule on $f(x,y) = \frac{xy^2}{x^2 + y^2}$?

The function is $f(x,y) = \frac{xy^2}{x^2 + y^2}$ and $0$ if $(0,0)$. So $f(x,y)$ is continous, the partial derivatives exist, and the partial derivatives are also continuous (the limit as $(x,y) ...
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2answers
66 views

Question about limit definition of partial derivative

I've seen it written two different ways: $$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x + h, y) - f(x,y)}{h}$$ and $$\frac{\partial f}{\partial x} = \lim\limits_{h ...
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1answer
293 views

Does there exist a continuously differentiable function with the following properties?

Does there exist a continuously differentiable function $f: [1,5] \rightarrow \mathbb{R}$, such that $f(1) \lt 0, f(5) \gt 3$ and $f'(x) \leq e^{-f(x)}$? Now do I just integrate it to get $f(x) ...
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1answer
30 views

Question about definition of derivative

$$\lim\limits_{\textbf{x} \rightarrow \textbf{x}_0} \frac{\|f(\textbf{x}) - f(\textbf{x}_0) - \textbf{T}(\textbf{x} - \textbf{x}_0)\|}{\|\textbf{x} - \textbf{x}_0\|} = 0$$ Does the $\textbf{T}$ part ...
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1answer
40 views

Help with definition of derivative

My textbook says the definition is this: $$\lim\limits_{\textbf{x} \rightarrow \textbf{x}_0} \frac{\|f(\textbf{x}) - f(\textbf{x}_0) - \textbf{T}(\textbf{x} - \textbf{x}_0)\|}{\|\textbf{x} - ...
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2answers
52 views

Line equation of a tangent line of $f(x) = x\cos(3x)$

I'm new here so maybe I'll need some help with formatting with MathJax, as well. So question asks for tangent line of $f(x) = x\cos(3x), x= \pi$ So: $$f(x) = x.\cos(3x)$$ $$f'(x) = ...
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1answer
130 views

Approximation of $x!$ - Proof needed

By drawing a graph of the geometric derivative of $x!$, $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}$, i guessed that $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}\sim_{+\infty}(x+1/2)$. ...
2
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1answer
66 views

Does the function differentiable?

Let $\alpha, \beta \ge 1 \in \mathbb{N}$ and: $$f(x) = \left\{ {\matrix{ {{x^\beta }\sin \left( {{1 \over {{x^\alpha }}}} \right)} \cr {0,x = 0} \cr } } \right.$$ I checked for ...
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0answers
22 views

Show that both these Prove $f$ is differentiable from the right at $0$

Let $f:[0,1) \rightarrow \mathbb{R}$ be continuous on [0,1) and differentiable on $(0,1)$. Suppose the limit $\lim_{x \to +0} f'(x)$ exists. Prove that $f$ is differentiable from the right at $0$. ...
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0answers
24 views

Limits of norms and deriviative as linear transformation

I'm self-studying Spivak's Calculus on Manifolds and he introduces the derivative by first looking at it as a linear transformation, $Df(a) = \lambda$, saying that for a differentiable function ...
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1answer
96 views

Is this notation good for the chain rule derivative?

When we take this derivative, for example: $$y = \log(\sin x)$$ We call $u = \sin x$, so we have: $$\frac{dy}{dx} = \frac{d y}{du}\frac{du}{dx} = \frac{1}{u}\cos x = \frac{\cos x}{\sin x}$$ But for ...
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3answers
44 views

necessary condition for the mean value theorem

Give an example which demonstrates that continuity is a necessary condition for the mean value theorem: I thought in this function: $$g(x) = \begin{cases} x + 1 & x < 1 \\[4pt] x - 1 ...
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2answers
28 views

Stationary point question

Find the coordinates of stationary points on the curve with the equation $(y-2)^2e^x=4x$ I differentiated to get $2(y-2)e^x\frac{dy}{dx}+(y-2)^2e^x=4$ $\frac{dy}{dx}=0$ $(y^2-4y+4)e^x=4$ What do ...
2
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3answers
74 views

Second derivative of $\arctan(x^2)$

Given that $y=\arctan(x^2)$ find $\ \dfrac{d^2y}{dx^2}$. I got $$\frac{dy}{dx}=\frac{2x}{1+x^4}.$$ Using low d high minus high d low over low squared, I got $$\frac{d^2y}{dx^2}=\frac{(1+x)^4 ...
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0answers
13 views

unimodality and continuous

i would like to ask question about unimodality of probability function ,from wikipedia http://en.wikipedia.org/wiki/Unimodal it says that In mathematics, unimodality means possessing a unique mode. ...
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1answer
74 views

How to show that this function is differentiable?

Let $$\phi: \mathbb{R} \rightarrow \mathbb{\mathbb{C}}, s \mapsto \int_2^{\infty} \frac{e^{isx}}{x^2\ln(x)}dx$$, I want to show that this function is differentiable everywhere. Unfortunately, it ...
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1answer
38 views

properties of third order derivative

I have a question in my assignment (about interpolation) which has the condition that the third-order derivative is continuous at $x_2$ and $x_{N-1}$. That is, $S'''(x_2)=S'''(x_{N-2})$. The question ...
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2answers
42 views

Calculate the energy in a circuit containing a resistor

A voltage peak in a circuit is caused by a current through a resistor. The energy E which is dissipated by the resistor is: Calculate E if Can anyone please give me some suggestions where to ...
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0answers
21 views

Partial derivatives of a function with respect to a vector of parameters

I'm reading a paper that tries to optimize a function. I'm having a little trouble understanding the derivation of the gradient of the function. Here's the optimization objective: $w$ is vector of ...
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1answer
23 views

Fréchet derivative and local maximum

I'm pretty confused with the idea of local maximum in function spaces. Normally having a null Fréchet derivative is a necessary but not sufficient condition for being a local maximum. Computing the ...
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0answers
31 views

Is it correct that $\frac{d\theta}{d\varphi'}=\left(\frac{d\varphi'}{d\theta}\right)^{-1}$?

Let $\theta$ be $k\times1$ and $\varphi$ be $k\times1$. Then $\frac{d\theta}{d\varphi'}$ is a $k\times k$ matrix with entries $\frac{d\theta_{i}}{d\varphi_{j}},\,\, i,j=1,...,k$. I was wondering if ...
3
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0answers
35 views

Prove the following expression is true.

Let $x_1,...,x_{n+1}$ be arbitrary points in $[a,b]$ and let $$Q(x)= \prod\limits_{i=1}^{n+1} (x-x_i)$$Now suppose $f$ is an n times differentiable function and tha P is a polynomial function of ...
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1answer
35 views

Definition of Limit of a Function

I'm a little confused by the definition of the limit of a function-on one hand I feel the definition suggests that your limiting variable is shrinking into a little delta ball- on the other hand when ...
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2answers
211 views

What is the mistake in this proof of product rule of differentiation?

I was trying to derive the product rule of differentiation which states: If $y=u\cdot v$, then, $y'=u'\cdot v+v'\cdot u$. So I assumed it like this: $y=u+u+u+\cdots$ ($v$ number of terms of $u$) ...
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1answer
12 views

Functions with symmetrical behaviour with respect to an axis or a plane

Suppose we have two functions with a symmetrical behaviour with respect to an axis. For the sake of simplicity, let $f(x)$ and $g(x)$ have a symmetrical behaviour with respect to the $y$ axis. A ...
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1answer
42 views

supremum of derivatives

Let $f$ twice continuously differentiable on $(a, \infty)$. Let $M_{0} = \sup f$, $M_{1} = \sup f'$, $M_{2} = \sup f''$. Show that $ (M_{1})^{2} \leq M_{0}M_{2}$. Also, How can this be modified ...
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2answers
56 views

Calculus - Derivatives of Polynomials

$f(x)=-3x^2-14x-5$ and $g(x)=\frac{1}{2}x^2-\frac{15}{2}x+29$ are parabolas on the same grid. The tangent to $y=f(x)$ at $P_1(x_1,y_1)$ and the tangent to $y=g(x)$ at $P_2(x_2,y_2)$, intersect at ...
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1answer
50 views

Tangent of a sine - can it disprove the marginal value theorem?

The marginal value theorem is partly explained by this text and graph: As animals forage in patchy systems, they balance resource intake, traveling time, and foraging time. Resource intake within ...
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1answer
38 views

How to solve the gradient of a matrix function

Let: $$A=\hat{r}+e^{j\theta}$$ Where $\hat{r}, e^{j\theta} \in \mathbb{C}^n$, and j = (-1)^(.5). Set $L \in \mathbb{R}^{n \ x \ n}$, define $S$: $$S=A^TLA$$ The gradient of $S$ is given by ...
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1answer
37 views

Derivative of $f(x)=\|Ax\|_2^2$

I'm trying to find the derivative of $f(x)=\|Ax\|_2^2$ where $A$ is some matrix and $\|u\|_2$ is the euclidean norm of $u$, $\|u\|_2 = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}$ I know how to do this by ...
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1answer
17 views

Finding coefficients of a third degree polynomial

The third degree polynomial $$-x^3 + ax^2+bx+c$$ has an maximum at $(2,10)$ and an inflation point at $(0,-6)$. Find the coefficients $a$ $b$ and $c$. Am I supposed to differentiate the polynomial ...
3
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0answers
113 views

The second derivative as a limit

It is well-known that if $f$ is twice differentiable at $a$, then $$ f''(a) = \lim_{h\to 0} \frac{f(a+2h)-2f(a+h) + f(a)}{h^2}. $$ See e.g. this question or this question. On the other hand, the ...
3
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1answer
78 views

How to prove l'Hospital's rule for $\infty/\infty$

I'm having trouble with this l'Hospital's rule wiki page(the proof of l'Hospital's rule): http://en.wikipedia.org/wiki/LHospital%27s_rule Well, in the case where the limit looks like $0/0$, it's ...
4
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2answers
59 views

Are $C^{\infty}$ completely defined by their derivatives?

This question has been on my mind for some time. Here's my process. Firstly, is it possible to construct a function such that it's defined with a different expression on different intervals, but that ...
2
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0answers
32 views

Is little-o preserved under integration and derivation of another variable?

Given an integrable function $g:\mathbb{R}\longrightarrow\mathbb{R}$, and a function $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ such that $f(x,y)=o(x^{-1})$ when $x\rightarrow\infty$, i.e. ...