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

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How do you differentiate $x_i^j\sum_i\sum_jx_i^j$?

I'm trying to find the jacobian of of a function that contains matrices, e.g. $$\dot{x_i^j} = \ \ x_i^j\sum_i\sum_jx_i^j + \sum_i\sum_jx_i^jy \ \ \ + \ ... \ = \ \ f(x_i^j,y) $$ Where $x_i^j$ is ...
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1answer
4 views

Angle between a line and a circle that it goes though

I just solved a task regarding the angle under which a certain line goes through a circle. The line naturally has two common points with the circle. It seems that the angle between them is the same in ...
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14 views

Analyze the variation of this function $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$ w.r.t. $x$

Please, I need to analyse the variation of the following function w.r.t. $x$ : $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$, where $E_1[a+b (x-1)]$ is the exponential integral, $a>0$, $b>0$ and ...
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1answer
13 views

Differentiation and PDE Theory

I have been given the following two definitions: 1) $D^ku$ is the set of all derivatives of order k of u 2) Let $\Omega$ be a non-empty subset of Euclidean space $\mathbb{R}^N.$ An expression of the ...
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1answer
12 views

Total differential related limit

I am trying to find whether total differential of $u(x,y)=y^{1/3}\arctan x$ exists at $(0,0)$, where partial derivatives are zero, so I want to know if $$\lim_{(h_1,h_2)\to\vec 0} ...
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1answer
21 views

Differentiating an Integral

Does anyone know any general approach for something like this: $$ \frac{d}{dx}\int_{-\infty}^{x}f(x,u)du\qquad\text{or}\qquad\frac{d}{dx}\int_{x}^{\infty}f(x,u)du\qquad $$ Basically, I'm trying to ...
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1answer
22 views

Coming Up With A Neutral Fixed Points Theorem

Question: If $f(x_0)=x_0,f'(x_0)=1$ and $f''(x_0)>0$, is $x_0$ weakly attracting, weakly repelling, or neither? (weakly attracting meaning $\exists\delta,\forall x\in ...
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1answer
30 views

Find the derivative of an integral.

Find the derivative of the following integral $$ F(x)=\int_x^{x^2}e^{t^7}dt $$ Find F′(x) given F(x). Normally I would show my attempt in working out the problem: however, I don't even know where ...
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7 views

On Differentiation Theorem and Radius of Convergence

For reference: http://www0.maths.ox.ac.uk/system/files/coursematerial/2014/2644/48/14MT-AnalysisI-sheet7.pdf 4.(b) For fixed $d \in \mathbb{R}$, let $f_d(x)=S(d+x)C(d-x)+S(d-x)C(d+x)$ By ...
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Conceptual question on differentiation in calculus?

In Calc class, my teacher told us that the only solution to $y' = y$ is $y = ce^x$, with $c$ being a real number. I am having difficulty understanding the only part. Is there a proof of this? Or am I ...
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2answers
64 views

Prove that a function is constant [duplicate]

I'm attempting to prove the following statement: Let $f:\mathbb{R}\to\mathbb{R}$ be a function and suppose that $|f(x)-f(y)|\leq (x-y)^2$ for all $x,y\in\mathbb{R}$, therefore f is constant. I ...
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0answers
33 views

Why aren't placeholders for arguments more common?

When learning about differentiation and integration, one often deals with functions, and it's common to use $D(x^2) = 2x$ as a function instead of $D(x\mapsto x^2) = (x\mapsto 2x)$, while it would ...
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2answers
26 views

Number of zeros of $ f^n $

Let $f:\Bbb R\to \Bbb R$ be infinetly differentiable function that vanishes at $10$ distinct points in $\Bbb R$.suppose $ f^{n} $ denote $n$-th derivate of $f$, for $n \ge 1$. Then which of following ...
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33 views

how to solve this chain rule problem

I'm having some difficulty solving this problem. The information I have is the following: $z=z(y_1,y_2,y_3,...,y_n)$ and $y$ is also composed of a function $y=y(x_1,x_2,x_3,...,x_n)$ given that ...
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2answers
38 views

Prove the existence of derivative

Let $f\colon \mathbb{R}\rightarrow\mathbb{R}$ be a continuous function on $\mathbb{R}$. Suppose $f(x)$ exists for all $x\neq0$, and $\lim_{x\rightarrow0}f'(x)$ exists. Show that $f'(0)$ exists. I ...
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1answer
42 views

Show that between any two real roots of the equation $e^x \cos{x}+1=0$, there is a root of the equation $e^x \sin{x}+1=0$

My steps are the following: Suppose $f(a)=f(b)=0$ Since $f$ is differentiable on $[a,b]$ and continuous on $(a,b)$ By Rolle's Theorem $\exists c\in(a,b)$ such that $$f'(c)=e^c(-\sin{c})+e^c ...
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0answers
35 views

Inflection point and 2nd derivative

Is it possible a function $f:\mathbb{R} \rightarrow \mathbb{R}$ to have an inflection point somewhere but that it is not two times differentiable at that point? If so, then can we have a form of that ...
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2answers
42 views

Find that the limit is $0$

I have to prove that the following limit is $0$: \begin{equation} \lim_{(x,y)\to (0,0)}\frac{\lvert x\rvert^2y^2}{x^2+y^4}=0. \end{equation} This is a part of an exercise where I have to study the ...
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2answers
49 views

How to prove that $f$ is differentiable at every point beside $x=-1$? [duplicate]

Consider $f(x) = |x+1|$. I want to show that for every $x_0\neq-1$ , $\lim_{h \rightarrow 0}\frac{f(x_0+h)-f(x_0)}{h}$ is exist. So far i just wrote the definition, and looked on 2 options:When ...
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0answers
36 views

$f$ is differentiable twice but not in $0$?

Let $f$ be differentiable twice in any punctured neighborhood of 0. Is $f$ necessarily differentiable at 0? How can I show or prove such a thing? Does it mean that the derivative is subtracted by $x$ ...
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1answer
26 views

solve equation with Intermediate value theorem…

set $a_1$,$a_2$,$a_3>0$ and $λ_3>λ_2>λ_1$ on $ℝ$. show that there are exactly two $x$’s for $a_1/(x-λ_1) + a_2/(x-λ_2) + a_3/(x-λ_3) = 0$ I tried use the intermediate value theorem but I ...
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34 views

Differentiable only at $x=0$ and $f'(0)>0$

Is there a function $f$ satisfy (1) Only differentiable at $x=0$ (2) $f'(0)>0$ but $f$ is not increasing state on $x=0$?
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3answers
48 views

Help with integration of $\frac{f'(x)}{[f(x)]^n}$.

How do I integrate an expression of the form $$ \frac{f'(x)}{[f(x)]^n} $$ with respect to $x$? Could I use some kind of recognition method, thus avoiding partial fractions? For example: $$ ...
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2answers
80 views

Find the limit without using Maclaurin series

Find a limit, $$\lim_{x\to 0} \frac{1-\cos{(1-\cos{(1-\cos x)})}}{x^8}$$ without using Maclaurin series. My attempt was to use L'hopital's rule but that's just too much, and chances of not making a ...
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0answers
27 views

The definition of continuously differentiable functions

When we say $f \in C^1$, we mean that f is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So ...
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2answers
105 views

What is the derivative of floor function? [on hold]

What is the derivative of the next equation? $$f(x) = \left\lfloor\frac{c}{x}\right\rfloor\ \ \ \ \ \text{ where }\lfloor\cdot\rfloor\text{ is the floor function.}$$ $c,x$ are positive integers ...
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38 views

Prove the inequality $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ by using derivative

The problem: show that $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ I tried to solve it with the derivative and the inequality $\sin(x) \le x$ for $x>0$ thanks for helpers
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1answer
22 views

Find a the value of a point on the tangent line

Suppose that the line tangent to the graph of $y = h(x)$ at $x = 3$ passes through the points $(-2, 3)$ and $(4, -1)$ with a slope of $-2/3$. Find $h(3)$. Hey guys, here's a question from my ...
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second derivative fail, classify the nature of the critical point

f(x,y,z)= $\frac{(x+y)^2}{2}+z^3$ the critical point I calculated is span{(1,-1,0)} the eigenvalue of the Hessian of point (1,-1,0) are 0,0,2, which means that this point is degenerate and the ...
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1answer
28 views

Draw $-dU(x)/dx$ for $U(x)$

It's been a little while since I've done any problems like this, but I just wanted to make sure I'm on the right track. Updated attempt:
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1answer
19 views

derivative of a function has only 2nd kind discontinuities

How would I be able to show the following claim If $f$ is differentiable with a finite derivative in an interval, then at all points $f'(t)$ is either continuous or has a discontinuity of the second ...
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1answer
44 views

Proof $\sum{ k{ x }^{ -k }=\frac { x }{ { (x-1) }^{ 2 } } }$

As the title says, I want to prove the following: $$\sum {k{x}^{-k}=\frac{x}{{(x-1)}^{2}}}$$ But I think I am doing something wrong. I start from the following: $$\sum{x^k} = \frac{x}{1-x} \implies ...
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2answers
39 views

Please explain the algebra in the last part of derivative of the sigmoid function

http://www.ai.mit.edu/courses/6.892/lecture8-html/sld015.htm how does this: $${1\over 1 + e^{-x}} \cdot {-e^{-x}\over 1 + e^{-x}}$$ become this: $${1\over 1 + e^{-x}} \cdot \left (1 - {1\over 1 + ...
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1answer
29 views

Matrix Differentiation using Kronecker operator issue

Let X an $n\times n$ variable matrix and given vectors and matrices $p_1$ ($1\times n$), $p_2$ ($n\times 1$), $\Omega$ ($n\times n$). What is the derivative of the function $f(X)=p_{1}X^{-1}\Omega ...
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2answers
64 views

Is there a better way of writing differentiation and integration?

Differentiation is commonly written simply with a prime mark and an equation, as $(x^2)' = 2x$. Although I find this confusing and think it'd better be written $D(x\mapsto x^2) = x\mapsto 2x$, as ...
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1answer
28 views

Derivates of periodic parametric cubic splines

My Problem is sort of solved, I overlooked, that paameters $B$ to $D$ are dependent on $x$ and $y$ one question remains, see bottom of question. I implemented a periodic parametric cubic spline, and ...
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1answer
16 views

Problems with vector vector derivative in optimization

I have a loss function of the followoing form: $L(\mathbf{a}) = \|\mathbf{b} - \mathbf{a}\|_2^2$ Where, $\mathbf{a}$ and $\mathbf{b}$ are vectors of dimension $d\times 1$. I need to calculate ...
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Derivative gradient power metric

I use the the following definition of gradient power metric of an image $I$ $M(I)=\sum_{i,j} \left|\frac{||I|*[-1, 1]|}{\sum_{i,j} ||I|*[-1, 1]|} \right|$ (I take $|I|$ bacause $I$ may have complex ...
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1answer
25 views

Smooth saturation function

I need a function similar to $$Saturation(x)=min(max(x, -1), 1)$$ except for I need it to be smooth with no jump in its derivatives. It seems $arctan$ is not a good candidate since I need it to keep ...
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question 3.29 from Folland Real Anyalysis

If $F$ in $NBV$ is real-valued, then show $u_F ^+=u_P$ and $u_F ^-=u_N$ where $P$ and $N$ are the positive and negative variations of F. (Use Exercise 3.28) Source: Folland Real analysis exercise ...
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1answer
13 views

Derivation of $f(x)=\prod_{m=0}^{n}(x-x_m)^{m+1}\tan(x), x_m=m\pi, M>0$

I have the following function: $$f(x)=\prod_{m=0}^{n}(x-x_m)^{m+1}\tan(x), x_m=m\pi, M>0$$ I would like to calulate the numeric root of: $n\pi, n\ge0.$ In order to do that, I want to use ...
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1answer
33 views

question 3.40 from Folland Real Anyalysis

Let $F$ denote the Cantor function on $[0, 1]$ (see $§1.5$), and set $F(x)= 0$ for $x<0$ and $F(x)=1$ for $x>1$. Let ${[a_n, b_n]}$ be an enumeration of the closed subintervals of $[0,1]$ with ...
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25 views

Calculate Derivative while Runge Kutta

I am thinking about writing a C++ code to solve an ODE using Runge Kutta method. As you know, RK method calculates the state space vector $X'$ in a few mid-points and uses these mid-points for ...
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1answer
16 views

Show that $\left|\left|\frac{d\hat{v}}{dt}\right|\right| = \frac{a\cdot\sin(\theta)}{s}$

As the title states, I'm trying to show that $$\left|\left|\frac{d\hat{v}}{dt}\right|\right| = \frac{a\cdot\sin(\theta)}{s}, $$ where $\hat{v}$ is the unit velocity vector of a particle $a = ...
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1answer
30 views

Meaning of partial derivatives of a vibrating string

Problem: Let $y(x, t)$ denote the vertical displacement of a vibrating string at a point $x$ on the string at time $t$. Suppose the string is stretched out along the $x$-axis, and the vibrations are ...
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20 views

Inverse function theorem and Implicit function theorem.

I have been trying to prove that implicit function theorem implies the inverse function theorem. Be $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\det[DF(x_0)]\neq 0$ for $x_0 \in ...
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0answers
20 views

Differentiable function only at $x=n$ where$ $n is an integer

Suppose $f:\mathbb R \to \mathbb R$ is only differentiable at integer points. Is this possible? If does, what kind of function is $f$?
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1answer
45 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
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1answer
50 views

What Did I Do Wrong When Solving For This 2nd Order Differential Equation? (answered myself)

$$ \frac{y''}{y'}+y' = f(x) $$ I set the following to be true: $$ y = \sum_{n=0}^{\infty} a_n x^n $$ $$ f(x) = \sum_{n=0}^{\infty} b_nx^n $$ Therefore: $$ y'' = y'(f(x)-y') $$ $$ ...