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Questions tagged [derivatives]

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

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452 views

How to prove this lemma related to Rolle's theorem

For any function $f$ denote by $Z(f)$ and $Z_o(f)$ the cardinalities of $f^{-1}(0)\cap[0,1]$ and $f^{-1}(0)\cap(0,1)$, respectively. Let $H=\{f\in C^\infty(\mathbb{R}): \text{supp}(f) = [0,1]\}$ From ...
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238 views

Number of zeros of the second derivative of composition of sigmoid-like functions

Consider the bounded, positive and monotonic functions $f(x) = \frac{x^k}{a^k+x^k}$ and $g(x) = \frac{b^h}{b^h+x^h}$ with $a,b>0,$ and $k,h > 1$ and defined for $x\in \mathbb{R}^+$. Using ...
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82 views

How to prove using elementary methods that this function is everywhere continuous but nowhere differentiable?

Let $f$ be the function defined on all of $\mathbb{R}$ by the formula $$ f(x) \colon= \sum_{n=0}^\infty \frac{1}{2^n} \cos \left( 3^n x \right). $$ How to show (rigorously but through elementary ...
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728 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ can ...
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163 views

Is there any continuous function that is only differentiable on $\mathbb{Q}$?

I am looking for a continuous function $f: \mathbb R \rightarrow \mathbb R$ so that $f$ is differentiable in $x$, if and only if $x \in \mathbb Q$. I already know there is no function that is ...
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120 views

Are there any other fields other than $\mathbb{R},\mathbb{C}$, rich enough to have analysis built on them?

I've been thinking about this, I don't know how to look up anything similar, so here I am asking a question. Specifically, is there any space $X$ with the following properties: Algebraic structure: ...
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237 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
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133 views

Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
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686 views

Differentiation under the Integral Sign

Let $X$ be an open subset of $\mathbb{R}$, and $Y$ be a measure space. Suppose that a function $f:X\times Y\rightarrow \mathbb{R}$ satisfies the following conditions: 1.$f(x,y)$ is a measurable ...
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80 views

If for each $x \in \mathbb R$ there exists $n$ such that $f^{(m)}(x)=0$ for all $m \ge n$ then prove that $f$ must be a polynomial

Let $f:\mathbb R \to \mathbb R$ be an infinitely differentiable function such that for every $x \in \mathbb R$ there exists $n$ such that $f^{(m)}(x)=0$ for all $m \ge n$. I need to prove that $f$ is ...
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109 views

Approximating a function without using its second derivative

So I am stuck in a rut. I have a function, and I am taking the Taylor Series. $$f(x+\ell)=f(x)+\ell f'(x)+\frac{\ell^2}{2!}f''(x)+\dots$$ Unfortunately, I know in advance that the second derivative ...
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131 views

Derivative of a f(x,y) with respect to g(x,y)

I've seen other posts about finding a derivative with respect to another function, but I didn't understand how it would work when the functions have more than one variable. I would like a general ...
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72 views

Is there any educational value into reading the original work of the authors who discovered certain theorems or concepts?

I am interested in reading original work of some authors of theorems or concepts in mathematics because I believe that there is also an educational value to this and it might help me understand better ...
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87 views

Hyper smooth and ultra smooth functions

So I was mixing smooth functions with fractional calculus when I came upon the following idea. We could have functions in $D^\alpha$ defined as follows: $f(x)\in D^1\iff\frac d{dx}f(x)=f'(x)$ exists....
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146 views

How to prove or falsify this inequality?

In STEP 2014 Paper II Question 2, an inequality is assumed for candidates to attempting the question about the approximation of $\pi $ $$\int_{0}^{\pi } (f(x))^2 dx \le \int_{0}^{\pi } (f'(x))^2 dx $$...
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39 views

limit of a region of integration in $\mathbb{R}^2$ approaches a line

I am trying to follow the derivation of derivatives in a paper published in some japanese journal but there seems to be a mistake in the proof. I will present the problem in 2D and in 2 variables so ...
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360 views

Prove that the Weierstrass-type function is nowhere differentiable

Given $0<\alpha\leq1$. Show that the function $$f(x)=\sum_{j=1}^\infty 2^{-j\alpha}\sin(2^jx)$$ is nowhere differentiable. I have solved the case $x=0$. Taking $t_l=2^{-l-1}\pi$, then $f(t_l)-f(0)=...
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615 views

“Angle-preserving” equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be angle-...
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Is there any geometric explanation of relationship between Integral and derivative?

It is said integral is anti-derivative, derivative is tangent of graph function in each point on the function and integral is the area of the region in the xy-plane bounded by the graph. I can not ...
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64 views

Generalisation of kth derivative to real values of k

The answer to this question is most likely no, but I'm asking anyway: Assume that $f\in C^n(\mathbb {R,R})$. Is their any natural generalisation of the map $$\{1,2,\ldots,n\}\to C(\mathbb{R, R})\\k\...
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787 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 ...
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1k 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 (...
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166 views

Are most physics books wrong about the covariant derivative and connection?

I have always read in many physics books that a valid way of intuitively introducing the covariant derivative and the connection was the following: (example in GR but same thing for gauge theories) ...
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83 views

(How) can one visualise the derivative of the function $A \mapsto A^{-1}$, where $A$ is a matrix?

In class, we have shown: Let $V$ be a finite-dimensional Banach-space, then the general linear group $$\mbox{GL}(V) := \{ A \in L(V, V): A \text{ is invertible } \}$$ is open in $L(V,V)$ and the ...
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186 views

Proving that $q(A,S)>0$ if $A>0$, $S<0$ and $p_i(A,S)>0$

Consider the polynomials $$p_1=3(56-56A+A^2)-112(-6+A)S-14(-36+A)S^2+112S^3+7S^4,$$ $$p_2=-112(-6+A)-28(-36+A)S+336S^2+28S^3,$$ $$p_3=-28(-36+A)+672S+84S^2,$$ $$p_4=672+168S,$$ $$p_5=168,$$ $$q=-168S-...
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143 views

Linearity of pushforward of tangent vectors and differentiability

Let $f:U\to V$ be a continuous map of open subsets of Euclidean spaces $U\subset\mathbb R^m,V\subset\mathbb R^n$. Suppose: For every differentiable curve $\gamma$ in $U$ based at $p\in U$ the ...
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290 views

Geometric Interpretation of the “Half Derivative”

I've been doing a little bit of research into fractional calculus involving fractional derivatives, and I was wondering what the geometric interpretations of such derivatives would be. As we know, ...
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149 views

Proof verification - A polynomial $P(x)$ has only real roots $\implies$ $P'(x)$ also has only real roots

Here's a problem that I've solved but I'm not very confident on my solution. Please check it there's any gap in my arguments. Also, is there a way to come up with a shorter proof ? Thank you. The ...
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119 views

Prob. 8, Chap. 5 in Baby Rudin: If $f^\prime$ is continuous on $[a, b]$, then $f$ is uniformly differentiable on $[a,b]$

Here is Prob. 8, Chap. 5 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $f^\prime$ is continuous on $[a, b]$ and $\varepsilon > 0$. Prove that there ...
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Are these relations possible to prove without defining this new kind of derivative?

I'm using these notations: 1.$log^n_xy$: For log with the base $x$ applied $n$ times to $y$. For example, $log^3y=log(log(log(y))$ all with the same base. 2.$^{n[x]}a$: For the power tower or ...
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53 views

Construction of $f$ so that $f^{(n)}(0)=(n!)^2$: does this work?

A question (Are derivatives actually bounded?) has been asked on Stackexchange as to whether there exists $f\in C^{\infty}$ such that $f^{(n)}(0)=(n!)^2$. Obviously $f$ is not analytic but the ...
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511 views

Derivative of remainder function wrt denominator

Given $f(x,y) = x \mathbin{\%} y = x - y \lfloor \frac{x}{y} \rfloor$, I want to find the partials ${{\partial f}\over{\partial y}}$ and ${{\partial f}\over{\partial x}}$ I understand there will be ...
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352 views

Proof verification: $\frac{d(\nu_1\times \nu_2)}{d(\nu_1 \times \nu_2)}(x_1,x_2)=\frac{d\nu_1}{d\mu_1)}(x_1)\frac{d\nu_2}{d\mu_2}(x_2).$

This is exercise 3.12 from Folland's Real Analysis. It took me a long times to come up with a solution to this problem, and I'd appreciate it if anyone could verify if my answer is correct. For $j=1,...
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160 views

Exterior Differential (and its Equivalent Differential Operator) of an Integral 0-Form

I am reading Witten's 1982 paper "Supersymmetry and Morse Theory," and while I am slowly learning the material as I read through the paper, I have come across an equivalence that, while it should be ...
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202 views

Possible correction to Exercise $5.15$ in Rudin's Principles

Here's Exercise $5.15$ in Rudin's Principles of Mathematical Analysis (Page $115$): Suppose $a \in \mathbb{R}^1$, $f$ is a twice-differentiable real function on $(a,\infty)$, and $M_0$, $M_1$, $...
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120 views

partial derivative of a facet normal wrt to one of its vertex

I am struggling to understand the derivation of an equation in a paper (A Bayesian Method for Probable Surface Reconstruction and Decimation, specifically Eqn. 16). Basically they define three ...
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116 views

Is there a differentiable function on a closed subset of $\mathbb{R}^n$ that cannot be continued differentiably on an open superset?

Let $A \subseteq \mathbb{R}^n$ be closed with no isolated points and $f:A \to \mathbb{R}^m$. Suppose that for every point $x_0 \in A$ we have (at least one) matrix $L_{x_0}$ such that $$ \lim_{x,y \to ...
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369 views

About the differential notation in measure theory

Is there any good reason for which integrating according to a measure includes a $\mathrm d$ as in $\int f\mathrm d\mu$ ? Or is it just a manner to keep formal consistency with the traditional ...
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404 views

chain rule for derivations

Off we go. So let $b:X\rightarrow Y$ be a function from $X$ to $Y$ endowed with as much structure as it needs to make sense of the question :) and $a:Y\rightarrow \mathbb R$ a function into the reals. ...
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320 views

Taylor Expansions in Spherical Coordinates (Generator of Rotations)

We can expand a smooth function $f:\mathbb{R}^3\to \mathbb{R}$ in a Taylor series: $$f((x^1,x^2,x^3)+(h^1,h^2,h^3))=f(x^1,x^2,x^3)+h_i\frac{\partial f}{\partial x^i}+h_ih_j\frac{\partial^2 f}{\partial ...
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130 views

Leibniz integral rule implementation

Can someone please explain to me why the following expression is true? I really tried to figure out how Leibniz integral rule works, but everytime I think I managed to figure out how to implement it, ...
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729 views

Uniform Differentiability

Let $f:\mathbb{R}^n \to \mathbb{R}$ be differentiable and such that $\nabla f$ is uniformly continuous. Show that $f$ is uniformly differentiable; that is, for any $\epsilon >0$, there is a $\delta ...
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344 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 $v_i$ What ...
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235 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 ...
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26 views

Inflection point at Vertical tangent?

In Wikipedia and many other places, it is stated that $f''(a)=0$ is a necessary condition for a function to have an inflection point at $x=a$. I was wondering, if a function had a vertical tangent, ...
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55 views

Differentiability and integrability of a function composed with itself

I am reviewing for an exam and came across a multi-part question that I am having a hard time with. We are asked to prove or disprove the following statements. If a statement is false, what additional ...
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41 views

Why is that $\int_a^b \frac{\partial f}{\partial x}(x,t)dt = \frac{\partial}{\partial x}\int_a^b f(x,t)dt$

This question is concerned with the integral with parameter, so let's assume that every function below is smooth. To find the formula for the derivative of an integral with parameter, say $$g(x) =...
4
votes
0answers
65 views

$\sum_{k=0}^n a_kx^k$ splits $\Rightarrow \sum_{k=0}^n \frac {a_k} {k!}x^k$ splits over reals

Suppose that $a_0, a_1, \ldots a_n \in \mathbb R$ and the polynomial $P(x) = \sum_{k=0}^n a_kx^k$ has all real roots. I'm supposed to show that $$ Q(x) = \sum_{k=0}^n \frac {a_k} {k!}x^k $$ also has ...
4
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0answers
39 views

Properties of a continuous function that satisfies $f(tx)=t^2 f(x)$

Let $f:\mathbb{R}^{n} \to \mathbb{R}$ be a continuous function such that $f(x) > 0$ for each $x \neq 0$ e, moreover, $$f(tx)=t^2 f(x)$$ for any $x \in \mathbb{R}^{n}$ and $t \in \mathbb{R}$. ...
4
votes
0answers
191 views

A rigorous yet intuitive summary of inflection and critical points for beginning calculus?

I haven't done these in awhile. While my analysis covered continuity but not differentiability, I have so far not revisited these in learning geometry or algebra. I am trying to help a calculus ...