7
votes
1answer
516 views

Where is the error in my proof that all derivatives are continuous?

I know that this can not be true due to counter-examples but I don't know where the error in my reasoning is. Assumption: If $f(x)$ is differentiable in $\mathbb{R}$ then the derivative $f'(x)$ is ...
0
votes
0answers
30 views

Differentiability of polynomials

Trivial question but I am confused with the notation If $p_{n-1}$ is a polynomial of degree $n-1$, is it $\in$ the differentiability class C^n$? Obviously if $p_n$ is a polynomial of degree $n$, ...
-1
votes
1answer
25 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...
1
vote
2answers
38 views

$\dfrac{\partial}{\partial x}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$

I'm trying to prove the following, interesting, relation: $\dfrac{d}{dx}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$ I tried to integrate by parts the RHS, but i don't ...
4
votes
1answer
79 views

Derivative in 0

I'm a highschool student and we don't learn maths in English. So please excuse me for my Math's English. I'm doing an exercise and I can't answer its final question. Can you help me? Thank you! Let ...
0
votes
2answers
40 views

Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
1
vote
1answer
29 views

Aftermath of Cauchy's mean value theorem

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
1
vote
0answers
32 views

Uniform convergence result in proof of second-derivative formula

This is a fairly basic analysis question. Consider a continuous function $f: \mathbb{R} \to \mathbb{R}$ which is twice differentiable at a point $x$. If necessary, also assume that $f \in ...
0
votes
1answer
14 views

Question about Peano form of the remainder

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
0
votes
3answers
81 views

A strange (!) behaviour of differentiability

I see by drawing diagram that $y=\max (0,\sin x)$ is not differentiable at some points. But $y=(\max (0,\sin x)) ^ 2$ is . How to explain/prove it ? Am I missing something easy ? If $f$ is not ...
1
vote
1answer
33 views

Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
0
votes
2answers
42 views

Derivative of y = $\sqrt{16x^2+5x+15}$

You are building a new house on a cartesian plane whose units are measured in miles. Your house is to be located at the point $(2,0)$. Unfortunately, the existing gas line follows the curve $y= ...
2
votes
1answer
21 views

Differentiability of product/composition of function

How will be the product and composition of two functions, where one is differentiable and another is just continuous, behave?I mean to say, if the product or composition is differentiable, then what ...
3
votes
0answers
53 views

Proving $f'(x)<0$ using sequential criterion of limit.

I'm trying to prove the following: Let $f:\mathbb R\rightarrow\mathbb R$ be a function twice differentiable such that $\forall x\in \mathbb R , f(x)>0$ $\forall x\in \mathbb R , f''(x)>0$ ...
4
votes
2answers
97 views

Failure of differential notation

Through the informal use of differentials, the product rule can be "proved" by writing $$d(fg) = (f + df)(g + dg) - fg = df\,g + f\,dg + df\,dg.$$ Neglecting the product of two differentials, we ...
1
vote
0answers
37 views

Is this an immediate consequence of the Straddle Lemma?

As main book, I'm using Bartle and Sherberts "Introduction to Real Analysis". In exercises of section 6.1 it's asked to prove the Straddle Lemma: Let $f:I\rightarrow\mathbb R$ be differentiable ...
1
vote
2answers
58 views

Differentiability of the Cantor Function

I know that the Cantor function is differentiable a.e. but I want to prove it without using the theorem about monotonic functions. I have already proved that $f'(x) = 0$ for all $x \in [0,1] ...
2
votes
1answer
67 views

Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
0
votes
1answer
26 views

relations between differential, partial derivative, directional derivative

I am a bit lost. Could you explain me relations between differential, partial derivative, directional derivative? I mean that I need some theorem and proofs that for example if differential exists ...
1
vote
1answer
51 views

Values of $6 + \int_a^x \frac{f(t)}{t^2} dt = 2 \sqrt{x}$

Let $f$ and $a$ such that $6 + \int_a^x \frac{f(t)}{t^2} dt = 2 \sqrt{x}$. I need to find the values of $f$ and $a$ that satisfies this condition. For this i tried: $F(x) = 6 + \int_a^x ...
2
votes
2answers
28 views

mean value property of derivatives in high dimensions

Let $E$ be a path-connected subset of $\mathbb{R}^n$ and $f$ a differentiable function on $E$. Prove or disprove: for any $x,y\in E$, there exists $z\in E$ such that $f(x)-f(y)=\nabla f(z)\cdot ...
0
votes
1answer
32 views

Newton derivative of the distance function in $\Bbb R^2$

If we consider the distance function $d$, where $d(x)=dist(x,\partial\Omega)=\inf_{y\in\Omega}\|x-y\|_2$, how would one calculate the derivative in some direction $v$, i.e. ...
1
vote
2answers
47 views

Extreme value problem, maximize ratio of volume to surface area

For a cylindrical can, how to choose the ratio of the height to the radius such that the ratio of the volume to the surface area gets maximized? The volume ist $V = \pi r^2 h$ and the surface ...
0
votes
1answer
48 views

$F(x) = \int_a^b \frac{x^2}{1+2\sin^3(t) + \sin^6(t) } dt$

Let $F(x) = \int_a^b \frac{x^2}{1+2\sin^3(t) + \sin^6(t) } dt$ i have to calculate the derivative of $F(x)$ with respect to $x$. Let $g(x) = \frac{x^2}{1+2\sin^3(x) + \sin^6(x)}$ then $g$ is ...
1
vote
0answers
92 views

Will antiderivative always be differentiable?

Suppose f(x) is continuous on [0,1]. Obviously, such a function will be integrable. Will antiderivative be always differentiable on (0,1)? The answer is "Yes" by the Fundamental Theorem of Calculus. ...
1
vote
1answer
34 views

$f$ a differentiable fucntion in $[a,b]$ with $f´(a) < C < f´(b)$

Let $f$ a differentiable fucntion in $[a,b]$, suppose the existence of a point $C$ with $f´(a) < C < f´(b)$ how can i deduce that given the function $g(x) = f(x) - C(x-a)$ then exist a pint ...
0
votes
2answers
45 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
1
vote
1answer
109 views

showing $\int _a^b\left(f'\left(x\right)\right)dx\:=\:f\left(b\right)-f\left(a\right)$

Let $f(x):[a,b]\to \mathbb R$, be differentiable on $[a,b]$ (and continuous) so that $f'(x)$ is integrable on $[a,b]$. I need to show that: $$\int _a^b\left(f'\left(x\right)\right)\mathrm dx = ...
1
vote
1answer
35 views

$F(x) = \int_0^x \sin{((x+t)^s)} dt$

Let $F(x) = \int_0^x \sin{((x+t)^s)} dt$ , how can i find the derivative with respect to $x$. First i tried to use the fundamental theorem of calculus that asserts that $$\text{if } F(x) = \int_a^x ...
1
vote
2answers
68 views

Local minimum implies local convexity?

Consider a real function $f$, and suppose it has a local minimum at $a\in \mathbb R$. It typically looks like What hypotheses can be added to $f$ so that there is some $\epsilon >0$ such ...
2
votes
1answer
33 views

if $f$ is differentiable at $x_0$ then the limit exists

Let $f$ differentiable at $x_0$. Show that the following limit exists $$ \lim_{h\rightarrow0} \frac{f(x_0+h)-f(x_0-h)}{h}$$ If $f$ is differetiable at $x_0$ then it's one-sided derivative exists ...
0
votes
1answer
54 views

$F(x) = \int_0^{x} t^2 e^{t^2}dt$

Let $y_0 = f''(2) + f'(1) + f(0)$ if $f$ is a real function defined by $f(x) = \int_0^{x} t^2 e^{t^2}dt$. How can I calculate the value of the expression $y_0$. I tried use the fundamental theorem ...
1
vote
1answer
81 views

Prove there's $x_0$ such that $f'(x_0)=0$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable at $\mathbb{R}$ and: $$\lim_{x\rightarrow \infty}\left( f(x)-f(-x) \right) = 0$$ Show there's $x_0$ such that $f'(x_0) = 0$. I tried to use ...
2
votes
3answers
46 views

$ F(x) = \int_0^2 \sin(x+l)^2\ dl$

Consider the function : $ F(x) = \int_0^2 \sin(x+l)^2\ dl$, calculate $ \frac{dF(x)}{dx}|_{x=0}$ the derivative of $F(x)$ with respect to $x$ in zero. Let $g(x) = \sin (x)$ and $h(x) = (x+l)^2$ then ...
6
votes
2answers
164 views

simple way to show $|| \partial_x \int_{B(x,\epsilon)} \frac{x-y}{|x-y|^3} f(y) dy||_{\infty} = O(||f||_{\infty})$ in $\mathbb{R}^3$

We are set in $\mathbb{R}^3$. Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a $C^1_0$ function, i.e. continuously differentiable with compact support. Let $\epsilon > 0$ be small. I need to show ...
0
votes
1answer
23 views

Real Analysis Question about Taylor's Thm & High order derivatives

The question is: Let $I$ be a subset of $\mathbb{R}$ be an open interval and let $f\colon I\rightarrow\mathbb{R}$ be differentiable on $I$, and suppose $f''(a)$ exists at $a\in I$. Show that ...
1
vote
2answers
24 views

Real Analysis. Derivative Question Help

The question is: Let $I$ be an interval. Prove that if $f$ is differentiable on $I$ and if the derivative $f'$ is bounded on $I$, then $f$ satisfies a Lipschitz condition on $I$. My attempt: As ...
1
vote
1answer
18 views

Real Analysis Use of Darboux's Theorem Help

The question I'm trying to answer is: Let $I$ be an interval and let $f:I\rightarrow\mathbb{R}$ be differentiable on $I$. Show that if the derivative $f'$ is never 0 on $I$, then either $f'(x)>0$ ...
2
votes
1answer
52 views

Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ z z' $?

This is a follow-up to Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ \dfrac{z'}{z} $?. It turns out that in that case, \begin{align} \text{$ y = ...
1
vote
1answer
21 views

Real Analysis Derivative Question Help

The question is: Let $g:\mathbb{R}\rightarrow\mathbb{R}$ be defined by $g(x)=x^2\sin(1/x^2)$ for $x\neq0$ and $g(0)=0$. Show that $g$ is differentiable $\forall$ $x\in \mathbb{R}$. Also show that the ...
5
votes
1answer
85 views

Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ \dfrac{z'}{z} $?

Can an arbitrary function $ y: \mathbb{R} \to \mathbb{R} $ always be expressed as $ \dfrac{z'}{z} $ for some differentiable function $ z: \mathbb{R} \to \mathbb{R} $, or are additional conditions on $ ...
0
votes
1answer
28 views

Verify if the function $f(x,y)$ is differentiable in $(x,y) = (0,0)$ and $(x,y) \ne (0,0)$

Verify if the function $f(x,y)$ is differentiable in $(x,y) = (0,0)$ and $(x,y) \ne (0,0)$ $$f(x,y) = \begin{cases} 2xy (\frac{x^2 - y^2}{x^+ y^2}) & x^2 + y^2 \ne 0 \\[4pt] 0 & other ...
0
votes
1answer
46 views

Pointwise derivative

Please I am struggling to find a proper definition of a pointwise derivative and also what is the difference between a pointwise derivative and the "classical" derivative. Can someone please point me ...
2
votes
1answer
49 views

Modifications of Weierstrass's continuous, nowhere differentiable functions

Recalling how nowhere continuous functions such as the Dirichlet function can sometimes be modified on a $\lambda$-null set of points (in this instance, a countable set) to become everywhere ...
0
votes
1answer
44 views

Directional derivatives exist for function neither continuous nor differentiable at the point they exist

It's an old past paper with no mark scheme, this always makes me a slightly afraid of exploring unaided. The function is: $f(x,y)=\frac{xy^2}{x^2+y^6}$ if $(x,y)\ne(0,0)$ else $0$. I am to show all ...
2
votes
1answer
51 views

Question about Differentials

I am reading the book "Advanced Calculus" written by Kaplan, and here is what I have: Suppose that $y(x)$ is a differentiable function at $x = x_0$. Then, we can write $y(x_0+\Delta x) = y(x_0) + ...
2
votes
2answers
50 views

Trying to solve a Taylor series problem

I have a Taylor series problem, well more precisely a Maclaurin series. I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$ Okay here goes: $$f'(x) = 3xe^{x^3} + 6x e^{{2x}^3}$$ ...
0
votes
2answers
40 views

Proving that the function f is constant, mean value theorem, derivatives

Having the following inequality, for a real-valued function $f$ which is twice differentiable: $f(a+h)-f(a)\geq f(a)-f(a-h)$ for any $a \in\mathbf{R}$, $h > 0$. and assuming that $f$ is bounded, ...
0
votes
1answer
24 views

computing directional derivate/ differentiability iff linear map exists

Let $\| \cdot \|$ be a norm on $\mathbb R^2$ and $S= \{ x \in \mathbb R^2 | \| x \| =1 \}$, $f: S \rightarrow \mathbb R$ a function with $f(-x) = -f(x)$ for all $x \in S$. Let $F: \mathbb R^2 ...
1
vote
1answer
90 views

Proof around Rolle's Theorem

Let $f(x)=\exp(\sin(2\pi x))$. I'm trying to prove that there is $a\in[0,1]$ such that, for $(n_r)$ a sequence of integers tending to $\infty$ (by this I mean $n_r$ tends to $\infty$ as $r$ tends to ...