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

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-1
votes
1answer
43 views

Proving a function $f$ is not differentiable at an unkown point $a$

Let's say I have an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and I want to prove that it is not differentiable at some unknown point $a$. Emphasis must be placed on the unknown part as that ...
0
votes
1answer
42 views

Elastic curves - What is wrong about my solution?

Given a curve $\gamma:\mathbb R\to\mathbb R^2$ with $\Vert\gamma'\Vert=1$ and curvature $\kappa(s)=\frac{c}{\cosh s}$, $c\in\mathbb R$, how can I show that $\gamma$ is an elastic curve for some ...
0
votes
1answer
47 views

Alternate definition of differentability at a point

Usually in most introductory Calculus courses, a definition of differentiability at a point $a$ is defined, as follows : A function $f$ is differentiable at $a$ if $f'(a)$ exists As a corollary ...
0
votes
0answers
20 views

Determine differentiability in region

Is there any test to indirectly determine if a function f is differentiable everywhere in a defined domain, for example $S=\{x|x\in\mathbb{R}\land x^2<4\}$? For example: let $f(x)=\frac{2}{x}$. We ...
2
votes
1answer
54 views

Where did I go wrong in finding maximum?

The question in my book is given as: If $x^2+y^2+z^2=1$ for $x,y,z$ belongs to all real numbers ($x,y,z$ are independent), then find the maximum of $x^3+y^3+z^3-3xyz$. What I tried: As all ...
0
votes
1answer
15 views

Left and right derivative of composition

Let $f(x)=y$ and $g(y)$ be two functions such tat $f'_+(x)$(right sided derivative exists) and $g'_+(x)$ (right sided derivative exists) let $h(x)$ be a function that is defined as $h(x)=g(f(x))$ is ...
3
votes
2answers
216 views
+50

Is there any solution to find a condition for $f(x)=a+bx^n+cx^2-dx>0$ to always hold true?

Okay, I am interested to know the criteria for a function to always hold $$f(x)=a+bx^n+cx^2-dx>0,$$ if it is given that $a, b, c>0$ and $n\in(-2,2)$ is some real number and $x>0$. My idea ...
2
votes
1answer
22 views

Sum of differentiable functions.

True/False Question : suppose that $f+g$ is differentiable at point $x_0$ therefore $f$ and $g$ are differentiable at $x_0$ I think this statement is false and I got a counter example : ...
1
vote
1answer
33 views

Local Immersion Theorem in $\mathbb{R}^n$ proof

I am trying to prove the following: Let $U \subset \mathbb{R}^n$ be open and $f \in C^1(U;\mathbb{R}^m)$. Let $x^\star \in U, \ y^\star = f(x^\star)$. Suppose that $\mathrm{d}f(x^\star)$ ...
-1
votes
1answer
49 views

Derivative Optimization Problem [duplicate]

I need help with finding the area of the largest rectangle in an ellipse from $y^2 + (x^2)/4 = 1$. I got it to y = $\sqrt{ 1 - (x^2)/4}$ but then I don't really know what to do, please help.
1
vote
2answers
23 views

Meaning behind directional derivative

My task was to find the directional derivative of function: $$z = y^2 - \sin(xy)$$ at the point $(0, -1)$ in direction of vector $u = (-1, 10) $. The result I found was $-21/\sqrt{101}$. But I ...
2
votes
1answer
51 views

Representation of the Fréchet derivative of $〈f,e_n〉$, where $f:H→H$, $H$ is a Hilbert space and $(e_n)_{n∈ℕ}$ is an orthonormal basis of $H$

Let $H$ be a $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $f:H\to H$ be Fréchet differentiable and $$f_n:=\langle f,e_n\rangle\;\;\;\text{for }n\in\mathbb N$$ ...
2
votes
1answer
32 views

Derivative formalism question

After seeing a lot of integrals going like $$\int f(x,y) \,dxdy = \int f(x,y)\,dA$$ I am wondering wether it is allowed to write something like this: $$\frac{d f(x,y)}{dA} = \frac{\partial^2 ...
0
votes
2answers
86 views

What is wrong with the argument that $\frac d{dx} \int_0^1 f(x)dx$ should always be $0$ for any $f(x)$?

What is wrong with the argument that $\frac d{dx} \int_0^1 f(x)dx$ should always be $0$ for any $f(x)$? My book used differentiation under the integral sign to evaluate an integral. The integral was ...
1
vote
0answers
17 views

Generalizing the notion of convexity for differentiable functions

Suppose $f$ is twice differentiable and its second derivative nonnegative in some interval $I$. We can "algebraically" characterize this by saying that $f$ satisfies, for all $t \in [0,1]$ and $x,y ...
3
votes
3answers
91 views

Derivative definition

Hey I have 2 derivative definition which were told in class. First one is $ \underset{x\rightarrow a}{\lim}\frac{f(x)-f(a)}{x-a} = f'(a) $ This on is pretty straight forward for me. The second one ...
2
votes
0answers
74 views

Derivative of this function

Let $f : S^{n-1} \subset \mathbb{R}^n \to \mathbb{R}^n \setminus \{0\}$ be a differentiable mapping, $n \geq 2$, and consider the function $F = \frac{f}{\|f\|} : S^{n-1} \to S^{n-1}$. I calculated the ...
5
votes
1answer
70 views

n-th derivative where $n$ is a real number?

We know that $$\frac{d^n}{dt^n} e^{at}= a^n e^{at}; \, n\in \mathbb N.$$ I want to know if the result is true if $n$ is a real number, i.e., $n\in \mathbb R$ ?
2
votes
1answer
27 views

Two methods of implicit differentiation don't correspond

I recently attempted a question on implicit differentiation twice. I differentiated using one method in the first attempt and then another method in the second attempt but they do not correspond when ...
2
votes
0answers
42 views

Condition for term-by-term differentiation of a non-convergent series

In a problem I have seen, a series $\sum_n u_n(x)$ with $$ f_n(x) = \frac {\log(1+n^4x^2)}{2n^2}$$ here $\sum u_n'(x)$ is not uniformly convergent, BUT If $f '(x) = \lim_{n\to\infty} f_n'(x) $ ...
1
vote
2answers
92 views

A function on [a,b] that is second differentiable and f'(a)=f'(b)=0

Let $f:[a,b]\rightarrow\mathbb{R}$ be secondly differentiable and $f'(a)=f'(b)=0$. Then there exits a point $c\in [a,b]$ such that $$|f''(c)|\geq\frac{4}{(b-a)^2}|f(b)-f(a)|.$$ I tried to prove it by ...
0
votes
2answers
41 views

Is $f''(x)=0$ sufficient for inflection point?

I'm a bit confused about $n$th derivative test.Is $f''(x)=0$ at a point sufficient to prove it is inflection point or not ?Or we need to check further if any higher odd derivative is $0$? And when ...
0
votes
0answers
47 views

Proof of higher order differentiation formula

How can one prove (maybe from first principle) the following formula used for higher order differentiation. $$\dfrac{d^{2}y}{dx^{2}}=\frac{d}{dx} \left( \dfrac{dy}{dx}\right) $$
1
vote
0answers
19 views

Find formula with Richardson Extrapolation based on centered difference formula

I'm preparing for my exams next week, and I'm making exercises as a preparation. Now, I'm asked to derive the following formula using Richardson Extrapolation based on the centered difference formula: ...
2
votes
2answers
67 views

Finite Difference Approximation of Derivative [closed]

I want to build a finite-difference approximation of this derivative: $\frac{\partial^2T }{\partial x^2}$ There are given an error of approximation: $O(\Delta x^{4})$ and nodal values of function:$ ...
-1
votes
1answer
57 views

If $\lim_{x\to 0} \frac{x^ncos(\frac{1}{x})}{tan^mx}$ … [closed]

Problem : If $\lim_{x\to 0} \frac{x^ncos(\frac{1}{x})}{tan^mx}$ is non differentiable at x = 0 then what does it mean : My approach : $f'(0) = \lim_{x\to 0} ...
2
votes
1answer
26 views

Let $f: [0, \infty ) \to \mathbb{R}$ be a continuous and strictly increasing function such that $f^4(x) =\int^x_0 t^2f^3(t)\,dt$ for all $x > 0$

Problem : Let $f: [0, \infty ) \to \mathbb{R}$ be a continuous and strictly increasing function such that $f^4(x) =\int^x_0 t^2f^3(t)\,dt$ for all $x > 0$. Find the area enclosed by $y = f(x)$, ...
4
votes
2answers
44 views

Finding a derivative given certain conditions.

Find $f'(0)$ if $f$ is a function such that $$1+f(x)+x^2(f(x))^3=11 \hspace{1cm}\text{and}\hspace{1cm}f(1)=2.$$ Here's what I've tried so far: If $f$ is differentiable around $0$, then the ...
0
votes
0answers
14 views

Stochastic gradient descent in neural network with logistic activation function

I am trying to derive the update rules for a unit of a neural network. To simplify, let's assume that need to perform a binary classification task on a dataset $\mathbf{X} = \{\mathbf{x}_i\mid ...
1
vote
1answer
31 views

How to find the derivative of matrix conjugation for unitary matrices at a point where the matrices commute?

Let $\text{SU}(2)$ denote the special group of $2 \times 2$ unitary matrices, that is, unitary matrices with determinant $1$. Define $f : \text{SU}(2) \times \text{SU}(2) \to \text{SU}(2) \times ...
3
votes
0answers
46 views

Existence of a zero point for a derivative

Could you please check correctness of my proof. Statement $a,b \in \mathbb R$, $f:[a,b]\to \mathbb R$ is a differentiable function, $f'(a)>0, f'(b)<0$, then $\exists \xi \in (a,b)$ such that ...
0
votes
3answers
39 views

Find Derivative using only chain rule [closed]

How can I find derivative of $\tan\left(\sqrt{x}\right)\cdot x^4$ using only chain rule?
0
votes
0answers
15 views

Verify the Green's function for Helmholtz equations

It is well known that $$ G(x)=\frac{1}{4\pi}\frac{\exp(ik|x|)}{|x|} $$ is the Green's function for Helmholtz equation $$ (\Delta+k^2)f=0 $$ in $\mathbb{R}^3$. My question is, given $v\in ...
0
votes
1answer
18 views

Proper way to find the critical points of a 2 variable function

I want to find the critical points of $g(x,y) = x^3 +y^3+3xy$ Do I need to find the points in which $\dfrac{\delta f}{\delta x} = 0 $ AND $\dfrac{\delta f}{\delta y} = 0$ or do I need to find the ...
0
votes
1answer
26 views

Show that the following function is discontinuous but all directional derivatives exist in (0,0) [closed]

Let $f:\mathbb{R^2} \rightarrow R$ such that $f = \left\{ \begin{array}{ll} 1 \quad \text{if} \quad x^2=y; x \neq 0 \\ 0 \quad \mbox{ otherwise.} \end{array} \right.$ I want ...
1
vote
1answer
24 views

Derivative of a variable times its summation

Say you want to calculate $$ \frac{\partial}{\partial x_i} x_i(a - b \sum_{i=1}^N x_i). $$ I assume the term $bx_i \sum_{i=1}^N x_i$ is derived using the product rule, but I am unsure what the ...
3
votes
2answers
68 views

Prove that the equation $x + \cos(x) + e^{x} = 0$ has *exactly* one root

Question : Prove that the equation $x + \cos(x) + e^{x} = 0$ has exactly one root This is what I thought of doing: $$\text{Let} \ \ \ f(x) = x + \cos(x) + e^{x}$$ By using the Intermediate ...
0
votes
3answers
43 views

Is there a way to combine functions so that you combine their derivatives?

Suppose $y,z$ are functions. What manipulation: "$?$" to the functions would yield the following? (if any) $$y?z=y\cdot z\\~\\ \frac {d(y?z)}{dx}=\frac{dy}{dx}\cdot\frac{dz}{dx}\\~\\ \frac ...
0
votes
2answers
33 views

A function that satisfies the $n$-th derivative where $x=0$ is $\frac{1}{n}$ [closed]

Is there a function that satisfies $f^{(n)}(0)=\frac{1}{n}$ for every positive integer $n$?
0
votes
1answer
23 views

Continuity vs differentiability versus directional derivatives

I'm having trouble with understanding the different concepts of continuity, differentiability and the existence of directional derivatives. I am given a function $f:\mathbb{R}^2\rightarrow\mathbb{R}, ...
1
vote
1answer
78 views

Why is $x^{1/3}$ not differentiable?

The problem says On $\mathbb{R}^1$consider $f(x)=x$ and $g(x)=x^{1/3}$ both $\mathbb{R} \to \mathbb{R}$. Consider atlases $\alpha_1=\{(\mathbb{R},f)\}$ and $\alpha_2=\{(\mathbb{R},g)\}$. Show that ...
1
vote
2answers
49 views
+50

Surjectivity of derivative of a vector valued function

Let $f:\mathbb R^3\to \mathbb R^3$ be a function such that $f(x,y,z)=f(x+y,0,x+z)$ for all $(x,y,z)\in \mathbb R^3$. I want to prove that $f^{'}(x)$ can never be onto for all point $x\in \mathbb R^3$ ...
0
votes
1answer
18 views

Show that piecewise function $f$ is $C^{\infty}$

I don't understand the first line of the solution If $f:\mathbb{R} \to \mathbb{R}$ defined by $f(x)={e^{-1/x}}$ if $x>0$ and $f(x)=0$ if $x \leq 0$ then show it is $C^{\infty}$. Well, it frst ...
2
votes
3answers
49 views

Why transform degrees into radians when computing linear approximation to find $\tan{44^\circ}$?

I am asked to find the linear approximation of $\tan{44^\circ}$. Why should I transform degrees into radians to do that? I understand that using degrees would give me a wrong solution (which would be ...
3
votes
2answers
83 views

Formula for the nth Derivative of a Differential Equation

I have the differential equation $$f'(x)=2xf(x)$$ With the initial condition that $f(0)=1$ I need to prove that the nth derivative evaluated at zero is equivalent to $n!/(n/2)!$ for even n. ...
-5
votes
0answers
50 views

Given $\lim _{x \rightarrow \infty} x f(x)=0 $ and $\lim _{x \rightarrow \infty} xf''(x)=0$ Prove that $\lim_{x \rightarrow \infty}xf'(x)=0$ [closed]

$f$ is twice differentiable and $f''(x)$ is continuous on $(0, \infty)$ , Suppose $\lim _{x \rightarrow \infty} x f(x)=0 $ and $\lim _{x \rightarrow \infty} xf''(x)=0$ Prove that $\lim_{x \rightarrow ...
0
votes
4answers
72 views

Where is $|xy|$ function differentiable

I'm trying to solve this problem: Let $f(x,y) = |xy|$. Find the sets of all points $(x,y) \in \Bbb R$ where $f$ is differentiable and compute the differential in those points. Can someone explain ...
0
votes
0answers
15 views

Differentiable version of Urysohn's lemma

Let $A,B$ be disjoint non-empty closed sets in $\mathbb R$ , then does there exist a differentiable function $f:\mathbb R \to [0,1]$ such that $f(A)=\{0\} , f(B)=\{1\}$ ? If the answer to the previous ...
1
vote
3answers
72 views

How to prove $r(x)>p(x)$?

Given two functions $r(x)$ and $p(x)$, both of which are defined on closed interval $x\in[a,b]$. Functions $r(x)$ and $p(x)$ also satisfy the following constraints: \begin{cases} r(a)=p(a)\\ ...
1
vote
2answers
25 views

limit of derivative and differentiable [duplicate]

Let f is differentiable on R. Following statement is true? If $\lim_{x\to a}f'(x)=\alpha$, then $f'(a)=\alpha$ (where $\alpha$ and $a $ are real numbers)