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

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1answer
23 views

Deriving the limit of a sequence?

Consider a function $\mu:\mathbb{R}\rightarrow\mathbb{R}$ differentiable at $0$. Now, consider the sequence $$ \sqrt{n}(\mu(\frac{h}{\sqrt{n}})-\mu(0)) $$ for $n \in \mathbb{N}$, $h \in \mathbb{R}$. ...
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2answers
74 views
+50

Show a function is Lipschitz

Suppose a real-valued function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by $$f(x) = \begin{cases} \hfill e^{-\frac{1}{\delta^2 - x^2} + \frac{1}{\delta^2}} \hfill & \text{ $|x| ...
4
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1answer
33 views

Does differentiability imply having bounded variation on some subinterval?

Suppose that $f:(a,b)\to\mathbb{R}$ is a differentiable function. Does it follow that $f$ has bounded variation on some subinterval $[c,d]\subset (a,b)$? Details and ideas Being differentiable ...
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1answer
48 views

$f$ convex strictly decreasing function , is $f'(x+\delta)-f'(x)$ convex

Assume you have a strictly decreasing convex differentiable function $f(x)$, $x \in \Bbb R^+$, I am wondering if the increment of the first derivative is also convex; i.e., $$g(x) = f'(x+\delta) - ...
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2answers
30 views

Evaluate $\lim_{x \rightarrow 1^+} (\ln\ x)^{x-1}$

Evaluate: $$\lim_{x \rightarrow 1^+} (\ln\ x)^{x-1}$$
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1answer
36 views

If $(u,v)$ is a point on $4x^2+a^2y^2=4a^2$,where $4<a^2<8$,that is farthest from $(0,-2)$ then $u+v$ is equal to?

If $(u,v)$ is a point on $4x^2+a^2y^2=4a^2$,where $4<a^2<8$,that is farthest from $(0,-2)$ then $u+v$ is equal to? My Approach: I took a parametric point $(t,4-4t^2/a^2)$.And then tried to ...
0
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1answer
18 views

How to use Rolle's theorem to verify the following?

How to use Rolle's theorem to verify the location of roots ? $f(x)=x^3+4/x^2+7$ has exactly one zero in ($-\infty$,$0$) I can do it without Rolle's theorem by finding the stationary point which is ...
0
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0answers
18 views

Application of average and instantaneous rate of change

John's business is currently selling 175 cookie boxes per day, but sales are dropping at a rate of 2 per day. he is currently charging $6 per box but to compensate for the dwindling sales , he is ...
1
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0answers
18 views

Prove $V(x)$ is an increasing function (involving PDF and CDF)

I need to prove the following: $V(x) = x + G(x)/g(x)$ is an increasing function where $G(x)$ is a CDF and $g(x)$ is the corresponding pdf. When I take the derivative, I get $$1 + g(x)^2/g(x)^2 - ...
1
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1answer
32 views

Clarke's generalized gradient formula computed on functions defined on open sets

In the book [1], Clarke et al. define the generalized gradient for a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ as follows. 8.1. Theorem (Generalized Gradient Formula). Let ...
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1answer
11 views

Relation implication

For $n \geq 0$ the following holds: $y^{(n+2)}(0) = -n^2y^{(n)}(0)$ Given the above relation, where superscript denotes the $n$th derivative with respect to $x$ and $(0)$ means function is evaluated ...
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4answers
80 views

hint to find the second derivative

Let $f:\mathbb{R}\to\mathbb{R}$ be a non-constant, three times differentiable function. If $f\left(1+\frac{1}{n}\right)=1$ for all integers n, then $f''(1)=$? by the given condition for n=0 $f(1)=1\\ ...
3
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1answer
33 views

Does continuous extension of a function and its densely defined derivative imply everywhere differentiability?

Let $V \subset \mathbb R^n$ be a closed set, and let $U \subset V$ be open as a subset of $\mathbb R^n$ and dense in $V$. Let $f:V \to \mathbb R$ and $G: V \to \mathbb R^n$ be continuous, with $G = ...
0
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1answer
14 views

Correct simultanous application of chain and product rule

For two continuous differentiable functions $g(x)$ and $h(x)$ we seek $$\frac{d}{dx} [g(x) h^{-1}(x)]$$ where $h^{-1}(x) = \frac{1}{h(x)}$. This asks us to apply product and chain rule in sequence, ...
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2answers
48 views

Where do I have to use Chain Rule of differentiation?

I have come across many examples of chain rule of differentiation while studying physics (eg. finding velocity of SHM,differentiating Kinetic Energy with respect to time etc.).But,I feel I lack the ...
4
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0answers
41 views

Find the derivative of the function $ y= x|\cos{\frac{\pi}{x}}|$

Function is defined as it follows : $x \neq 0$ and $f(0)=0$ My work is: $\frac{d}{dx}(x|\cos{\frac{\pi}{x}}|)$ = $|\cos{\frac{\pi}{x}}|$ + $x(\frac{d}{dx}|\cos{\frac{\pi}{x}}|)$ = ...
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0answers
50 views

Dini derivative of the maximum of two continuous functions

Given two locally Lipschitz continuous functions $f_i:\mathbb{R}^n \to \mathbb{R}$, $i\in \{1,2\}$, let $f:\mathbb{R}^n\to \mathbb{R}$ be given by $$ f(x)=\max\{f_1(x),f_2(x)\}. $$ Is it true that the ...
0
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3answers
40 views

Can you have many integrations for a single function? (sorry if the terminology is wrong)

Forgive me for the amateur question (first time here), but it's bugging me! OK, for example, integrating 1/5x would yield what? ln(5x)/5 +C OR ln(x)/5 +C ? Both of these when differentiated form ...
1
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2answers
48 views

Limits & Derivatives - Derivative of a function

We are learning "derivative of a function" under the chapter of "Limits and Derivatives". The following is the definition we have in our book I have the following example in my book. I'd like ...
2
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2answers
32 views

derivative of arctan(u)

Im trying to find the derivative of $\arctan(x-\sqrt{x^2+1})$ here are my steps if someone could point out where I went wrong. $$\begin{align} \frac{\mathrm d~\arctan(u)}{\mathrm d~x} \;& =\; ...
0
votes
1answer
38 views

Is $\displaystyle\lim_{h \to 0} H_n(f(h), g(h)) = H_n(\displaystyle\lim_{h \to 0} f(h), \displaystyle\lim_{h \to 0} g(h))$ true for all $n$?

Consider the limit $\displaystyle\lim_{h \to 0} H_n(f(h), g(h)), $ where $H_n(a, b)$ denotes the $n$th hyperoperation $H_n(a,b) = a \uparrow^{n-2}b$ with both $f(x)$ and $g(x)$ being continuous and ...
0
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1answer
25 views

Tangent meets curve again

If the tangent at the point $(16,64)$ on the curve $y^2=x^3$ meets the curve again at at $Q(u,v)$ then $uv$ is ? If found the tangent to the curve at $(16,64)$ but then I cannot find $uv$.Give your ...
1
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2answers
51 views

Is there any way to differentiate such function?

Let $S$ be a set. If I had a bijection $f$ mapping each element $n\in \mathbb{N}$ to an element $s \in S$ such that: $$s = f(n) = \sum^{n}_{k=1} {1\over k}$$ Is the function differentiable in ...
3
votes
1answer
238 views
+200

The root of summation function

This is a calculation I need for my statistics project Big edit: simplify the function $f(x)$ a lot. Define for $f(x)$, $x\geq 0$, $$ f(x):=\sum_{k=1}^\infty ...
1
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1answer
29 views

Partial derivative of x - is quotient rule necessary?

Let $$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to determine if the given function is harmonic. I know that the 2nd partial derivative with respect to $x$ should, when added to the 2nd partial ...
4
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3answers
65 views

why $|x|$ in $\frac{d}{dx}\sec^{-1}x=\frac{1}{|x|\sqrt{x^2-1}}$

I derived $$\frac{d}{dx}\sec^{-1}x$$ as follows: Let $$z=\sec^{-1}x$$ Then $$x=\sec z$$ differentiating both sides w.r.t $x$ we get $$1=\sec z \tan z \frac{dz}{dx}$$ so we get ...
1
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1answer
19 views

Riemann Sum-esque limit involving Radicals

Me and my friend were trying to evaluate the following limit: $$ I\equiv\lim_{N\to\infty}\sum_{k=0}^N\sqrt{\frac{1}{N^2}+\left(f\left(\frac{k+1}{N}\right)-f\left(\frac{k}{N}\right)\right)^2} $$ it was ...
4
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1answer
42 views

$\frac {\partial}{\partial t}T$ vs $\frac d{dt} T$.

Suppose we have a function $T_1=F(x,y,t)$. Now suppose that $x=g(t),y=h(t)$, so we have a new $T_2=F(x(t),y(t),t)$, so then we have that $\frac \partial{\partial t} T_2=F_t$ and $\frac d{dt}T_2=F_x ...
-2
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0answers
23 views

Derivation of a matrix function

what is the derivation of the following function with respect to U: $F = {\left( {UX{U^T}} \right)^{ - 1}}UY{U^T},\,\,\,\,\,U \in {\mathbb{R}^{m \times n}},\,X,Y \in {\mathbb{R}^{n \times n}}$ ...
0
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0answers
12 views

A $C^2$ $f$ such that for every $x \in \mathbb R^n$, $t \in \mathbb R$, $f(tx)=t^2f(x)$. [duplicate]

I am trying to do the following exercise: Suppose $f:\mathbb R^m \rightarrow \mathbb R^n$ is $C^2$ and for every $x \in \mathbb R^n$, $t \in \mathbb R$, $f(tx)=t^2f(x)$. Show that there exists a ...
0
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0answers
15 views

Given $f:[a,b] \rightarrow R$, if $x'$ is a local minimum of $f$ and $x'<b$ then there exists a sequence $x_n$ converging to $x'$ with $x'<x_n<b$

I'm trying to understand the demostration of the folowing lemma: Is a function $f:[a,b] \rightarrow R$ differentiable in a local extrema $x'$ then $f'(x')=0$ Demostration: $x'$ is a local ...
0
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1answer
17 views

Find the versors of the Frenet Triad

I have the following curve: $C(t)=(\frac{t^5}{5}, \frac{t^3}{3}, t ^ 2).$ The problem is to find $T$, $N$ and $B$ of the Frenet Frame. I know the fact that $\vec{B} = \vec{T} \times \vec{N}$. I've ...
3
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5answers
77 views

How to evaluate $\lim\limits_{x\to 0+}\frac 1x \left(\frac 1{\tan^{-1}x}-\frac 1x\right)$?

How to evaluate $\lim_{x\to 0+}\dfrac 1x \Big(\dfrac 1{\tan^{-1}x}-\dfrac 1x\Big)$ ? I used L'Hospital's rule but with no success . Please help . Thanks in advance
2
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1answer
37 views

Directional derivative understanding

[Beginning multivariable question.] I have just been introduced to a theorem that says $$D_uf(x)=\nabla f(x)\cdot u.$$ So in the two-dimensional case, $$\nabla f(x,y)= \langle f_x(x,y),f_y(x,y)\rangle ...
0
votes
2answers
39 views

Derivative of function involving absolute value

Could anyone help me with differentiating $|x|^5$ and $\frac{|x|^3}{(1+x^2)^8}$? I used the way we differentiate $|x|$ via substitution, i.e. enter link description here It fails on the two ...
0
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1answer
27 views

A cylindrical reservoir

A reservoir has the shape of a vertical cylinder with height $3m$ and radius $4m$ and it is filled with water. i) Let x be the height in meters measured from the bottom of the reservoir. The weight ...
2
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0answers
35 views

Which of the following functions is not everywhere differentiable?

Today, I took a FAMAT test and had this question: Which of the following is not everywhere differentiable? $A) \sin\left(\frac 1{x^2 + 1}\right)$ $B) \ln(x^2)$ $C) \arctan(x)$ $D) \sqrt{1 + ...
1
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1answer
46 views

Is $(x^2+y^2+x)dx+xydy$ the same as ${dy\over dx}(x^2+y^2+x)+{dx\over dy}(xy)$ ? Is this just a different notation?

Is $(x^2+y^2+x)dx+xydy$ the same as ${dy\over dx}(x^2+y^2+x)+{dx\over dy}(xy)$? Is this just a different notation?
1
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0answers
28 views

derivative of a projection matrix

The projection onto a parametrised vector $v(\lambda)$ is $P_v = \frac{vv^{T}}{v^{T}v}.$ Its complement is $$P = I-\frac{vv^T}{v^{T}v}.$$ I've got an expression containing this complementary ...
1
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1answer
18 views

Taylor Series in Fractional Calculus

I recently studied fractional calculus, namely the possibility to define fractional derivatives of some functions, like $$\frac{\text{d}^{1/2}}{\text{d}x^{1/2}}\ f(x) ~~~~~~~~~~~~~ ...
1
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1answer
20 views

4th order mixed leibniz derivative

How exactly is the order of mixed partials read in Leibniz notation? In Lagrange notation, we just read from left to right. $$f_{xyzz} = (\frac {\partial} {\partial z}(\frac {\partial} {\partial ...
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0answers
52 views

Why the differential equations have a wave behavior?

The differential equation for string: $$\frac{1}{c^2} \frac{\partial^2 f}{\partial t^2}=\frac{\partial^2 f}{\partial x^2} \tag{1}$$ I have inital condition: $$f(x)=\begin{cases}20x, & 0\le ...
1
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4answers
63 views

Is there a way to evaluate the derivative of $x$! without using Gamma function?

Taking the factorial function $x!$ I wonder if there is a method to find the first derivative of this function without making any use of the Gamma function (or related integral representations of the ...
-1
votes
0answers
22 views

Calculus (Differentiation) [duplicate]

Let $f$ be a function that satisfies the condition $f(x+y)=f(x)f(y)$ for all $x$ and $y$. a) Prove that $f(x)$ is not equal to zero. b) Assuming that $f(x)$ exists when $x$ is a real number, use the ...
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4answers
24 views

Feynman lectures, Volume I, chapter 13-4

While reading Feynman lectures on Physics, volume I, Chapter 13-4, I found following assumption, which I don't understand: Then, since $r^2 = \rho^2 + a^2$, $\rho\,d\rho = r\,dr$. Therefore ... ...
0
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1answer
21 views

Problem calculating $(g\prime)^{-1}(y)$.

I have some problem understanding how to calculate the inverse of a function. I have an example below: Calculate the following: $(g\prime)^{-1}(y)$. The $y$-value is: $y(s)=g\prime=2s-1 $. ...
1
vote
1answer
78 views

$f\in C^1(\mathbb R)$ , having finitely many zeroes and $f'$ changes sign at exactly two of these points , solutions of $f(x)=y$ for given $y$?

Let $f:(0,1) \to \mathbb R$ be a continuously differentiable function having finitely many zeroes and $f'$ changes sign at exactly two of these points , then is it true that for any $y \in \mathbb R$ ...
0
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1answer
41 views

higher order partial derivative notation (Leibniz)

Which one of the following two are correct? $$ f_{xy} = \frac {\partial} {\partial y} (\frac {\partial f} {\partial x}) = \frac {\partial ^2 f} {\partial xy}$$ or $$ f_{xy} = \frac {\partial} ...
1
vote
1answer
27 views

Proving P(x) > 0 given a condition.

$P(x)$ is a polynomial function such that, $P(1) = 0, P′(x) > P(x), ∀ x > 1. $ Prove that $P(x) > 0, ∀ x > 1.$ I was trying to do by taking the P(x) in the denominator and then ...
1
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2answers
34 views

$(r-1)^{th}$ derivative of $x^{k+r-1}$

EDIT: added $x^k$ in final answer I want to find: \begin{align} \frac{d^{r-1}}{dx^{r-1}}\left(x^{k+r-1}\right) \end{align} Writing out the first few terms and what I think is the last term we get: ...