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

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How would we solve $\lim_{x\to\infty} \frac{x-\sin x}{x+\cos x}$ by using L'Hôpital's rule?

Problem here is at one point after differentiating both numerator and denominator, I get $$\lim_{x\to\infty} \frac{x-\sin x}{x+\cos x} = \lim_{x\to\infty} \frac{1-\cos x}{1-\sin x} $$ after that ...
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1answer
37 views

Taylor series expansion of $f(x)=\frac{sinx}{x-\pi}$ at $x=\pi$

I was solving it and on one step I need to find the 2nd Derivative of $f(x)$, I am getting -1/3, but according to book it's -1/6.Please help me out here.
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1answer
33 views

Lie brackets definition

Let $v,w$ be vector fields on a smooth manifold $M$ (i.e. $v : M \rightarrow TM = \lbrace (p, v_p) : p \in M, v_p \in T_p M \rbrace$). The Lie brackets of $v,w$ are defined as $$ [v,w](f)|_p = ...
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2answers
53 views

Exercise on differentiable function

$f\colon \mathbb{R} \rightarrow \mathbb{R}$ a differentiable function and $m \in \mathbb{R}$ for which $f'(x)\ge m$ for every $x \in \mathbb{R}$. Let $Z= \{x\in \mathbb{R} \mid e^{\sin(x)} = f(x)\}$ ...
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0answers
38 views

Finding the derivative for the following functions

Question:Using the rules of finding derivatives, find the derivative $f’$ of the following functions: (a) $f(x) = 2x +8$ (b) $f(x) = a + bx + cx^2$ (c) $f(x) = 120$ My answer I used rules and my ...
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2answers
44 views

How can I solve this using the Lagrange method?

This is what I keep doing but the answer seems to be wrong every time.
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50 views

How can I find the gradient of this function? $f(r,t)=r^3\cos(t).$

$$f(r,t)=r^3\cos(t).$$ Is it not like this: $<3r^2,-\sin(t)>$
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66 views

Multivariable Calculus - Calculating Derivative Matrix

I'm working with Munkres' Analysis on Manifolds. From chapter 2 (this isn't a homework question): Given $f: \mathbb R^2 \rightarrow \mathbb R^2 : f(r,\theta)=(r\cos(\theta),r\sin(\theta))$, ...
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1answer
28 views

continuity with lipschitz dertivative

Can somone explain me what does it mean to say a "function is continously differentiable with a Lipschitz derivative near the limit point". I dont understand the technical jargoans. Thanks
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2answers
244 views

Prove $|\sin x - \sin y|< |x-y|$

Problem 15-24 of Michael Spivak's Calculus (first edition) is Prove that $|\sin x - \sin y|< |x-y|$ for all numbers $x$ and $y$. The same statement, with $<$ replaced by $\leq$, is a very ...
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3answers
104 views

Manipulation of differentials

In solving some differential equations in physics, one has to do some peculiar manipulation of differentials to wind up with a desired function. My professor has warned up that doing so, "Will make ...
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1answer
41 views

Lagrange's Theorem exercise

Be $u : \mathbb{R} \rightarrow \mathbb{R}$ a $C^2$ function. Proof that exists a $x \in \mathbb{R}$ with $0<x<2$ for which $u(2)-2u(1)+u(0)=u''(x)$ Applying Lagrange's Theorem I showed: ...
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3answers
92 views

Why does f' = 0 gives the min or max?

I understand how to calculate it, but I am just curious, why actually it works? Do we have a proof that it always works?
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2answers
486 views

If a function has no critical points then how can I find where the function is increasing or decreasing?

Recently, I have discovered some problems that have no critical points i.e. $$f'(x) \not = 0$$ For example, if we have the exponential $e^x$ divided by some other function squared i.e. $(x+2)^2$. ...
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2answers
74 views

how to solve $\frac{df}{dx}$ derivative against $dy$?

If I take derivative of $\frac{df}{dx}$ against ${dy}$, will it be $\frac{d^2f}{dxdy}$. I am a little confused. So to solve this we will look for those values which have variable $y$ in them in ...
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1answer
102 views

Numerically calculate the second “left hand” derivative

The Problem I have a series of measurements for which I have to calculate the first and second derivative numerically in a "live" fashion, i.e. using only previous data. This is easy for the first ...
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0answers
16 views

derivative of $\nabla_{\lambda_i^{(1)}} \mathrm{Tr}\left((\Sigma_1 + \Sigma_2)^{-1} \cdot V^{(1)}_i V^{(1)^\mathrm{T}}_i \right) $

As stated in the title, I seek to find the derivative of Equation (1) with respect to the eigenvalues of the first covariance matrix, $\Sigma_1$. \begin{equation} \nabla_{\lambda_i^{(1)}}\; ...
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1answer
26 views

Differentiability in $\mathbb R^n$

Let $U\in \mathbb{R}^n$ be open, and let $f:U\to \mathbb{R}^m$, and let $a\in U$. Let $\|\cdot\|'$ be a norm on $\mathbb{R}^n$, and let $\|\cdot\|''$ be a norm on $\mathbb{R}^m$. Prove that $f$ is ...
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2answers
98 views

is there a trick or shortcut for this derivative

For the following function S(z), I would like to know the nature of it's derivative. I calculated the first derivative wrt to z but it assumes monstrous proportions when it comes to the second ...
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2answers
44 views

Show that $f$ is increasing when $\alpha=0$ and decreasing when $\alpha=1$

Let $$f(x)=(x+\alpha)\log\left(1+ {1 \over x}\right)$$ Show that $f$ is increasing when $\alpha=0$ and decreasing when $\alpha=1$ on $[1,\infty)$ The derivative is: $$f'(x)=\log\left(1+{1 \over ...
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What function can be differentiated twice, but not 3 times?

In complex analysis class professor said that in complex analysis if a function is differentiable once, it can be differentiated infinite number of times. In real analysis there are cases where a ...
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1answer
59 views

Tangent vector to a curve on a manifold

If one has a curve $\sigma : (-1,1) \rightarrow M$, where $M$ is a smooth manifold, the tangent vector in $\sigma(0)$ is usually defined as $$ \sigma'(0) (f) = \dfrac{d f \circ \sigma}{dt} \Big|_0,$$ ...
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3answers
86 views

Convex functions - two questions

I have two questions regarding convex functions: First question: Let f be convex function on closed interval [a,b]. Prove that f has maximum in x=a or x=b. I understand that $\forall ...
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1answer
68 views

Compute a derivative of a function defined by an integral

If $h$ is continous, and $f$ and $g$ are differentiable and $$F(x)=\int \limits_{f(x)}^{g(x)} h(t) dt$$ then f is differentiable and $F'(x) = h(g(x))g'(x) - h(f(x))f'(x)$
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61 views

Two functions that are derivatives of each other

Is there a pair of functions such that: $D f(x) = g(x)$ $D g(x) = f(x)$ The functions are not the same function (without this 0 and $e^x$ would work) In case that there isn't such pair of ...
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2answers
61 views

Solve this limit (Maclaurin or differentiate?)

I have this assignment where I should calculate the limit below: $$ \lim_{x\to0}\frac{\sin 2x}{x\cos x} $$ I can use l'Hospitals rule (because it is a "zero divided by zero"-case) and therefore ...
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2answers
81 views

I'm having trouble understanding this derivation

In my notes I came across an equation: $$\ddot\theta = \frac{\mathrm d\dot\theta}{\mathrm dt} = \frac{\mathrm d}{\mathrm dt}\left(\frac{v\sin(\theta)}{r}\right) = \frac{\dot v \sin (\theta) }{r} + ...
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1answer
54 views

Show the derivative is bounded

Let $f: (0, \infty)$ such that: $$f'(x) = \frac{1}{{2\sqrt x }}\left( {\sin \frac{1}{x} + 1} \right) + \sqrt x \left( { - \frac{1}{{{x^2}}}} \right)\cos \frac{1}{x}$$ Now, We can bound it by: ...
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2answers
39 views

Derivative of $x^2e^{-x}$

So I have to function $x^2e^{-x}$. Do I derivative that like $f' g+g' f$ or $f' g'$? I'm not sure because it is derivative over x so if you can help me.
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1answer
88 views

There exist such f, f'

My question is: There exist a function $f = f(x)$ that satisfy $$\lim_{x \to +\infty} f(x) = +\infty$$ $$\lim_{x \to -\infty} f(x) = +\infty$$ $$\lim_{x \to +\infty} f'(x) = +\infty$$ $$\lim_{x \to ...
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3answers
206 views

Derivative of $(\ln x)^{\ln x}$

How can I differentiate the following function? $$f(x)=(\ln x)^{\ln x}.$$ Is it a composition of functions? And if so, whic functions? Thank you.
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1answer
67 views

differentiate arctan (maclaurin?)

I have this assignment: Differentiate this expression: $$ f(x) =\arctan \frac{x-1}{x+1} $$ There is also known that $-1 < x$ (Why is that important?). I do not know how to solve this problem... ...
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0answers
41 views

How to find $n^{\text{th}}$ derivative of $x^{1/x}$

How does one find the $n^{\text{th}}$ derivative of $x^{1/x}$? Using wolfram alpha we have some solutions, but most of them are in terms of funny functions and have some weird restraints. ...
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1answer
68 views

check if complex function is differentiable

The question is to check where the following complex function is differentiable. $$w=z \left| z\right|$$ $$w=\sqrt{x^2+y^2} (x+i y)$$ $$u = x\sqrt{x^2+y^2}$$ $$v = y\sqrt{x^2+y^2}$$ Using the ...
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2answers
186 views

Second derivative of binomial distribution

I try to prove that according to binomial distribution $P(X=k)={n \choose k}p^k(1-p)^{n-k}$ the maximum probability $P(X=k)$ is achieved at maximum likelihood, i.e. $p=\frac{k}{n}$. Let's apply ...
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1answer
134 views

Find the derivative of $\int_x^{x^2} e^{-t^2}dt $

Hey guys this was given to me as an exercise question and its really confusing. I'm not really sure where to start with this one, and I am assuming that the derivative isn't just $e^{-t^2} dt$. ...
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2answers
60 views

If I derive $ e^{-2x} $, where does x go?

So I have the function $$ e^{-2x} $$ and if I derive this I thought that I should get $$ -2xe^{-2x} $$ But the $x$ disappears, why? Is it an inner derivative and because of that, I also have to ...
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2answers
54 views

Find maximum of a function 4

I would appreciate if somebody could help me with the following problem Q: Find the maximum of $$f(x) =\frac{x-\ln(1+x)}{x\ln(1+x)};\quad(x>0)$$
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7answers
137 views

How do I differentiate $\sin^2x$?

I thought that because this is true: $$ \sin^2x=(\sin x)^2,$$ I could differentiate the expression like this: $$ \frac{d}{dx}\sin^2x=2\cos x.$$ But I am supposed to get $$ \sin(2x) \quad ...
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1answer
78 views

Function with a constant infinite order derivative, infinite final value, 0 initial value, and graph that resembles geometric growth

Please forgive my vocabulary & usage because I'm only a math amateur, so I'll try to describe this the best I can. Does such a function exist that has an infinite order derivative with a constant ...
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2answers
95 views

Show $f$ is differentiable and $f'$ is continuous.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by: $$f(x) = \left\{ {\begin{array}{*{20}{c}} {1,x = 0} \\ {\frac{{\sin (x)}}{x},x \ne 0} \\ \end{array}} \right.$$ Prove $f$ is ...
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1answer
78 views

Show a function is not bounded

Let $f(x):(0,\infty)\rightarrow \mathbb{R}$, $f$ is differentiable. It is given that: $\mathop {\lim }\limits_{x \to \infty } f'(x) = L > 0$. Show $f$ is not bounded. I've seen a proof ...
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1answer
331 views

Find the derivative of $F(x) = \int_0^x xf(t) dt$

this was given as an exercise in my first year honours math class. I can't seem to wrap my head around why this is not equal to $xf(t)$. Any help is appreciated! heres the question: Find the ...
2
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2answers
54 views

Suppose $f$ is real-valued function defined on $[1,\infty)$ with $f(1)=1$.Suppose, moreover, that $f$ satisfies…

Suppose $f$ is real-valued function defined on $[1,\infty)$ with $f(1)=1$.Suppose, moreover, that $f$ satisfies $$f'(x)=\dfrac{1}{x^2+f^2(x)}$$. Show that $f(x)\le 1+\pi/4$ for every $x\ge 1$. My ...
2
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1answer
39 views

$g(z) = \int_{0}^{2\pi}f(e^{i\theta})\frac{e^{-i\theta}}{e^{-i\theta}-\bar{z}}d\theta$ is antiholomorphic

I 've encountered this fact: if $z \in D(0,1) $ and $f$ is continous on $\partial D(0,1) $ then $$g(z) = \int_{0}^{2\pi}f(e^{i\theta})\frac{e^{-i\theta}}{e^{-i\theta}-\bar{z}}d\theta$$ is ...
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101 views

Finding velocity in optimization problem

Given $s=-16t^2+192t+144$, what is the velocity when $s=0$? This is part of a larger optimization problem which I solved, except for this last part. The critical point occurs at $t=6$, so after ...
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1answer
69 views

Frechet Derivative: Why bounded (linear) operator?

Why do we want the frechet derivative to be a bounded linear operator? (This meant more as a collecting ideas - I know bounded operators behave fine but that would exclude alot of examples such as the ...
2
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1answer
145 views

Differentiability vs Analyticity

What makes the crucial difference between the reals and the complex numbers is that the complex numbers are algebraically closed. So while going through all the proofs that "being holomorphic implies ...
2
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2answers
87 views

Find greatest value of $y(x) = (0.9^x)(300x + 650)$

Question and attempt $y(x) = (0.9^x)(300x + 650)$ Estimate at what x value that y reaches its maximum value The only way I could think of would be to use derivatives, so I tried it: $y'(x) = ...
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3answers
47 views

Under certain conditions, prove: If $\lim_{x\to\infty}f'(x) = 0$ and $\lim_{n\to\infty} x_n = \infty$ then $\lim_{x\to\infty}f(x_{n+1}) - f(x_n) = 0$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable with $\lim_{x\to\infty} f'(x) = 0$. Furthermore let $(x_n)_{n\in\mathbb{N}}$ be a sequence of real numbers with $\lim_{x\to\infty}x_n = ...