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

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0
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1answer
40 views

Specify the values of $p$ and $p'$ for a polynomial

Problem 10-26 from Spivak's Calculus, 4th edition: Let $a_1, \dotsc, a_n$ and $b_1, \dotsc, b_n$ be given numbers. If $x_1, \dotsc, x_n$ are distinct numbers, prove that there is a polynomial ...
1
vote
1answer
56 views

Limit of the derivative of a function

Under what conditions is true: If $$\lim_{x\rightarrow\pm\infty}\Phi(x,y)\rightarrow 0$$ then $$\lim_{x\rightarrow\pm\infty}\frac{\partial}{\partial x}\Phi(x,y)\rightarrow 0$$ Some time ago I ...
3
votes
1answer
102 views

Derivative of Binomial Coefficient wrt k

I've got $\binom{2N}{N-x}$ and I'd like to take the derivative with respect to x. I know that I can take the derivative of $\binom{n}{k}$ w.r.t. n using logarithmic differentiation, but that's not ...
0
votes
2answers
47 views

Differential of a shifted function

If I'm given the differential equation: $$\frac{d(12-24f(t))}{dt} = 5$$ How do I rearrange this so that it looks like a normal first order linear differential equation? e.g, so it looks something ...
0
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1answer
32 views

Concavity of a parametric curve: a formula for $d^2y/dx^2$

I am going through old math texts and this problem is suddenly giving me problems. We have two functions, $y(t)=t^3-3t$ and $x(t)=t^2$, and the the question asks for concavity of the curve. It ...
3
votes
4answers
88 views

Differential equation which has following solution $y=\frac{1}{1+\exp(ax)}$

Is there any linear differential equation which has following solution $$y=\frac{1}{1+\exp(ax)}$$ $a$ is constant. something like: $$ y'' + by' +cy + \alpha = 0$$ where $b$, $\alpha$ and $c$ are ...
1
vote
0answers
83 views

Zeros of the derivatives of a finite Blaschke product.

Let $B$ be an $n$ degree finite Blaschke product. By considering the level curves of $B$, one can show that $B'$ has $n-1$ critical points in the disk (counting multiplicity). Is anything known ...
4
votes
1answer
60 views

Where do I make mistake on this derivative containing e^x^2

My brother is preparing for the university and asked me the following multiple choice question. $$\frac{d}{dx}(x^3 * e^{x^2})$$ a) $e^{x^2}*x^2*(1+2x)$ b) $e^{x^2}*x^2*(3+2x)$ c) ...
1
vote
2answers
73 views

Summation of infinite series

If we know the series sum given below converges to a value $C$(constant) $$\sum_{n=0}^{\infty}a_n =C \tag 2$$ Can we generate following in terms of C. values of $a_n$ will tend to zero as n goes to ...
0
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1answer
41 views

Partial derivatives of $xy^2/(x^2+y^2)$ at the origin

I noticed that this is a big black hole in my understanding of partial derivatives at the point. I don't know how to count it: $$ f(x,y) = \frac {xy^2}{x^2+y^2} $$ $$ \frac {df}{dx}(0,0)=\lim_{t\to ...
2
votes
1answer
38 views

Solve 2 connected ODEs describing a domain

This problem confused me for a long time. I have 2 ODEs which describe part of our domain. They are connected at middle: $$ \frac{d^2}{dx^2} u = -a, x<x_0 $$ $$ \frac{d^2}{dx^2} u - \frac{u}{b^2}= ...
1
vote
4answers
69 views

Differential equation with the solution of $(1+ax/2)\exp(-ax)$

Is there any linear differential equation which has following solution $$y=(1+ax/2)\exp(-ax)$$ $a$ is constant.
0
votes
1answer
48 views

Two Strictly Convex Functions with Contact of Order 1

Let $f,g: \mathbb{R}\rightarrow \mathbb{R}$ be two strictly convex functions, where $f$ is differentiable, $g$ is smooth, and $f\geq g$. Suppose that for some $x_0\in \mathbb{R}$: ...
1
vote
1answer
30 views

Differentiable Strictly Convex Function on Interval

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable, strictly convex function. Let $I\subset \mathbb{R}$ be a closed, bounded interval such that $f'(x) \neq 0$ on $I$. Is $f$ strongly ...
0
votes
1answer
67 views

How do the existence and $L^p$ integrability of weak derivative affect the smoothness of Sobolev functions?

In the definition of Sobolev spaces, let us say the space $W^{1,p}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^n$. What contributes more to the smoothness of a function $f$: 1-The fact ...
0
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0answers
49 views

question on differentiable and continious function

How should the function $f(x)=x\operatorname{sgn} x$ be defined at $x=0$ so that it is continuous there? Is it then also differentiable? How should the function $g(x)=x^2 \operatorname{sgn} x$ be ...
1
vote
1answer
255 views

Finding the derivative using the definition?

Calculate the derivate of the given function directly from the definition of derivative, and express the result using differentials $$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ when $f(x)= ...
1
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0answers
39 views

how to differentiate an integral

the integral is of the form below $$ \frac {d {\int y(x, t)h(x) dx}}{dy(x, t)} $$ what does the differentiation give? $h(x)$ and what about $$ \frac {d {\int y(x, t)h(y(x,t)) dx}}{dy(x, t)} $$ ...
3
votes
3answers
70 views

show that $f^{(3)}(c) \ge 3$ for $c\in(-1,1)$

Let $f:I\rightarrow \Bbb{R}$, differetiable three times on the open interval $I$ which contains $[-1,1]$. Also: $f(0) = f(-1) = f'(0) = 0$ and $f(1)=1$. Show that there's a point $c \in (-1, 1)$ ...
0
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2answers
30 views

Determine the equation for the tangent in a point on a curve

I am supposed to determine the equation for the tangent in point (4,1) to the curve: $$5\sqrt{x}=2\sqrt{y}(x+y^2)$$ I think that I should differentiate the expression and then put the values (4,1) ...
2
votes
3answers
42 views

Zero point when $f'(x)\gt c$

Suppose that the function $f:\mathbb R\to\mathbb R$ is continuously differentiable and that there is a positive number $c$ such that $f'(x)\ge c$ for all points $x$ in $\mathbb R$. Prove that there is ...
0
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1answer
69 views

Differentiation problem involving chain and product rules: $y=(3x+2)^2 e^{5x} + \sin (3x)$

I am just stuck on 2 questions. I have managed to complete one however I keep finding various answers to it using online calculators so I'm not sure if it's correct. The other I'm stuck on. I could ...
1
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3answers
47 views

The inflection points of $f(x)=(x^2-4x+1)e^{-x}$

I got the function $f(x)=(x^2-4x+1)e^{-x}$. The task is to find the inflection points. The correct answer is $x=4-\sqrt{5}$ and $ x=4+\sqrt{5} $. I got the second derivative to $f(x)$. But when I ...
0
votes
2answers
50 views

What is this lower number?

I was taught that the lower number in math would be the base, but you can't have base 0 (can you?) I'm looking at some derivatives and it looks something like this. $$x^2_0$$ Sorry for the stupid ...
0
votes
1answer
41 views

Calculus formula doubt

I am having a confusion in some of the formulas of differential and integral calculus. If $y=\ln x$, then $dy/dx=1/x$ and integral of $\tan x$ is $\log|\sec{x}|$ and also similarly of $\cot x$ and ...
0
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0answers
98 views

Generalized Leibniz Rule

Leibniz Rule states that, $$(f\cdot g)^{(m)}(x)=\sum_{k=0}^m \binom{m}{k} f^{(m-k)}(x)g^{(k)}(x).$$ Writing this with differentiation denoted by $D$, we might say $$D^m (fg) = \sum_{k=0}^m ...
2
votes
2answers
152 views

Is $\tan(x)$ differentiable for $x\in ( -\pi/2 , \pi/2 )$

This is an assignment question and in class we taught the definition that: A function $f(x)$ is differentiable if we can find $f(x+h) - f(x) = Kh +h E(x,h)$, where $K=f'(x)$ and $E(x,h) ...
0
votes
1answer
52 views

Absolute continuity and derivatives of integrals

I am preparing for a comprehensive at the end of the month, so I would appreciate any input I could get on this solution. I am pretty confident if the first part, but I think the second answer could ...
0
votes
1answer
87 views

Two methods of finding a function $f$ such that $Mdx+Ndy=0$ on the curves $f(x,y)=c$

this problem is from my class,i did one way and got one answer,professor did it in another way and got another answer.question is:Find $f(x,y)=constant$ where differential equation is ...
0
votes
1answer
44 views

Calculating the value of an integrals derivative given then value of the integral

I am given the following informations about a function: $$f\in C^1(\mathbb{R}),\quad f(3)=7,f(7)=13,\quad \int_{3}^{13}f'(x)\,dx=12$$ and i need to find the value of $$\int_{7}^{13}f'(x)\,dx.$$ A ...
4
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1answer
46 views

How to show that $\int\limits_{-\infty}^{+\infty}(n-1)\Phi(x)^{n-2}\phi(x)^2dx$? decreases in $n$?

I was working on a research project that involves taking the integral of $$(n-1)\int\limits_{-\infty}^{+\infty} \Phi\left(x\right)^{n-2}\phi\left(x\right)^2dx,$$ where $\Phi(.)$ is the CDF for ...
0
votes
2answers
36 views

Derivation of deformation formula in physics textbook

There is a derivation of a deformation formula for rocks in one of my textbooks which I don't quite follow. As the problem is mathematical, I've decided to post it here The derivation goes as ...
2
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0answers
62 views

Proof of Cauchy-Riemann equations using differentials as quotients?

In my analysis 2 book the proof goes like this: If a complex function $f = P(x,y) + iQ(x,y)$ is differentiable at a point $z$, then $$ \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} ...
1
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0answers
23 views

A question about absolute continuity and differentiability of $f$

Let $f$ be an absolute continuous on $[0,1].$ Suppose that there exists a continuous function $p:[0,1]\rightarrow R_{+}$ and $\lim_{x\rightarrow 0}p(x)=0$ such that for any Lebesgue points $\xi, ...
0
votes
1answer
46 views

Show that the approximation to $f$'($x_0$) has discretization error $O$($h^2$)

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h$ and $x_{1} = x_0 + h$ with $h$ > 0. Given a smooth function f, show that the approximation to $f'(x_0)$ given by the centered ...
2
votes
1answer
61 views

How to find $g'(0)$ if $g(x)=\sin(f(x^2+x)-2x)$ and $f$ satisfies $|f(x)|\le x^2$ for all $x$?

The problem goes as follows: Let $f(x)$ be a function such that $|{f(x)}|\le x^2$ $\forall x \in [-1,1/7]$. The first part of the problem is prove that $\lim_{x\rightarrow0} ...
1
vote
1answer
31 views

How to normalise equations of the form $dy/dx=B$ and $d^2y/dx^2=A$?

So I am trying to normalise equations of the form, $$dy/dx=B \mbox{ and } d^{2}y/dx^{2}=A$$ If I define $y^{*}$ as; $$y^{*}=By \Rightarrow dy^{*}/dy=B $$ Is it also then true that, $$d(dy^{*})/dy = B ...
2
votes
2answers
70 views

Direction of Greatest Increase

Problem: Find the direction of greatest increase at $P$. $$f(x,y)=4x^2+y^2+2y$$ $$P=(1,2,12)$$ Solution: The greatest increase in $f(x,y)$ at $P$ can be attained by moving in the direction of ...
6
votes
4answers
218 views

How to maximize or minimize $f(x)=ax^2+bx$?

I am trying to self-study calculus from the Internet. I have learnt things mostly from MIT OCW site and also from other sites. However, I am stuck on this simple problem: Find the maximum/minimum ...
3
votes
3answers
73 views

Using the chain rule with a composite function

I'm a little confused on this homework problem and I could use some explanation if anyone has seen something like it before. The question is: Use the Chain Rule to find $\frac{dy}{dt}$ at $t = 9$ ...
0
votes
3answers
48 views

Find the derivative of $\frac{{(x^3)^{4/3}}}{(2-x)^{4/3}}$

I tried to solve it using the chain rule first and then doing the quotient rule after. However, I end up with $\frac{24x^2-8x^3(x^3)^{4/3}}{3(2-x)^{7/3}}$ My professor said it's wrong. Kindly explain ...
3
votes
1answer
63 views

Derivative: $f_x, f_y, f_{xy}$ of function - $f(x,y)$

Let's say $f(x,y) = x^2 + 2xy +y^2$ $f'_x = 2x + 2y$ $f'_y = 2y + 2x$ $f'_{xy} = 2x + 2y$ ? Am I right about the third?
4
votes
0answers
32 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
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3answers
96 views

Evaluate $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at $t=1$

I need to find a "nice" formula for the evaluation of $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at t=1, where $d_j \in \mathbb{N}$. I have already proved ...
0
votes
2answers
251 views

Optimization problems: Finding the optimal path

I'm still trying to get the hang of optimization problems in calculus and I'm looking for a little help. I'm having trouble finding equations to model the following problem: I'm fairly sure I need to ...
1
vote
1answer
71 views

Proving power rule for $x^n$ with arbitrary positive $n>0$

How to prove that $\frac{d}{dx} x^n = nx^{n-1}$ for every $n>0$ (possibly fractional)? Context It was already shown that $\frac{d}{dx} x^n = nx^{n-1}$ for positive integer $n$. My friend ...
1
vote
0answers
53 views

Differentiation and integration of a series

If I have a power series $$\sum _{k}^{\infty }f(x)$$ and I differentiate it I get according to my current knowledge $\sum _{k}^{\infty }f(x)'$,however when I look at a power series defined by $$\sum ...
5
votes
1answer
441 views

Combination of linear functions that give the derivative operator

Let $D$ be the derivative operator and $C^\infty$ the set of functions derivable once. Here $f^n=f\circ f\circ\cdots\circ f\text{, }n\text{ times}$ It can be easily shown that there exists ...
0
votes
2answers
94 views

Ordinary Chain Rule Confusion

Let $f$ be the function defined in Q1, and let $g$ be a function such that $g^\prime(x)=\sin(\sin(x+1))$ and $g(0)=2$. Find $(f\circ g)^\prime(0)$ and $(g\circ f)^\prime(0)$. For $x\neq0$, the ...
0
votes
3answers
138 views

Assumptions in Word Problems.

My dilemma has been that I am confused on how we make mathematical assumptions in WORD problems. Suppose you are given a related-rates word problem. (Q#) Air is being pumped into a spherical balloon ...