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

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5
votes
1answer
70 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
0
votes
2answers
41 views

Is the quotient rule needed in this case?

I need the partial derivative w.r.t. $r_{20}$, that in this function is only in the denominator, do I need to use the quotient rule? $\dfrac{\partial f}{\partial r_{20}} = \dfrac{r_{00}*u/w + ...
1
vote
0answers
50 views

Given a function $f$, find the largest $n$ such that $f(x)/x^n$ can be defined at $x=0$ to become differentiable there

Let $f(x) = \ln\left(\frac{x^2}{2}+1\right)+\cos x -1$. Find the largest $n\in\Bbb{N}$ such that there is $C\in\Bbb{R}$ such that: $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C ...
1
vote
2answers
26 views

Confirm right model

Have a Khan problem I've been working under the "Related Rates" category. GIVEN: A 2 meter tall boy "h" is rollerskating away from a 5 meter lantern at constant dx/dt = 2 meters per second. How ...
8
votes
4answers
126 views

How to find the derivative of a function defined by an integral? Namely, $f(y)=\int_0^{y^2} e^{-x^2y^2}dx$

Find at each point of its domain the derivative of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ $$f(y)=\int_0^{y^2} e^{-x^2y^2}dx$$ $$$$ Is the domain of the function $\mathbb{R}$ because of ...
3
votes
2answers
46 views

Prove that $f(a) \leq f(x) \leq f(b) $

If the following data are given, prove that $f(a) \leq f(x) \leq f(b) $ f is differentiable on [a,b] and f'(x) $ \geq 0 \forall x \in (a,b) $ Is the following argument correct? $f'(x) \geq 0 ...
-2
votes
2answers
32 views

Function with second derivative [closed]

I got the function $f(x)=(x+1)^{11}(2-x)$. I got the derivative to $3(7-4x)(x+1)^{10}$? What´s the second derivative of $f(x)=(x+1)^{11}(2-x)$ ?
1
vote
2answers
44 views

Finding derivative

$\lim\limits_{x\to\ 2}\frac{f(x)-f(2)}{x^2-4}=4$ where $f(x)$ is defined on $\Bbb R$ and $g(X)=\frac{f(x)e^x}{1-x}$. What is $g'(2)$?
0
votes
1answer
28 views

$f$ differentiable on $[a,b]$, but not Lipschitz

Question 11-37(d) of Spivak's Calculus, 4th ed., asks If $f$ is differentiable on $[a,b]$, is $f$ Lipschitz of order $1$ on $[a,b]$? The phrase "differentiable on $[a,b]$" is a little ...
3
votes
3answers
85 views

What is $\frac{d^n}{dx^n} \frac{e^{\lambda x}}{x}$?

I was wondering whether there is an explicit way to say what the derivative of $\dfrac{d^n}{dx^n} \dfrac{e^{\lambda x}}{x}$ for $n \in \mathbb{N}_0$is, where we assume that $\lambda \neq 0$.
2
votes
1answer
32 views

Strict local extremum without $f'$ “changing signs”

Let $f:\mathbb{R}\to \mathbb{R}$. Is it possible that $f$ has the following properties: $f$ is differentiable in a neighborhood of $a\in \mathbb{R}$ $a$ is a strict local minimum There is no ...
0
votes
0answers
10 views

Nomenclature for the function appearing in Carathéodory's criteria of differentiability

In my previous question Concerning Carathéodory's criteria of differentiability and a proof that differentiable implies continuous I stated the criteria as follows: There exists a function $\phi$ ...
1
vote
0answers
14 views

A question on the procedure of finding the matrix of a linear transformation of a polynomial and a combination of its derivatives

I'm trying to self-study Linear Algebra from Linear Algebra Done Wrong, but the book doesn't have solution manual so my question might be extremely easy, apologize in advance: The question is for the ...
1
vote
2answers
41 views

Using complex derivative to shows that a function is constant

If we know $\frac{\partial f}{\partial z} \equiv f'(z)=0$ where $f(z)=u(x,y)+iv(x,y)$ why do we need to check the Cauchy Riemann equations are all equal to zero, before concluding that $f$ is ...
0
votes
1answer
39 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
49 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
20 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
27 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
votes
1answer
21 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
85 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
26 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
55 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
64 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
votes
1answer
36 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
35 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
67 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
38 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
17 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
32 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
votes
0answers
37 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 ...
2
votes
1answer
183 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
vote
0answers
31 views

Weird derivative computation

I found the following formulas in a control theory textbook : $$s(x,t)=\left(\frac{d}{dt}+\lambda\right)^{(n-1)}\varepsilon $$ where $\varepsilon(t)=T\left(\frac{e(t)}{p(t)}\right)$ and ...
1
vote
0answers
32 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 $$ \frac {d {\int y(x, t)h(y(x,t)) dx}}{dy(x, t)} $$ ...
1
vote
1answer
27 views

Multivariable-calculus, derivative and second derivative [closed]

I got the function $f(x,y)=\ln \sqrt{x^2+y^2}$. The task is to find the derivative function and the second derivative function. How do I get there?
2
votes
3answers
63 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
votes
2answers
28 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
41 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
votes
1answer
47 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
vote
3answers
41 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
45 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
38 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
votes
0answers
46 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
49 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
38 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
66 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
40 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
votes
1answer
42 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
24 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
votes
0answers
33 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
vote
0answers
16 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, ...