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

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19 views

Are these two compositions of two functions differentiable?

Assuming $U=\{x\in\mathbb{R}^2:x_1^2+x_2^2<1\}$ is the open unit circle in the plane and $f,g:U\rightarrow\mathbb{R}^2$ two functions with $f(0)=g(0)=0$. $f$ is Fréchet-differentiable in $0$, and ...
2
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2answers
51 views

Matrices derivative

I have a linear product of matrices, I did solve most of it, however, I stop at this component $(X^T W^T D W X)^{-1}$. Given that $X$ is $n \times p$ matrix and $D$ is $n\times n$ matrix. $W$ is a ...
0
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0answers
19 views

Differential of square map in Lie group away from identity

I've looked everywhere for this specific example but couldn't find it. Probably simple but I only need it for a small application and my Lie theory is very rusty. Let $G$ be an arbitrary Lie group ...
1
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4answers
43 views

When can I use the natural log to help solve an integral?

Why is it okay to do this: $\int \frac{1}{x-2}dx = \ln(x-2)$ but not this: $\int \frac{1}{1-x^2}dx = \ln(1-x^2)$
3
votes
1answer
30 views

diffeomorphism inbetween two subsets of $\mathbb{R}^2$

Consider the function $$f: \mathbb{R}^2 \to \mathbb{R}^2, \space\space f(x, y) := \pmatrix{x(1-y) \cr x y}$$ Now first, why is $f$ continuously differentiable? Then, I want to prove that $f$ ...
1
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2answers
29 views

second derivative of the composition of two multivariable functions

Let $U \subset \mathbb{R}^n$ be open, and let $\gamma: \mathbb{R} \to U$ and $f: U \to \mathbb{R}$ be to functions that are differentiable at least twice. I want to show that $\frac{d^2}{dt^2}(f ...
2
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4answers
111 views

What is wrong in my $f'(x)$?

We have $f:\mathbb{R}\rightarrow\mathbb{R}, f(x)=\frac{x^2-x+1}{x^2+x+1}$ and we need to find $f'(x)$. Here is all my steps: ...
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0answers
14 views

Hesse matrix is negatively semidefinite if a function has a local maximum

Let $U \subset \mathbb{R}^n$ be open, and let $f: U \to \mathbb{R}$ be at least twice continuously differentiable. Also, we assume $f$ has a local maximum $a \in U$. I now want to show that the Hesse ...
0
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0answers
53 views

First derivative of energy function [closed]

Given the following energy function $E(d)$ (also found here on page 3): $$ E_d = \sum_{x,y \in \Omega} \left(d_{x,y} - \hat{d}_{x,y}\right)^2 + \lambda \sum_{x,y} \left( ...
3
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5answers
89 views

Differentiate the Function $f(x)= \sqrt{x} \ln x$

Differentiate the Function $f(x)= \sqrt{x} \ln x$
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1answer
65 views

How to calculate this derivative $D^{\alpha}f(x)$?

Let $v\in\mathbb{R}^n$ be a fixed vector, and $f$ a function given by $f(x)=\cos(x\bullet v)$, where $x\bullet y$ is the dot product. What is the derivative $D^{\alpha}f(x)$ for $x\in\mathbb{R}^n$ ...
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2answers
34 views

Calculus: simpler way of showing that derivative is negative?

I want to show that $\frac{1-(1-\beta)^N}{\beta}$ is strictly decreasing in $\beta$ for $\beta \in (0,1)$ and $N \geq 2$. My approach so far is as follows: I take the derivative with respect to ...
0
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2answers
43 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ The correct answer is $\frac ...
3
votes
1answer
36 views

Existence of differentiable functions on $\mathbb R$ whose derivative is constant on the complement of uncountable set but not everywhere

Let $ A $ be a countable subset of the set of real numbers and $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f'$ is constant on $\mathbb R \setminus A$ , then I know that $f'$ is ...
1
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0answers
38 views

How to take derivative of $F(u)=\sum_{i=1}^{N} \int f^2(x) u_i^q(x) dx $

I have to find the derivative of a function. Could you help me to find it $$F(u)=\sum_{i=1}^{N} \int_{\Omega} f^2(x) u_i^q(x) dx $$ where $q \ge 1$, $f(x): \Omega \to R$, $u_i$ is membership ...
1
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2answers
39 views

Find maximum of a function

I want to find the maximum of a function. $$ d = \frac{35}{3} + \frac{7}{3}\sin( \frac{2\pi}{365}t ) $$ I don't know if I applied the chain rule correctly. $$ w = \frac{2\pi}{365}t $$ $$ w' = ...
1
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3answers
51 views

Slopes of inverse functions

I have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ...
0
votes
1answer
53 views

Lebesgue differentiation

Some days back I was doing the lebsegue integration. I was amazed by the integration's ability to maximize the potential of Reinman integration. Are there any new differentiation (except the metric ...
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3answers
29 views

Constructing Polynomial Function from Set of Points and Slopes

I only have a basic knowledge of calculus but I would like to know if it's possible to, given a set of points each with their own slopes, construct the simplest (or any) polynomial function that ...
11
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3answers
992 views

Are polynomials infinitely many times differentiable?

Are polynomials infinitely many times differentiable? If so, does it only mean that at some point we reach 0 and then we keep on getting 0? Thank you!
-1
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2answers
126 views

If $f(x+y^3)=f(x)+[f(y)]^3$ and $f'(0)\ge0$, what can $f(10)$ be? [closed]

A real valued function satisfies the condition: $f(x+y^3)=f(x)+[f(y)]^3$ for all real $x$, $y$. If $f'(0)\ge0$ how to find $f(10)$ ?
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4answers
70 views

Derivative of the function $y = 2^{\sqrt{\tan x}}$

How to find derivative of the following function: $y = 2^{\sqrt{\tan x}}$ , $y' = ?$ I did the following $$\frac{d}{dx}2^{\sqrt{\tan x}} = 2^{\sqrt{\tan x}}\ln{2}(\sqrt{\tan}x)'$$ and stopped here. ...
-2
votes
5answers
97 views

How to find $f(\frac{1}{\sqrt3})$ and $f'(1)$? [closed]

It is given that $$f(x)+f(y)=f\left(\frac { x+y }{ 1-xy } \right)$$ for real values of $x$ and $y$ $(xy \neq 1)$ and $$\lim _{ x\to0 }{ \frac { f(x) }{ x } }=2$$ How do we find ...
3
votes
1answer
53 views

Derivative by Definition of $\frac{\sin^2(x)}{e^x-1}$

I have to prove the derivative by definition of $$\frac{\sin^2(x)}{e^x-1}$$ $$f^{\prime}(x)=\lim_{\Delta x \to 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}}$$ $$\large f^{\prime}(x)=\lim_{\Delta x \to ...
0
votes
1answer
42 views

Derivative of a trigonometric function

What is the derivative of $$\cos^2 a (\tan a - \tan b)$$ Please anyone explain in detail. The differentiation is with respect to $a$. I tried to obtain the answer using chain rule, but didn't get it. ...
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votes
1answer
41 views

Chain rule differentiation [closed]

Can any one show me the steps to differentiate $v^2$ according to chain rule? Why is derivative of $v^2$ found out by chain rule and not by exponent formula?
-2
votes
1answer
33 views

Positive derivative on [0,1] implies a continuous derivative on [0,1]

If a real-valued function F defined on [0,1] is differentiable with positive derivative f everywhere on [0,1], can we conclude that f is continuous?
2
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3answers
59 views

derivative integral $\int_0^{x^2} \sin(t^2)dt$

I want to know how I derivative this integral: $$\int_0^{x^2} \sin(t^2)dt$$ what are the steps to derivative it?
0
votes
3answers
67 views

n-th derivative test.

Let $f(x)$ be a function such that it is $n$ times differentiable and $f^{'}(a)=f^{''}=(a)f^{'''}=(a)....=f^{n-1}(a)=0$ and $f^{n}\ne0.$ The $n^{th}$ derivative test tells us about the concavity of ...
1
vote
2answers
33 views

What is the cosine of angle of intersection of following functions?

1st Function: $\displaystyle 3^{x-1}\log x$ 2nd Function: $\displaystyle x^x-1$ How to find the cosine of angle of intersection of these two curves? Their $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ are not ...
1
vote
3answers
40 views

Multiplicity of a root of a polynomial

:) It's true that, if a polynomial has a root (let's say, k, for example) with multiplicity n (n>1, for n integer), then it's true that the derivate polynomial have k as a root with multiplicity ...
4
votes
1answer
39 views

If a function is left- and right-differentiable everywhere, how much can the one-sided derivatives disagree?

Just for fun, I was proving some results about convex functions the other day. I was able to show that for a convex set $E\subseteq\Bbb R,$ if $f:E\to\Bbb R$ is convex, then $f$ is left- and ...
0
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0answers
40 views

Second Order Differentials: Using $y = A + Bxe^x$

I've went over some of my math work which I'm currently doing at Uni and came across a rather confusing example. The example I went over is based on Second Order Differentials. So basically what I ...
7
votes
5answers
971 views

Proof of the derivative of ln(x)

I'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + h) ...
4
votes
2answers
106 views

Function that is continuous and its differential is continuous

Let $ f: \mathbb{R} \rightarrow \mathbb{R}$ . Show that $f$ is continuously differentiable if and only if, for every $x \in \mathbb{R}$ there exists a $l \in \mathbb{R}$ with the property that ...
2
votes
2answers
57 views

Proving that $\lim_{x {\to} \infty}f(x)=\infty $

Given $\ f(x)$ that is differential in $\ (x_0,\infty), f'(x)\ge a,$ for every $\ x> x_0$ and $\ a>0$, trying to show that $\lim_{x\to\infty}f(x)=\infty$. So far I've tried using Mean Value ...
0
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2answers
28 views

How do I find the equations of line tangent to a unit circle that have a slope of 1?

I'm supposed to find the equations of the lines tangent to a circle of radius $1$ and centered at the origin that have a slope of $1$. I know these things: this is a unit circle, the equation for a ...
0
votes
1answer
72 views

How can I find $f'(0)$ of this function?

I need to find $f'(0)$ if $$f(x)={x^2\sin x-\cos\left(3x\right)\over e^{-3x}+1}$$ How do I do this? When I tried using the quotient rule it became messy very quickly so I thought that there must be ...
2
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1answer
36 views

What is the proper use of Leibniz notation for one-sided derivatives?

The only notation I've seen has been restricted to either Lagrange's prime notation or Euler's $D$. Here are some of the variants: $$f'(a^+):=\lim_{x\to a^+}\frac{f(x)-f(a)}{x-a}$$ ...
0
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0answers
16 views

Proving multi-variable differentiability using the limit definition

I'm doing advanced calculus and I find it challenging to solve multi-variable limits while proving differentiability, more specifically 2 variable limits. could you show me how do I solve this limit?: ...
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3answers
49 views

In proving the product rule, how do we know to add and subtract f(x+h)g(x) from the numerator in the derivative definition?

I watched two YouTube videos to try to get a proof that makes sense, but in both videos, the authors said something to the effect of "add and subtract f(x+h)g(x)" without a good explanation as to how ...
2
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0answers
26 views

Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
3
votes
1answer
160 views

What is the minimum degree of a polynomial for it to satisfy the following conditions?

This is the first part of a problem in the high-school exit exam of this year, in Italy. The differentiable function $y=f(x)$ has, for $x\in[-3,3]$, the graph $\Gamma$ below: $\Gamma$ exhibits ...
1
vote
1answer
36 views

application of inverse function theorem (first I thought implicit function theorem and then corrected it), how to continue?

Let $f=(f_1,f_2,f_3):\mathbb{R}^2\to\mathbb{R}^3$ continuously differentiable, $\det\begin{pmatrix} D_1f_1 & D_2f_1 \\ D_1f_2 & D_2f_2 \end{pmatrix}\not=0$. How to prove: In every point ...
3
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2answers
28 views

$\|f(x)-f(y)\|\ge c\|x-y\|$ for all $x,y\in U, c>0$-> continuously differentiable inverse function $g:f(U)\to U$

Let $U\subset\mathbb{R}^n$ open, $f:U\to \mathbb{R}^n$ continuously differentiable, $\|f(x)-f(y)\|\ge c\|x-y\|$ for all $x,y\in U, c>0$. Why is $\det(Df(x))\neq 0$ for all $x\in U$ and $f\colon ...
1
vote
2answers
68 views

On proving the total differential.

I am following an open-course on multi variable calculus provided by MIT taught by Denis Auroux. The question I am about to ask is from this lecture. In the lecture Denis Arnoux gives a sketch proof ...
0
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2answers
30 views

Domain of derivative on open interval is open

Let $f : (a, b) \to \mathbb{R}$. Suppose that the derivative $f'$ exists at every point of a set $E \subseteq (a,b)$. Is it true that the domain $E$ of $f'$ is open? And if it is not true, is it true ...
0
votes
1answer
53 views

Can ∂x and ∂y in a derivate be seen as ∂ times x or ∂ times y?

I'm watching some tutorials on machine learning and know just enough calculus to have an intuition on what a derivative is, but that's it. But this question is bugging me so much that now I'm pretty ...
0
votes
1answer
24 views

What is $\nabla\cdot A\nabla u$ for $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$?

Let $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$. How can we compute $\nabla\cdot A\nabla u$? I assume we need to apply some kind of product rule, but I ...
2
votes
1answer
28 views

What is $\nabla Au$ for $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$?

Let $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$. How can we compute $\nabla Au$? I assume we need to apply some kind of product rule, but I wasn't able to figure out ...