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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0answers
9 views

If the derivative is zero on [a, b] so the function is constant - using Heine-Borel?

I know the proof using MVT but I was wondering if it can be proofed using Heine-Borel Lemma, that "Every open cover of close interval has a finite subcover". (without compactness, simple as that). ...
1
vote
2answers
37 views

Is 'a' differentiable in f when f is a product of a differentiable and non-differentiable function?

Recently, I was studying differentiable and non-differentiable functions and I wondered whether this "conjecture" of mine is true: 1) "If $f(x)$ is a function that is the product of $g(x)$ and $h(x)$ ...
-5
votes
0answers
40 views

Need help for calculating two derivatives analytically

During building a Jacobian matrix for a numerical simulation I need to calculate following two derivatives where I hesitate about the correct answer: $$\partial( \partial X/ \partial\theta)/\partial ...
0
votes
0answers
17 views

Estimating the derivative of the difference function from that of a function

Suppose that $f$ is a twice differentiable function in an interval $(N,2N)$. We write $f_1(n,h)=f(n+h)-f(n)$, i.e., $f_1$ is the difference function. Then, a proof I'm reading estimates that if ...
2
votes
3answers
97 views

Derivative of $(-1)^x$

I'm taking a summer calc 2 class and we're getting into alternating patterns. I was interested in seeing the graph of $(-1)^x$ so I typed it into my TI-84 for $y = (-1)^x$. Surprisingly, the graph is ...
0
votes
1answer
34 views

Continuously differentiable operator

I consider an operator $A:H^1_0\to H^1_0$ defined by $$Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$$ where $$ G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to know what ...
2
votes
2answers
42 views

Why is 1 not a critical point for this function?

For the function $f(x) = \frac{x^2}{x-1}$, why is $1$ not a critical point, along with $0$ and $2$? Don't critical points include discontinuities?
1
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2answers
29 views

Solving the logistic equation

Please help to solve the following logistic equation: $$y'=y \cdot (1-y), \ ∀(t>0 \land y(0)>0)$$
2
votes
4answers
54 views

Can we take out a constant while differentiating?

In the solved example above, rather than taking $a^2x^4$ together and differentiating $a^2 = 0$, we differentiated $x^4$ and took out $a^2$. Why? Couldn't we have differentiated $a^2$ and gotten the ...
1
vote
1answer
29 views

How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$

Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is: $$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$ where $u_i(x)$ is a value that indicate ...
-12
votes
1answer
50 views

Please, find derivative and type how to solve [on hold]

$F(x,y)=(3xy^3-x^2y^2)^3$ $F'_x(x,y)= ?$ can someone help me((
6
votes
2answers
2k views

Very high order derivative

I'm doing some basic calculus exercises on higher derivatives. But I'm stuck at a problem. The question is to find the 1469th derivative of $f(x)=x^{532}-5x^{37}-4$. I've read something about using ...
0
votes
0answers
15 views

Derivation of energy function

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( ...
2
votes
1answer
28 views

derivative of a recursive vector-valued function

I have a recursive vector-valued function $$\mathbf{y}(t)=\mathbf{W}\mathbf{y}(t-1).$$ To compute the derivative of $\mathbf{y}(t)$ with respect to $\mathbf{W}$, do I need to use the product rule? ...
1
vote
1answer
40 views

Verify $\frac {\partial B} {\partial T} =$ $\frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)}$$ The correct answer (I believe) ...
1
vote
0answers
12 views

Derivative of sum of two functional derivatives with different ranges

I have a functional of the the following form, $(o<a<1)$ : $F(g(x)) = \int_0^a \! g(x) x \, \mathrm{d}x. + \int_a^1 \! (g(x)-k)x^2 \, \mathrm{d}x. $ I want to find $ \frac{\partial ...
0
votes
0answers
27 views

Question about differentiability, definition and consequences.

Let $E,F$ be normed spaces, we say $f:E \to F$ is differentiable in $x_0\in E$ if there exist $Df(x_0) \in \mathcal{L}(E,F)$ such that $$\lim_{h\to 0}\frac{f(x+h)-f(x)-Df(x_0)(h)}{\|h\|}=0$$ or ...
0
votes
1answer
39 views

Prove or Disprove a Property of $f$

Let $f:[0,1]\mapsto\mathbb{R}$ be a differentiable function, prove or give a counter-example that there exists a $c\in[0,1]$ such that $\frac{4[f(1)-f(0)]}{\pi}=(1+c^2)f'(c)$ attempt: I tried to ...
-1
votes
0answers
27 views

5 point derivative filter [on hold]

n many of the paper it is said that 5 point derivative filter transfer function is given by $$H(z) = \left(\frac{1}{8T}\right)(-z^{-2} - 2z^{-1} + 2z + z^{2}).$$ But no one has given detailed ...
0
votes
1answer
47 views

How to find the value of $2g(1)+2f(1)-h(1)$?

If $$\lim_{ m\to\infty }{ \frac { x^{ m }f(1)+h(x)+1 }{ 2x^m+3x+3 } }$$ is continuous at $x=1$ and $g(1)=\lim_{ x\to0}(\ln x)^{ 2/\ln(x) }$ then how to find the value of $2g(1)+2f(1)-h(1)$? Assume ...
0
votes
3answers
56 views

Differentiate the Function $g(x)= \ln\ \frac{a-x}{a+x}$

$$g(x)= \ln\ \frac{a-x}{a+x}$$ $$\frac{dy}{dx}\ =\frac{d}{dx}\ \ln \frac{a-x}{a+x}$$ $$g'(x) = \frac{1}{\frac{a-x}{a+x}}\cdot\frac{1}{1}\ \ln\ \frac{a+x}{a-x}$$ $$g'(x)= \frac{a+x}{a-x}$$ This ...
0
votes
0answers
9 views

Continuity/Differentiability of parametric function

I'm having a problem in understanding the nature of this function:Continuity and Differentiatiability of the following function parametrically defined. $x=2t-|t-1|$ and $y=2t^2+t|t|$ Will it be ...
-1
votes
4answers
53 views

How to prove $f(x)$ is differentiable at $x=0$ [on hold]

A real valued function satisfies $$|f(x)| \leq x^{2}\quad \forall \quad x\in R $$ then how to prove f(x) is differentiable at x=0 ?
3
votes
4answers
342 views

Am I using the chain rule correctly?

I'm supposed to find $y'$ and $y''$ of this function: $$y=e^{\alpha x} \sin\beta x$$ This is what I have done so far: $$y'=e^{\alpha x}\sin\beta x\cdot \alpha x'\sin\beta x\cdot \sin'\beta x \cdot ...
0
votes
0answers
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
votes
1answer
39 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
votes
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
vote
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
29 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 $f$ ...
0
votes
2answers
23 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
votes
4answers
110 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: ...
1
vote
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
votes
0answers
48 views

First derivative of energy function [on hold]

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
votes
5answers
85 views

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

Differentiate the Function $f(x)= \sqrt{x} \ln x$
0
votes
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 $x\in\mathbb{R}^n$ ...
0
votes
2answers
33 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
votes
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
35 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
vote
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
vote
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
vote
3answers
50 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
50 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 ...
1
vote
3answers
28 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
votes
3answers
985 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
votes
2answers
125 views

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

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 $f(10)$ ?
2
votes
4answers
67 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
95 views

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

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
52 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
41 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. ...
-4
votes
1answer
40 views

Chain rule differentiation [on hold]

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?