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
9 views

$f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$, then $x \cdot \nabla f(x) = 2 f(x)$

Assume that $f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$ for all $x\neq 0$ and $\alpha > 0$. Then I want to prove that $x \cdot ...
0
votes
0answers
8 views

If $f: U \rightarrow \mathbb{R}^n$ differentiable such that $|f(x)-f(y)| \geq c |x-y|$ for all $x,y \in U$, then $\det \mathbf{J}_f(x) \neq 0$

Let $f: U \rightarrow \mathbb{R}^n$ be a differentiable function on an open subset $U$ of $\mathbb{R}^n$. Assume that there exists $c>0$ such that $|f(x)-f(y)| \geq c |x-y|$ for all $x,y \in U$. ...
5
votes
2answers
140 views

Is a differentiable function always integrable?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
0
votes
3answers
52 views

n-th order derivative of a function

Find the n-th derivative of the following functions: $y = x\sqrt{1+x^2}$ $y = \dfrac{x}{\sqrt{x-x^2}}$ All help will be appreciated. Thank you!
2
votes
1answer
19 views

Directional derivative of a function

Feel like I may have gone wrong somewhere with this question: Find the directional derivative of the function $f(x,y) = \displaystyle\dfrac{2x}{x-y}$ at the point $P(1, 0)$ in the direction of the ...
3
votes
2answers
66 views

Whats the derivative of $\sqrt{4+|x|}$ using first principle

Here is my attempt: $$f(x)=\sqrt{4+|x|}$$ $$f`(x) = \lim_{h\to0} \frac{\sqrt{4+|x-h|}-\sqrt{4+|x|}}{h}$$ multiplying by the conjugate: $$\lim_{h\to0} ...
1
vote
1answer
22 views

Differentiation of multivariate function with respect to another multivariate function [on hold]

How do I calculate the derivative $\frac{df(x,y)}{d(xy)}$, given that $x,y\neq 0$ and assuming that the derivative exists?
0
votes
1answer
22 views

Point of inflection and third derivative

In my textbook there is a confusing statement. If $f'''(ξ)=0 $ and $f''(ξ)\ne0$ then $ξ$ is inflection point. However this confuses me as it is contrary to book example and this. Also in class notes ...
0
votes
1answer
12 views

Differentiating a function that includes vectors using the chain rule

I am trying to differentiate the function: $$g(x) = f(3\vec k + x(\vec l + \vec k))$$ where $\vec k$ and $\vec l$ are in $\mathbb R^n$ and $x$ is in $\mathbb R$. I think I need to use the chain ...
0
votes
1answer
34 views

Evaluate the integral, and then take the derivative of it.

I'm mostly curious as to if the way I've went about solving this is correct, or if there is a more simple way to get the answer. So I first evaluated the top section And when I did that I got ...
0
votes
1answer
15 views

w = x² - y² + 3z² direction with no change in w

Consider w = x² - y² + 3z². At (1, 1, 1), what is the fastest rate of change for w? What is a direction along which there is no change in w? I know how to do the first part, since the fastest rate of ...
-3
votes
0answers
27 views

Am I doing it wrong? [duplicate]

$$\frac{\partial }{{\partial b}}\left( { - a\sum {{x^b}} } \right) = - a\sum {\frac{\partial }{{\partial b}}\left( {{x^b}} \right)} = - a\sum {\frac{d}{{db}}\left( {{x^b}} \right)} = - a\sum ...
0
votes
0answers
18 views

How should i go about proving an expression of this kind?

Lets say i have a complete bell polynomial that is equal to a summation such that $$ B_n(d_1,d_2,\cdots,d_n) = \sum_{k=0}^{n}[g(x)^{-k} h(k)] $$ Where $d_n = \frac{d^n}{dx^n}[f(x)\ln(g(x)]$ and ...
0
votes
0answers
31 views

Help with solving a derivative [duplicate]

$\frac{\partial}{\partial b}\ln{L}=\frac{n}{b}+\sum{\ln{x}}-\frac{\partial}{\partial b}(-a\sum{x^b})$ I couldn't derivate the last part of this function
2
votes
1answer
32 views

If $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$, then $f$ is a diffeomorphism

Suppose that $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ is a differentiable function such that $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$. (Note that $\mathbf{J}_f$ is ...
-2
votes
2answers
64 views

How to differentiate a series? [on hold]

I need help with this function. Can this function be differentiated? $\frac{\partial}{\partial b}\ln{L}=\frac{n}{b}+\sum{\ln{x}}-\frac{\partial}{\partial b}(-a^b\sum{x^b})$ I dont know how to last ...
0
votes
0answers
8 views

Matrix derivatives for the HJB and ARE relationship

How does one take the derivative of these matrix equations? (Backround:{My professor used them in the proof showing that the Hamilton-Jacobi-equation equivalently solves the free end-point ...
-1
votes
1answer
42 views

differentiating the ln function [duplicate]

I tried it but I couldn't solve it. I have to differentiate w.r.t a and b: $$\ln(L)=n\ln(a) + n\ln(b) + (b-1) \Sigma \ln(x) + (-a^b\Sigma x^b)$$ Sorry for bad English.Thanks for helping.
-1
votes
1answer
26 views

Behavier of function and its derivatives at infinty

If $\lim_{t\to \infty}(\phi(t))=0$ and $\lim_{t\to \infty}(\phi''(t))=0$ then can we say $$ \lim_{n\to \infty}(\phi'(t))=0$$ Can we have a $\phi(t)$ such that $\lim_{t\to \infty}(\phi(t))=0$ but ...
3
votes
1answer
25 views

Directly proving continuous differentiability

Let us say that we want to prove that a function $f: I \to \mathbb{R}$ defined on an open interval $I$ is continuously differentiable on $I$. One way to do this is to establish that $f'(x)$ exists at ...
2
votes
0answers
28 views

Derivatives 1, 2 and 3 and limits

A question for you. Show that if $\lim_{x\to+\infty} x\,f(x)=0$ and $\lim_{x\to+\infty} x\,f''(x)=0$ then $\lim_{x\to+\infty} x\,f'(x)=0$ Thanks ;)
0
votes
2answers
39 views

Can this inequality be solved with Mean value theorem

As my sub-assignment I have to solve inequality: $$ \ln\left(\frac{1}{x} + 1\right) -\frac{1}{x + 1} > 0 $$ If I understood MVT correctly, I should set $g(x)=\ln\left(\frac{1}{x} + 1\right) ...
2
votes
1answer
40 views

Does all function's domain stay the same\expands as we derivate them?

Lets define a funciton $f(x)$ with a domain of, lets say $a>x>b$. If I derivate this function, it's domain will always stay the same or expand? Or it can be "reduced"? Is that mean that $f'(x)$ ...
2
votes
1answer
97 views

How to find 50th derivative of $\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$?

I need to compute 50th derivative of $$\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$$ Of course I would not compute 50 derivatives. I want to find a certain regularity. And what I have: As ...
1
vote
1answer
35 views

How to show that $\lim_{x\to \infty}f'(x)=0$

Let $f$ be a real-valued, bounded, twice differentiable function defined on $(0,\infty)$ with $f'(x)\ge 0$ and $f''(x)\le 0$. Show that $$\lim_{x\to \infty}f'(x)=0$$ I understand $f: (0,\infty) ...
0
votes
1answer
21 views

Missing something about second derivative tests

I'm studying second derivative tests, concavity and inflection points in khan academy ...
0
votes
1answer
45 views

How to take the derivative of a matrix with respect to itself?

Could someone please explain how to take the derivative of matrix with respect to itself? $$\frac{\partial \textbf{X}}{\partial \textbf{X}}$$ where $\textbf{X}$ is an M x N matrix
0
votes
0answers
21 views

Do these statements prove this formula?

$$ \frac{d^n}{dx^n}[g(x)^{f(x)}] = g(x)^{f(x)} B_n(d_1,\cdots,d_n) $$ Calling $$ d_n = \frac{d^n}{dx^n}[ln(g(x))f(x)] $$ Since faa di bruno's formula states $$ \frac{d^n}{dx^n}[f(g(x))] = ...
0
votes
0answers
26 views

What is the point of reflection of this function

$$y = 3x(x+5)^{2/3}$$ Is there some kind of trick to simplify it?
1
vote
0answers
34 views

Proof Verification for $n2^{n-1} = \sum\limits_{k=1}^n k\binom{n}{k}$

$(1+x)^n = \sum\limits_{k=0}^n\binom{n}{k}x^k$ by binomial theorem $\frac{d}{dx}(1+x)^n =\frac{d}{dx}\sum\limits_{k=0}^n\binom{n}{k}x^k$ $n(1+x)^{n-1} = \sum\limits_{k=1}^n k\binom{n}{k}x^{k-1}$ ...
0
votes
1answer
16 views

Inverse of a function involving a Jacobian.

Why is it true that if the inverse of both $ \tilde{f} $ and $ f $ exists then: $$ \tilde{f}\left(\vec{x}\right) = [Df(x_{0})]^{-1} f(\vec{x}) $$ $$ \implies \tilde{f}^{-1}(\vec{x}) = ...
2
votes
0answers
29 views

Differentiability and Monotonic Functions

I just read proof from Royden of theorem: 'Every Monotonic functions are differentiable almost everywhere.' But proof use Vitali Covering Lemma. But Vitali Covering Lemma is based on fact if we assume ...
6
votes
4answers
94 views

Show that $\lim_{x \to +\infty}\left(f(x)+f'(x)\right)=0 \Rightarrow \lim_{x \to +\infty} f(x)=0$

How to show that $\lim_{x \to +\infty}(f(x)+f'(x))=0 $ implies $\lim_{x \to +\infty} f(x)=0$?
0
votes
0answers
18 views

Derivative of the detrminant map

Question : For $ v = (v_1, v_2) \in \mathbb R^2$ and $ w = (w_1, w_2) \in \mathbb R^2$, consider the determinant map $det : \mathbb R^2 \times \mathbb R^2 \rightarrow \mathbb R$ defined by $det ...
0
votes
1answer
20 views

prove integration formula relating to derivatives

Could any one help me solve this problem ? it is from Apostol's calculus volume 1
2
votes
0answers
33 views

Help differentiate long equation

I need to differentiate the following equation twice with respect to $\alpha$. It is a profile log likelihood equation, where I need the derivatives to get the information matrix. The equation is: ...
2
votes
2answers
27 views

Finite difference method

I wanted to ask something regarding the finite difference approximation. I used the finite difference to calculate the numerical derivatives of my function. The finite difference is given by the ...
3
votes
4answers
79 views

Derivation of the integral

Evaluate $$\large\frac{d}{dx}\int_{0}^{\large\int_0^{e^x}{\cos (s)\,\mathrm ds}}\sec(t^2)\,\mathrm dt$$ I got the answer to be $$e^x\cdot\sec(\sin^2(e^x))\cdot \cos(e^x)$$ but do not know if ...
0
votes
0answers
15 views

Maximum volume of an open box with a square base?

A box with a square base and an open top is to be made. You have 1200cm^2 of material to make it. What is the maximum volume the box could have? Here's what I did: 1200 = x^2+4xz; where x=length of ...
3
votes
1answer
75 views

If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$

Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints. I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} ...
0
votes
0answers
39 views

problem with deriving continuity equation

I am studying Aerodynamics, to be more precise, the fundamentals of Aerodynamics. The first law is the continuity equation, for which it is explained in the book that I am using. However, I wished to ...
0
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0answers
17 views

Parametric derivative of $x^2+y^2+sin(4x)+sin(4y)=4$.

I am trying to parametrize $x^2+y^2+sin(4x)+sin(4y)=4$. I need to find a way of taking the intersections between $x^2+y^2+\sin(4x)+\sin(4y)=4$, and $\tan(nx)$, as n increases from $0\le{n}\le{2\pi}$. ...
0
votes
0answers
14 views

Find a vector field $\mathbb{Y}$ satisfying $L_{\mathbb{X}}\mathbb{Y}=\mathbb{Z}$

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $\mathbb{X}=(1,y)$. Let $\mathbb{Z}$ be the vector field on $\mathbb{R}^2$ given by $\displaystyle \mathbb{Z}(x,y)= \bigg( ...
0
votes
1answer
17 views

$\frac{d}{dx} f(x)$ piecewise defined

my function is defined as follows: $\frac{1}{x} $if $x \not = 0$, $ 1 $ if $ x=0$. Does the $\frac{d}{dx} f(x)$ in $x=0$ exist?
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0answers
45 views

urgent help please solve this eqaution [on hold]

Solve the following equation: $$f' '(x) + 4 f(x)= 0$$ where $f(0)=1$ and $f'(0)=-1$
1
vote
4answers
56 views

Trig differentiation

Prove that there is a constant C such that $$ \arcsin{\frac{1-x}{1+x}} + 2\arctan (\sqrt{x}) = C $$ for all $x$ in a certain domain. What is the largest domain on which this identity is true? What ...
0
votes
0answers
21 views

Find the lie derivative of a particular integral

Let $\mathbb{X}=(1,y)$ be a vector field on $\mathbb{R}^2$. Let $\Phi_t$ be the flow of $\mathbb{X}$. The flow of $\mathbb{X}$ I have calculated to be $\Phi_t(x,y)=(x+t,ye^t)$ Given a function ...
0
votes
0answers
12 views

How to take derivative $||W^TWx-x||_2^2$ with respect to $W$? [on hold]

It bothers me for a long time... How to take derivative $||W^TWx-x||_2^2$ with respect to $W$? I could not get a very efficient representation for the gradient... Help.........
1
vote
1answer
29 views

Maple - Substitute into an expression involving derivative

I'm working with some matrices including derivatives like d/dt(x(t)). I need to replace this whole expression {d/dt(x(t))} with something like xdot. I have tried to use "subs", but it seems to refuse ...
0
votes
2answers
26 views

find the derivative of a function with more than one variable

I have a function $g(a)=f_i(x+a(y-x))$ where a$\in$$\Re$ and x,y$\in \Re^d$. How can I find the first and the second derivative of this function? The second part of the exercise is asking me to use ...