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

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Second derivative of function in matrix form

Given an equation \begin{equation*} log(L_{c}(n|Z^{*}n)) =log\left(\frac{\displaystyle\prod_{k=1}^{K}\frac{(m_{k}^{*})^{n_{k}}\exp(-m_{k}^{*})}{n_{k}!}} ...
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3answers
52 views

Finding Derivatives $f(x)={1\over x+1}$

I'm using the Limit Definition to find the derivative, $$f'(x)=\lim_{\Delta x \to 0} {f(x+\Delta x) - f(x) \over \Delta x}$$ $$$$ Now, I want to find the derivative for the function, $$f(x)={1 \over ...
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10 views

Show that the function $B(A,y)=Ay : M^{3x3}xR^3 \to R^3$, where $M^{3x3}$ is the space of square matrices 3x3, is a bounded bi-linear function.

And also find the first derivative of the function. $\|B(A,y)\| \over {\|A\|\|y\|}$$\leq 1$ therefore bounded? And first derivative$ B(x+H1,y+h2)-B(x,y)= A*h2 + H1*y+H1h2$ now i thnik i can take ...
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2answers
48 views

Prove that f(x)=C1sinx +C2cosx for constant C1 and C2…

It's given that f is differentiable twice and that $f''+f=0$ I have to show that $f(x)=C_1\sin(x) +C_2\cos(x)$ for constant $C_1$ and $C_2$. There is also a hint: using the given data, prove that ...
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2answers
65 views

prove $f$ is a constant [duplicate]

Lets's say we have a differentiable function $f:[a,b]\to \mathbb{R}$ with $f^\prime\equiv0$ How do I show that $f\equiv C$ by using the mean value theorem?
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4answers
62 views

$k'=k$ only for $e^x$ [duplicate]

How can one prove without using anything but differentiation, that $e^x$ is the only function with $f'=f$? Clearly I can prove that $(e^x)'=e^x$, and $0'=0$, but how can one show that no other ...
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36 views

How to solve this “Coupled Nonlinear Differential Equations”

First, I'm sorry to bring the formula as picture. I'm not that experienced with formula tags in written. BTW, I need your help with solving this "coupled nonlinear partial differential equations". ...
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2answers
304 views

Optimization—Finding the Area of the Largest Isoceles Triangle

I managed to solve $(a)$. Since the area of a triangle is determined by $\frac{1}{2}$ base $\times$ height, and we already know the height, we just have to solve for the base. Using Pythagorean ...
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1answer
66 views

Matrix Derivative of this Equation

I'm trying to solve this minimization problem: $$ \min_{\Theta} \frac{C_1}{2} \sum_{j=1}^{N-1} \|\vec{\theta_{j+1}} - \vec{\theta_j}\|^2 ,$$ where $\Theta = (\vec{\theta_1}, \vec{\theta_2}, \ldots, ...
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23 views

What does the matrix derivative of this equation look like?

I'm trying to solve this minimization problem: $$ \min_{\Theta} \frac{C_1}{2} \sum_j^N ||\vec{\theta_j}||^2 $$ where $\Theta = (\vec{\theta_1}, \vec{\theta_2}, ..., \vec{\theta_N})$. (FYI, it's ...
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1answer
36 views

partial derivative of $f(X(t),t)$ with respect to $t$

Suppose that $f(x,t) = x^2$. Clearly, $\frac{\partial f}{\partial t} = 0$. However, let us now consider $f(X(t),t) = X(t)^2$. The book I am reading claims that $\frac{\partial f}{\partial t}(X(t),t) ...
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0answers
26 views

Derivatives : trouble to understand formulas

My teacher gave us some useful formulas, but honestly I don't know how to understand it. gradient of a scalar field : $d_{x}i{V^{i}}f(M)\varepsilon ^{i}$ gradient of a vector field : ...
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1answer
36 views

laplace transformation solve heaviside d.e. $y''+2y'+y=2(t-3)U(t-3)$ given $y(0)=2$ and $y'(0)=1$

$y''+2y'+y=2(t-3)U(t-3)$ given $y(0)=2$ and $y'(0)=1$ I did the transformation and obtained $Y=e^{3s}(\frac{1}{s^2}-\frac{2}{s}-\frac{1}{s^2}+\frac{2}{s+1})+(\frac{3}{(s+1)^2}+\frac{2}{(s+1)})$ This ...
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2answers
59 views

Is it differentiable at $x=(0, 0)$?

Let $ \displaystyle f(x, y)=\frac{x^3-y^3}{x^2+y^2} $ be a multivariable function. Examine if it is differentiable at $x=(0,0)$. I proved that the limit of the partial derivatives at $x=(0, 0)$ are ...
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1answer
45 views

laplace transformation solve heaviside d.e. $y''+4y=U(t-4)$

$y''+4y=U(t-4)$ so that $y(0)=3$ and $y'(0)=-2$ I have applied the transformation in both terms obtaining $Y=\frac{3s^2+10s+1-e^{4s}}{s(s+4)}$. How can i solve it?
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14 views

use laplace transformation to solve $y^{iv}-16y=0$, being $y(0)=1$, $y'(0)=0$, $y''(0)=0$, $y'''(0)=0$

Folowing the process, i came to $Y=\frac{s^3}{s^4-16}$ However, when trying to write the fraction as a sum of other fractions,the system is undetermined. ...
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1answer
56 views

Regarding the derivative of the $j$-invariant

Is anyone aware of a formula for the derivative of the $j$-invariant $j(\tau)$ with respect to $\tau$? Here, $\tau$ is in the upper half-plane. I would image there are probably quite a few formulae ...
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2answers
48 views

A question about two common definitions

Two definitions make me puzzled ! 1. The definition of $\textbf{Functions Differentiable at a Point}$: A function $f$ defined in a neighborhood $(x_{0}-\delta,x_{0}+\delta)$of a point $x_{0}$, ...
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8answers
265 views

Intuitively, why should the coefficient of the derivative of $x^n$ be $n$?

I am able to differentiate $x^n$ with respect to $x$ from first principles using the definition of differentiation. Also it seems natural that the gradient of a finite polynomial will be one order ...
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183 views

Why isn't $f(x) = x\cos\frac{\pi}{x}$ differentiable at $x=0$, and how do we foresee it?

Consider $$f(x)=\begin{cases} x\cos\frac{\pi}{x} & \text{for} \ x\ne0 \\ 0 & \text{for} \ x=0. \end{cases} $$ Its difference quotient ...
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1answer
152 views

For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $

Suppose that $f$ is continous on $[0,1]$ , differentiable on $(0,1)$ , and $f(0)=0$ and $f(1)=1$.For every integer $n$ show that there must exist $n$ distinct points ...
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2answers
72 views

Function such that $f^{(n)}(0)=\frac{(n!)^2}{(2n+1)!}$

I was trying to solve another problem and come up with the problem if there is a function with closed form such that $$f^{(n)}(0)=\frac{(n!)^2}{(2n+1)!};(n\ge1).$$ I tried to check the condition for ...
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1answer
52 views

Visualization of relation between integration and derivative operations [duplicate]

If I have a function $f(x)$ and I find the derivative I will get $f'(x)$. Furthermore, if I do the integration of the derivative $f'(x)$, as a result I will get again my original function $f(x)$. ...
2
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3answers
102 views

Derivative of a continuous funtion

Let $g:R\to R$ be a continuous function with $g(x+y)=g(x)+g(y), \forall x,y\in R.$ Find $\frac{dg}{dx},$ if it exist.
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2answers
78 views

Determine all real polynomial solutions y of a differential equation

Determine all real polynomial solutions y of a differential equation $$y'(x) = 5x^7 + 4x^5 + 3x^3 + x + 8$$ for all real numbers $x$. Any hints for starting this would be greatly appreciated.
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3answers
93 views

inverse laplace transformation of $\arctan(\frac{4}{s})$

inverse laplace transformation of $\arctan(\frac{4}{s})$ using I was trying use 12 but i couldn't arrive to a solution
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1answer
49 views

Differentiabily of a complex valued function

The function $f(z)=|z|^{2}+i.\bar{z}+1$ is differentiable at (a) $i$ (b) $1$ (c) $-i$ (d) no point of $\mathbb C$. We know the derivative of $\bar z$ does not exists at any point. So the ...
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0answers
70 views

laplace transformation $\cos^2(3t)$ and $\sin(5t)cos(2t)$

it is asked to transform $\cos^2(3t)$ and $\sin(5t)cos(2t)$ using the results from i think the process might be similar for both of them but i don't know wich result to use. can you help me? ...
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0answers
61 views

Practical calculation of the derivative of the integral of a function

In this Question, I wanted to make sure that $$\frac{d}{dt} \left(\int_0^{t} \phi(t)dt \right) = \phi(t),$$ and provided $\phi(t)$ is continuous, and that a derivative of the function exists, it was ...
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2answers
76 views

Laplace tranform of $t^{5/2}$

It is asked to transform $t^{5/2}$. I did $t^{5/2}=t^3\cdot t^{-1/2}$. Then followed the table result $$L\{{t^nf(t)}\}=(-1)^n\cdot\frac{d^n}{ds^n}F(s)$$ However i got $\frac{1}{2} \cdot\sqrt\pi ...
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3answers
207 views

What is the derivative of the integral of a function?

Is this correct ? $$ \frac{d}{dt} \left( \int_0^t \phi(t)dt \right) = \phi(t) $$ If not, how can I recover $$ \phi(t) $$ knowing only $$ \int_0^t \phi(t)dt $$ ?
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2answers
34 views

derivative of $\frac{d}{dn}(1+\epsilon/2n)^n.$

I need to show that derivative of $\frac{d}{dn}(1+\frac{\epsilon}{2n})^n > 0.$ I use formula $(a^x)' = a^x\ln x.$ For now i have: $\frac{d}{dn}(1+\frac{\epsilon}{2n})^n = ...
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1answer
79 views

if $f(x)$ is differentiable at a x, prove that: $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$

If $f(x)$ is differentiable at x, I need to prove that $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$ exist and is finite. so if $f(x)$ is differentiable at a $x$, the difference quotient exist for this ...
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1answer
303 views

Optimisation problem - circle and square

A piece of wire of length $20$cm is cut into $2$ parts. the first part is bent into a circle of radius $r$ in cm, the second into a square of side length $s$ in cm. a) write down an expression for ...
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1answer
34 views

Does $g'$ need to be continuous for $g(x_0) = 0$, $g'(x_0) \neq 0$ to imply $g$ changes sign in a neighborhood of $x_0$

The following theorem holds: Theorem: Let $g:\mathcal{A} \rightarrow \mathbb{R}$ be differentiable and let $x_0 \in \mathcal{A} $. If $g(x_0)=0, \; g'(x_0)\neq 0$ then $g$ changes sign at a ...
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1answer
49 views

diferential equation system differential operators method

$x'-3x+2y=t$ $y'+2x=e^t$ it is asked to solve by the mentioned method $\Delta(D)=D(D-5)$ $\Delta_1=1-e^t$ $\Delta_2=-2t-2e^t$ $yD^3(D-5)(D-1)=0$ $xD^2(D-5)(D-1)=0$ When solving for the ...
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4answers
48 views

Derivative of a composition of functions

The problem is as follows: Find $g^\prime (2),$ given that $g(x) = f(x^2 + 2)$ and $f(e^x) = \log(\sqrt{x}).$ The answer turns out to be: $\displaystyle \frac{1}{3\log6}$ I tried to use the chain ...
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1answer
67 views

How derivative relates to roots of original function

Assume $f$ is differentiable on $\mathbb{R}$. Show that for any $ k \in \mathbb{R}$, $f' + kf$ has a root between any two distinct roots of $f$. I am completely stumped on this. What are some good ...
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1answer
38 views

About derivative of the inverse function

I think I misunderstand something about derivative of the inverse function. Say we are transforming from (x,y) to (r, $\theta$), this requires calculating $\frac{\partial x}{\partial r}$. $$x=rcos ...
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1answer
66 views

Partial derivative with matrices

I have reforumulated my problem of computing some quantities $\mathbf{a}\in R^{m}$ from $\mathbf{b}\in R^{n}$ in a matricial form: $$\mathbf{b} = (C\odot(\mathbf{1}_{n}\cdot \mathbf{a}^{T}))\cdot ...
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3answers
164 views

How to solve given expression?

We know that the derivative $f'(1)=3$. $$ \lim_{h \to 0} \frac{f(1-5h^2)-f(1+3h^2)}{h^2(h+1)}=? $$ I try to solve it by applying L'Hôpital's rule, but answer was incorrect. Since $f$ is ...
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1answer
48 views

Construct a complete 3rd order ODE with constants coefficients knowing 2 particular solutions and one particular solution of the homogeneous equation:

Construct a complete 3rd order ODE with constants coefficients knowing 2 particular solutions of this equation: $y_2=\ln(x)$ $y_1=x+\ln(x)$ and one particular solution of the homogeneous equation: ...
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2answers
41 views

reduction order method $xy''-(1+x)y'+y=x^2e^{2x}$, $y_1=1+x$

reduction order method $xy''-(1+x)y'+y=x^2e^{2x}$, being $y_1=1+x$ a solution of the homogeneous equation. I made y=u(1+x) and got $u''(x^2+x)-u'(x^2+1)=x^2e^{2x}$ Then i did $u'=w$ and obtained ...
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2answers
56 views

If a differentiable function approaches $-\infty$ as a limit from the positive side, must its derivative simultaneously approach $\infty$?

Can we say that if $g: (0, \infty)\rightarrow\Bbb{R}$ is a differentiable function and $\lim \limits_{x \to 0+}g(x)= -\infty$, then $\lim \limits_{x \to 0+}g'(x)= +\infty$ is always true? I ...
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1answer
97 views

Solving Partial Differential Equation $ \Delta u = 4 $

How can I solve this equation: $ \Delta u = 4 \\ u(x,x)=2x^2 \\ u_x(x,x) = 2x$ where $u=u(x,y)$ using substitution: $ \Phi ^{-1}(s,t) = (x-y,y) $? My attempt to solve this: $v=u \circ \Phi ...
3
votes
1answer
58 views

Axes-intersections of normal tangents to an ellipse

Question: What values can $x_T$,$y_T$,$x_N$, and $y_N$ take on? Let $T$ and $N$ be the tangent and normal lines to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ at any point on the ellipse in the ...
0
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2answers
102 views

if f'(x) is odd then f(x) is even?

Im trying to prove but every proof I encounter can also prove that if f'(x) is even then f(x) is odd and this is not correct (x^3 + 1 for example) thanks!
2
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1answer
50 views

reduction order method $(2-x)y'''+(2x-3)y''-xy'+y=0$, $y_1=e^x$

$(2-x)y'''+(2x-3)y''-xy'+y=0$, $x<2$ being $y_1=e^x$ a solution for the homogeneous equation. making $y=ue^x$ i came to $u''+u'''=0$ making $u''=w$ , $w'+w=0$ this way $w=e^{-x}*c_1$ and ...
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0answers
22 views

Clarifications about SDEs, Differentials & Derivatives

A general SDE look like the following: $$ \mathrm{d}\psi=a\mathop{}\!\mathrm{d}t+b\mathop{}\!\mathrm{d}W,\tag{1} $$ where $\psi:t\mapsto y = \psi(t)$ is the solution, while $a$ and $b$ can be both, ...
0
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2answers
41 views

About normal derivative

Let $w:\Omega \subset \mathbb{R}^n \longrightarrow \mathbb{R}$, where $\Omega$ is an open, bounded, connected set, $w \in C^2(\Omega)\cap C(\overline\Omega)$ and $x_0 \in \partial \Omega$ such that ...