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

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Maximizing Likelihood Function.

Let, $$\mathbf y_i = \mathbf X_i\mathbf\beta + \mathbf Z_i\mathbf b_i+ \mathbf\epsilon_i,$$ where $\mathbf y_i\sim N(\mathbf X_i\mathbf\beta, \Sigma_i=\sigma^2\mathbf I_{n_i}+\mathbf Z_i \mathbf ...
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1answer
35 views

$n$th derivative of function $\frac{1}{(1-2x)^2}$

I am trying to find the $n$th derivative of the function $\frac{1}{(1-2x)^2}$. The first three are simple but I can't see a schema right now. \begin{align*} y^{\prime} & = \frac{4}{(1-2x)^3}\\ ...
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1answer
20 views

What is the full width of a peak of the function $F(X)=\frac{1+\cos((2N+1)πX)}{1+\cos(πX)}$

With $$1 + \cos \theta = 2 \cos^2 \frac{\theta}{2},$$ the function becomes $$f_n(x) = \left( \frac{\cos \frac{(2n+1)\pi x}{2}}{\cos \frac{\pi x}{2}} \right)^2.$$ It peaks at odd X integer values. ...
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2answers
43 views

How to find the set of values $S$ where $f$ is not differentiable?

Let's assume we are given an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and for the purposes of this question, let's assume we know nothing about the differentiability of $f$, i.e. we have no ...
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1answer
29 views

Derivative of x|x| at 0

I am trying to show that $f(x) = x|x|$ is differentiable for all $x \in \mathbb{R}$. By computing the prime derivative I get: $$f'(x) = |x|+x(|x|)'$$ I know that $(|x|)' = \begin{cases} 1 \ ...
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1answer
163 views
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Is there any solution to find a condition for $f(x)=a+bx^n+cx^2-dx>0$ to always hold true?

Okay, I am interested to know the criteria for a function to always hold $$f(x)=a+bx^n+cx^2-dx>0,$$ if it is given that $a, b, c>0$ and $n\in(-2,2)$ is some real number and $x>0$. My idea ...
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1answer
44 views
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Representation of the Fréchet derivative of $〈f,e_n〉$, where $f:H→H$, $H$ is a Hilbert space and $(e_n)_{n∈ℕ}$ is an orthonormal basis of $H$

Let $H$ be a $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $f:H\to H$ be Fréchet differentiable and $$f_n:=\langle f,e_n\rangle\;\;\;\text{for }n\in\mathbb N$$ ...
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2answers
81 views

Formula for the nth Derivative of a Differential Equation

I have the differential equation $$f'(x)=2xf(x)$$ With the initial condition that $f(0)=1$ I need to prove that the nth derivative evaluated at zero is equivalent to $n!/(n/2)!$ for even n. ...
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3answers
93 views

Real Analysis question on FTC, Integral

Let $g:[0,1] \rightarrow \mathbb R$ be a continuous function and assume that $$ \int_{0}^{1} g(x) \phi'(x) dx = 0 $$ for all continuously differentiable functions $\phi: [0,1] \rightarrow ...
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2answers
43 views

Question on Rolle's theorem involving roots

Use Rolle's theorem to show that $f(x)=x^3-\frac{3}{2}x^2+\lambda$, $\lambda \in \mathbf{R}$ never has 2 zeroes in $[0,1]$. I started by assuming that $\exists$ $2$ zeroes in$[0,1]$ Then ...
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1answer
22 views

Determine the Fourier series considering the derivative of a function

Let $f\left(x\right)=x^2+1$ on the interval $\left[-\pi,\pi\right]$, which is extended periodically to $\mathbb{R}$. I have calculated the Fourier series of $f$ to be ...
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4answers
647 views

Continuity of Derivative at a point.

Is it possible that derivative of a function exists at a point but derivative does not exist in neighbourhood of that point. If this happens then how is it possible. I feel that if derivative exists ...
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1answer
36 views

Differentiation Involving Determinant.

I have to compute the following differentiation : $$\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times ...
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1answer
71 views
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On an injective ring homomorphism from the ring of continuous functions to the ring of differentiable functions

Let $\phi : C \to D$ be an injective ring homomorphism such that $\phi(1)=1$, where $1$ denotes the constant function $1$ and $C,D$ are the rings of continuous and differentiable functions on ...
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0answers
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Problem finding the tangent plane and the normal line of an surface [on hold]

Good night, I have a serious problem when I try to find a tangent plane for the following surface at the point $P$: $$x^{2}+y^{2}+z^{2}=6, \hspace{4mm} P=(-1,-2,3).$$ I make this: $\nabla ...
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2answers
54 views

Derivative of $\log |AA^T|$ with respect to $A$.

What is the derivative of $\log |AA^T|$ with respect to $A$ where $|A|$ denotes the determinant of matrix A?
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4answers
667 views

Double obstructing wall problem, what is the optimal walk path and length?

Every day, you walk from point A to point B which are exactly $2$ miles apart straight line distance, however, each day, there is a $50$% chance of there being an obstructing wall perpendicular to the ...
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0answers
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Factoring $R(ry')'-y(rR')'=[r(Ry'-R'y)]'$

In a problem this formula was used and I'm not seeing how this factor using the chain rule was derived. Other than calculating the derivative of the two that someone else already solved and showing ...
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2answers
26 views

Relative Extrema - First-derivative test of : $f(x)=x^5-5x^3-20x-2$

Find the relative extrema of the function by applying the first-derivative test: $$f(x)=x^5-5x^3-20x-2$$ So I found the $f'(x)$ $$f'(x) = 5x^4-15x^2-20$$ Now, I'm trying to find the critical ...
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2answers
37 views

Partial Derivative of $xy^2+yz^2+xyz+x^2y^2z^2=5$

Someone can tell me what the Partial Derivative of $\frac{d^2z}{dy^2}$ of function $z(x,y)$ if it`s look like this: $$xy^2+yz^2+xyz+x^2y^2z^2=5$$ I try to solve the first derivative: ...
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2answers
62 views

When is $\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$ a valid approximation?

It is often said that when the change in e.g. $\Delta x$ is small than we can make the approximation: $$\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$$ But it is not enough to say $\Delta x$ is small ...
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1answer
41 views

If $f\colon\mathbb{R}\to\mathbb{R}$ satisfies $\lvert f(x)\rvert\le x^2$ for every $x\in\mathbb{R}$, then $f$ is differentiable at 0.

If $ f\colon \mathbb{R} \to \mathbb{R}$ satisfies $\lvert f(x)\rvert\le x^2 $ for every $x \in \mathbb{R} $, then $f$ is differentiable at $0$. The solution provided uses delta-epsilon to prove ...
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1answer
74 views

Pushforward of a vector

The push forward of vectors allows to transform the components of a vector $X$ to be pushed along a map $h:\mathcal{M}\to\mathcal{N}$ between the manifolds $\mathcal{M}$ and $\mathcal{N}$. This is ...
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1answer
38 views
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Surjectivity of derivative of a vector valued function

Let $f:\mathbb R^3\to \mathbb R^3$ be a function such that $f(x,y,z)=f(x+y,0,x+z)$ for all $(x,y,z)\in \mathbb R^3$. I want to prove that $f^{'}(x)$ can never be onto for all point $x\in \mathbb R^3$ ...
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1answer
41 views

How the derivatives are different if sign changes.

I have this expression $$\frac{1}{(1 - x) ^ 2}$$ I need the derivative of this expression. So I calculated it, no big deal. However something has crossed my mind. Mathematically $(1 - x) ^ 2 = (x - 1) ...
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Monotony and convexity of $U(t) = w(t) - t. w'(t)$

Let $\mathcal{D} = (\mathbb{R}^{+*})^2$. $c \in ]0,1[$. Moreover we have $\theta < 1$ and $\theta \ne 0$. We consider the following function (which is called CES or constant elasticity of ...
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1answer
40 views

To check differentiability of function [on hold]

A function $f: [0,3]\rightarrow \Bbb{R}$ is defined by $$f(x) = |x| + |x-1| + |x-2| + |x-3|\quad \forall x \in [0,3]$$ The number of points in $[0,3]$ where $f$ is not differentiable is
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3answers
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$\frac{d}{d\theta}\mu=?$, where $\theta=\log\frac{\mu}{1-\mu}$ [on hold]

How can I solve the following differentiation : $$\frac{d}{d\theta}\mu,$$ where $\theta=\log\frac{\mu}{1-\mu}$ ?
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1answer
346 views

Tetrahedron volume in the first octant

The surface is given: $xyz = 2$ It is in the first octant so $x > 0, y > 0, z > 0$. The tangent plane taken at any point of this surface binds with the coordinate axes to form a ...
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Translating Logistic Regression loss function to Softmax

I currently have a program which takes a feature vector and classification, and applies it to a known weight vector to generate a loss gradient using Logistic Regression. This is that code: ...
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4answers
126 views

Is the function $y(t)$ is a solution of the equation $y'=\sin(yt)$?

Is the function $y(t)$ a solution of the equation $y'=\sin(yt)$? any thought to start me up? I'm not sure what is the question asking. EDIT: Someone tell me if I'm correct or not . If I'm finding ...
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1answer
27 views

Mean-value Theorem $f(x)=\sqrt{x+2}; [4,6]$

Verify that the hypothesis of the mean-value theorem is satisfied for the given function on the indicated interval. Then find a suitable value for $c$ that satisfies the conclusion of the ...
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2answers
76 views

Why does $\lim_{h \to 0^-} \frac{f(x+h) - f(x)}{h} \neq \lim_{h \to 0} \frac{f(x-h) - f(x)}{h} $

I realize that the only reason one-sided limits arise is as a result of the $\epsilon-\delta$ definition of a limit, applied to the real field $\mathbb{R}$, and that one-sided limits aren't even well ...
2
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2answers
392 views

Second derivative of a composite function

Say, we have three Banach spaces $X, Y, Z$ and $g:X \to Y, \ \ f:Y \to Z$ are twice (Fréchet) differenciable. The question is: what is $(f \circ g)''$? Since $(f \circ g)'':X \to ...
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1answer
10 views

Finding the Components of a Hessian Matrix of a Quadratic Form

I'm trying to find the Hessian form of the following quadratic form: $f(x,y) = x^2y+y^2+xy$. I know that it's in the form of a matrix and that the elements of $H_f(a)_{i,j}=\dfrac{\delta^2f}{\delta ...
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2answers
37 views

Differentiate a Function (Help me Solve?!)

Find $\dfrac{d}{dx}$ for: $C(1+Ae^{-bt})^{-1}$ I have tried and arrived at: $-C(1+Ae^{-bt})^{-2}$ however that is not the correct answer.
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5answers
153 views

Finding the value of x at which the tangent to the curve is parallel to the x axis

I have thoroughly searched up how to attempt this question. However, I am not sure if my answer is correct or if I even attempted the question correctly. Assistance would be greatly appreciated! ...
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1answer
33 views

Local Immersion Theorem in $\mathbb{R}^n$ proof

I am trying to prove the following: Let $U \subset \mathbb{R}^n$ be open and $f \in C^1(U;\mathbb{R}^m)$. Let $x^\star \in U, \ y^\star = f(x^\star)$. Suppose that $\mathrm{d}f(x^\star)$ ...
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4answers
58 views

Dividing derivatives by derivatives

We are often taught that $$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}$$ Why are we allowed to say this? What about the case of higher derivaitves, i.e. ...
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2answers
38 views

Extrema Where the Derivative is Undefined

Say we are given the derivative of a function say, $$f'(x)=\begin{cases} 5 & x<3 \\ -5 & x>3 \end{cases}$$ Notice that the derivative has opposite signs on either side of $x=3$, so you ...
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0answers
72 views

Derivative of this function

Let $f : S^{n-1} \subset \mathbb{R}^n \to \mathbb{R}^n \setminus \{0\}$ be a differentiable mapping, $n \geq 2$, and consider the function $F = \frac{f}{\|f\|} : S^{n-1} \to S^{n-1}$. I calculated the ...
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1answer
33 views

Prove Two Functions are Simultaneously Continuous

Let $f,g,h: \mathbb{R} \rightarrow \mathbb{R}$ so that $f$ is differentiable, $g,h$ monotone and $f'=f+g+h$. Prove that $g$ is continuous in $x_0$ iff $h$ continuous in $x_0$. My ...
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1answer
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Derivative Optimization Problem [duplicate]

I need help with finding the area of the largest rectangle in an ellipse from $y^2 + (x^2)/4 = 1$. I got it to y = $\sqrt{ 1 - (x^2)/4}$ but then I don't really know what to do, please help.
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4answers
78 views

Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable

Problem : Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable? For the sake of generality, let's assume that it is unknown to us that $|x|$ is not differentiable at $x = 0$ ...
0
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1answer
41 views

Elastic curves - What is wrong about my solution?

Given a curve $\gamma:\mathbb R\to\mathbb R^2$ with $\Vert\gamma'\Vert=1$ and curvature $\kappa(s)=\frac{c}{\cosh s}$, $c\in\mathbb R$, how can I show that $\gamma$ is an elastic curve for some ...
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1answer
43 views

Proving a function $f$ is not differentiable at an unkown point $a$

Let's say I have an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and I want to prove that it is not differentiable at some unknown point $a$. Emphasis must be placed on the unknown part as that ...
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1answer
47 views

Alternate definition of differentability at a point

Usually in most introductory Calculus courses, a definition of differentiability at a point $a$ is defined, as follows : A function $f$ is differentiable at $a$ if $f'(a)$ exists As a corollary ...
2
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1answer
54 views

Where did I go wrong in finding maximum?

The question in my book is given as: If $x^2+y^2+z^2=1$ for $x,y,z$ belongs to all real numbers ($x,y,z$ are independent), then find the maximum of $x^3+y^3+z^3-3xyz$. What I tried: As all ...
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0answers
20 views

Determine differentiability in region

Is there any test to indirectly determine if a function f is differentiable everywhere in a defined domain, for example $S=\{x|x\in\mathbb{R}\land x^2<4\}$? For example: let $f(x)=\frac{2}{x}$. We ...