Questions tagged [derivatives]

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

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41 views

Let $f:[0,1] \to \mathbb{R}$ be continuously differentiable function

I was practicing calculus today and stumbled across this problem. I have tried solve using properties of inequalities but it doesn't get me far. Let $f:[0,1] \to \mathbb{R}$ be continuously ...
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2answers
15 views

Getting value of x by differentiating a given equation

If we consider an equation $x=2x^2,$ we find that the values of $x$ that solve this equation are $0$ and $1/2$. Now, if we differentiate this equation on both sides with respect to $x,$ we get $1=4x.$ ...
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2answers
12 views

Find parametric equations for the midpoint $P$ of the ladder

The following problem appears at MIT OCW Course 18.02 multivariable calculus. The top extremity of a ladder of length $L$ rests against a vertical wall, while the bottom is being pulled away. ...
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Is the 1st definition about the differentiability of $f : A \to \mathbb{R}^m$ equivalent to the 2nd definition?

It is convenient to define a function $f :\mathbb{R}^n \to \mathbb{R}^m$ to be differentiable on $A$ if $f$ is differentiable at $a$ for each $a \in A$. If $f:A\to \mathbb{R}^m$, then $f$ is called ...
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29 views

Directional derivative along the intersection of two surfaces

How can i find the intersection curve between these two surfaces $$ \left\{ \begin{array}{cc} 2x^2 + 2y^2 − z^2 &= 50\\ x^2 + y^2 -z^2 &= 0 \end{array} \right. $$ I need it to find the ...
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26 views

Can a tangent to a curve can also be its normal?

This question is related to applications of derivatives.I know the answers is yes but I can’t visualize the diagram
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1answer
35 views

Derivative determinant function of a matrix

Basically this is the problem I am trying to solve, I differentiated it but am pretty sure it isn't correct, used the formula: $$\frac{f(I+h) - f(I)-Ah}{h}$$ where A is the derivative and I is the 2 ...
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1answer
37 views

Is this function differentiable at x=1? [closed]

I have tried to prove differentiability using two different formulas but the results are different. Which is the correct way? $$\begin{array}{l} f(x)=\left\{\begin{array}{ll} 5 x-4 ; & 0<x \...
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1answer
25 views

Find a function satisfy certain condition

Find function $f:\mathbb{R}\to\mathbb{R}$ such that $f(2)=2$ and $$\sum_{i=1}^nc_i\frac{\partial(x_1^2+\dots+x_n^2)^{\frac{f(y)}{2}}}{\partial x_i}=\frac{\partial(x_1^2+\dots+x^2_n)^{\frac{f(y)}{2}}}{...
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1answer
22 views

Complex derivative of Hadamard product inside Frobenius norm

I'm trying to find the complex derivative of $$||R - P \circ \gamma \gamma ^H||_F ^2$$. with respect to $\gamma$. I saw the post regarding the real counterpart of the same question here. However, ...
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1answer
37 views

Central Difference Approximations

Hi Guys I was going through the different approximations which can be used for differentiation such as the forward difference, the backward difference and lastly the central difference approximations. ...
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34 views

Differential Equation using finite difference method

I am working on the following question $$y''+8(\sin^2 \pi y) y=0$$ where the initial conditions are $$y(0) = y(1) = 1$$ Now by the finite difference method i have made the substitution for $y''$ ...
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33 views

Why is electric potential function in free space infinitely differentiable?

Electric potential function in free space of a continuous charge distribution $\rho'$ distributed over volume $V' \subset \mathbb{R}^3$ is denoted by: $\psi (x,y,z): \mathbb{R}^3 \setminus{V'} \to \...
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22 views

Using Lipschitz to Prove Solutions are Continuable on R

$x ^ { \prime } = \cos \left( x ^ { 2 } \right)$ Given the above equation, I need to show that it determines a dynamical system. So, since this cannot be directly solved, I tried using Lipschitz to ...
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22 views

Solving $2 \alpha k=\omega^{2} \mu {\epsilon}^{''}+\omega \mu \sigma,k^{2}-\alpha^{2}=\omega^{2} \mu {\epsilon}^{'}$

I need help in this question please , i didn't understand it efficiently to solve it Solve for $k$ and $\alpha$ \begin{array}{l} 2 \alpha k=\omega^{2} \mu {\epsilon}^{''}+\omega \mu \sigma \\ k^{2}-...
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MCQ on Cauchy problem

$y u_x-xu_y=0,u=g $ on $ \Omega $ has a unique solution in neighborhood of $\Omega$ for every differentiable function g: $\Omega \rightarrow R$ if 1.$\Omega =\{(x,0):x>0\}$ 2.$\Omega =\{(x,y):x^...
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2answers
23 views

Is $dX/dt=X(t)$ the correct ODE for $X(t)=e^t$?

For a school project for chemistry I use systems of ODEs to calculate the concentrations of specific chemicals over time. Now I am wondering if $$ \frac{dX}{dt} =X(t) $$ the same is as $$ X(t)=e^...
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What is the maximum of the function in this feasibility?

g(x,y)=xy2+x2−4xy. The region is only feasible if y≥−1, x≤3 and y≤x. I found the minimum in (2,2) but the other points were saddle points. Is there any maximum extrema in this region?
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3answers
55 views

Evaluation of $\lim_{x \to 0} \frac{\sin(3x)-4\sin(2x)+5\sin x}{x^5}$ without l'hopital rule [closed]

$$\lim_{x \to 0} \frac{\sin(3x)-4\sin(2x)+5\sin x}{x^5}$$ How do you evaluate this limit without using l'hopital a tiring 5 times?
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Gradient descent with non-Lipschitz continuous gradients

In general, we know that for strongly convex functions for which we can compute the Hessian and find the Lipschitz constant $L$ of the gradient, gradient descent will converge provided that the step ...
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1answer
25 views

Is monotonicity a necessary condition for the inverse function theorem?

A textbook I was reading, Introduction To Real Analysis By Robert G. Bartle (page 169) states that the inverse theorem is defined as: Let $I$ be an interval in $\mathbb{R}$ and let $f: I \...
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19 views

Prove that every power series centered at $a\in\textbf{R}$ whose radius of convergence is $R$ is differentiable in $(a-R,a+R)$.

Let $f:(a-R,a+R)\to\textbf{R}$ be the function \begin{align*} f(x) := \sum_{n=0}^{\infty}c_{n}(x-a)^{n} \end{align*} Then the function $f$ is differentiable on $(a-R,a+R)$, and for any $0 < r < ...
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1answer
57 views

Differentiation of an integral depending on a parameter

Let $f(t):=\int_0^{\pi/2} \arccos\frac{t-\tan^2x}{t+\tan^2x}\,dx$, for $0\leq t\leq 1$. I would like to differentiate $f$ with respect to $t$ by taking the partial of the integrand: $$ f'(t) = \int_0^...
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0answers
75 views

Compute the derivative of the determinant function on 2x2 matrices

Using the definition of the derivative and limit, compute the derivative of the determinant function on 2x2 matrices at the identity (Which we consider as a subset of $\mathbb{R}^4$ under the ...
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1answer
17 views

Help with differentiation chain rule with tensors in backpropagation

Say, we're given $N$ feature vectors $\mathbf{x}_i \in \mathbb{R}^{D \times 1}$ and assembled into a matrix $X \in \mathbb{R}^{D \times N}$. We also have a matrix $W \in \mathbb{R}^{D \times D}$, $W = ...
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0answers
20 views

Derivative for a piecewise defined function and continue its definition.

$f: \mathbb{R} \rightarrow \mathbb{R}$ $ f(x) = \begin{cases} 2e^{-x}-1, & \text{for } x \leq 0 \\ x^2+1, & \text{for } 0 < x < 2 \\ x^3-3, & \text{for } x \geq 2 \...
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1answer
34 views

rate of change in natural log $\ln(x)$

I have confusion with the rate of change in natural logarithm, as we know that, in analytical manner \begin{gather*} y\ =\ \ln( x)\\ \\ \frac{dy}{dx} \ =\ \frac{1}{x}\\ \end{gather*} or in a ...
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2answers
48 views

Show that $f(x) = |1-x^2|^3$ is a differentiable function.

$f: \mathbb{R} \rightarrow \mathbb{R}$ $x \rightarrow |1-x^2|^3$ Show that f is a differentiable function and calculate its derivative. Check whether f if is a continuous function. $f'(x_0)=lim_{x\...
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2answers
39 views

How many derivatives does $ f=\sum_{k=0}^\infty e^{-k}\cos(kx) $ have?

How many derivatives does $f$ have? $$ f=\sum_{k=0}^\infty e^{-k}\cos(kx) $$ To be honest, I'm stuck right at the beginning. So, I would appreciate any clue!
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1answer
21 views

Calculating sum of series using derivative of a function

We're given the following problem: "We know that $\frac{1}{1 - x} = \sum_{k=0}^{\infty} x^k $ for $ -1 < x < 1 $. Using the derivative with respect to $x$, calculate the sum of the following ...
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18 views

Minimization of Quartic Function

Hi I am dealing with an optimization problem with quartic function: $$x = \mathop{argmin}\limits_{x\in \mathbb{R}+} x^4 + (\frac{\alpha}{2} - 2y) x^2 - d\alpha x +y^2 + \frac{\alpha d^2}{2}$$ , where $...
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0answers
40 views

Derivative of a unit vector

Consider a vector function $r: \mathbb{R} \to \mathbb{R}^n$ defined by $r(t)$. We use $\hat{r}$ to denote its normalized vector, and $\dot{r}$ to denote $\frac{d}{dt}r(t)$. We know that the derivative ...
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1answer
11 views

How to prove that $\wp''$'s zeros are not at half-periods?

This is an exercise adapted from Apostol. The problem is stated as Prove that $$\wp''\left(\frac{\omega_1}{2}\right)=2(e_1-e_2)(e_1-e_3)$$ where $\omega_1,\omega_2$ generates the lattice for $\...
2
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1answer
32 views

$f(0)=f(1)=0$, $f(x)=\frac{f(x+h)+f(x-h)}{2}$ implies $f(x)=0$ for $[0, 1]$

Question: Suppose $f$ is continuous on $[0, 1]$ with $f(0)=f(1)=0$. For $\forall x\in (0, 1)$, there $\exists h>0$ with $0\le x-h<x<x+h\le1$ such that $f(x)=\frac{f(x+h)+f(x-h)}{2}$. Show ...
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4answers
52 views

What is the reason for the implication $2x \lt 2y \Rightarrow 2x + \sin x \lt 2y + \sin y$?

I'm trying to prove that the function $$ f: \mathbb{R} \rightarrow \mathbb{R} $$ $$ x \rightarrow 2x + \sin x $$ is strictly increasing and then injective. So for $x,y \in \mathbb{R} \phantom{2} $ $...
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0answers
14 views

Caratheodory differentiablity vs. continuously differentiable

Here is the definition of Caratheodory differentiability: Let $I\subseteq \mathbb{R}$ be an open interval, $a\in I$, $f:I\to\mathbb{R}$. We say $f$ is Caratheodory differentiable at $a$ if there ...
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1answer
39 views

Sum rule for infinite sums

Why does the sum rule of differentiation fail sometimes for an infinite sum.(In terms of the derivation of the sum rule.)An example where it fails is here: Interchanging the order of differentiation ...
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2answers
85 views

Constant function composition

I have the following Problem: Let $g \in C^1(\mathbb{R^2};\mathbb{R})$. Show that an injective $f \in C^1 ((-1,1);\mathbb{R^2})$ exists, so $g \circ f$ is constant. The hint asks if there is a $(x,...
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35 views

Definition of differential equations [closed]

If we consider the LDE to be of the form $$a_n(t)y^{(n)}(t)+a_{n-1}(t)y^{(n-1)}(t)+......+a_1(t)y^{(1)}(t)+a_0(t)y(t)=g(t)$$ and a non-LDE includes terms which have the dependent variable as ...
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1answer
78 views

Homework Assignment, Function Totally Differentiable, Correct Solution?

We are given the following function: \begin{align*} f: \mathbb{R}^2 &\rightarrow \mathbb{R} \\ f(x,y) &= \begin{cases} {x^3y^4 \over x^6 + y^4} \ & (x,y) \neq (0,0) \\ 0 \ &(...
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1answer
64 views

Inverse of Laplace transform of $\mathscr{L}\{f(t)\}=\frac{1}{s^{2}}\tanh\left(\frac{s}{2}\right)$

I can't find its inverse transform, I had thought in $$f(t)=\begin{cases} t \space\space\space\space\space\space\space\space\space\space\space\text{ if }0 \leq t<1,\\ -t+2 \space\text{ if }1\leq t&...
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1answer
17 views

Uniqueness of the Frechet Derivative: the role of $x \in int_X(T)$

I'm currently trying to learn some functional analysis as a way to improve my ability to read economic theory papers. I've come across what I thought was a simple proof but on reflection I don't think ...
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0answers
33 views

Interchange of derivatives and integrals

I am trying to justify the following exchange of derivatives and integrals. Suppose that $s, \theta \in [a, b] \subset \mathbb{R}$. Let $\pi_\theta(s) = \max [0, \theta - s]$, so it is continuous and ...
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1answer
45 views

Why isn't $f(x)=0$ ever mentioned as a solution to $f'(x)=f(x)$?

I know that $f(x)=e^x$ is the accepted and useful solution to $f'(x)=f(x)$, but why isn't $f(x)=0$ ever mentioned as a solution as well? Is it simply because it's not useful?
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0answers
23 views

Definition of slope of parametric tangent

Let $x = u(t)$ and $y = v(t)$ be a parametric representation of a curve. If we assume $u$ is one-one on some open interval in $t$, we can define $f = v \circ u^{-1}$. Then the curve $(u(t), v(t))$ on ...
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3answers
47 views

Show that not exists any polynomial function such that $f(x) = \log (1+x)$. [duplicate]

Does anyone have any idea on that problem? Let $f : \mathbb{R} \to \mathbb{R}$ be a polynomial function. Show that not exists any $f$ such that $f(x) = \log (1+x)$. It's easy to show that $a_0 = 0$...
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0answers
29 views

Help with differentiability proof for piecewise function

I was able to prove that $f(x)$ is continuous only when $x$ is $0$ or irrational. But I'm unable to prove that it is not differentiable at any point (which is what the book's answer says). Can someone ...
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1answer
87 views

Why can $e^x$ be defined as the unique function $f(x)$ such that $f(x)=f'(x)$ and $f(0)=1$?

The definition that $e^x$ is the unique function $f(x)$ such that $f(x)=f'(x)$ and $f(0)=1$ has two problems for me: How is $e^x$ the unique function that satisfies this property? $ke^x$ also has ...
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0answers
28 views

Applicability of L'Hôpital's rule

The derivation of L'Hôpital's rule requires Cauchy's theorem, which, in turn, requires the following conditions: two functions $f(x)$ and $g(x)$ must be continuous and differentiable in an interval $[...
2
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1answer
69 views

How to visualize Euler's number?

I am interested if there is geometric meaning (using graphs) of $(1 + \frac{1}{n})^n$ when $n \rightarrow \infty$. Also, is there visual explanation of why is $e^x = (1 + \frac{x}{n})^n$ when $n \...

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