Questions tagged [derivatives]
Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
24,412
questions
1
vote
0answers
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 ...
0
votes
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 $1=4x.$ ...
0
votes
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.
...
0
votes
0answers
11 views
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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
0
votes
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 ...
-1
votes
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?
$$\begin{array}{l}
f(x)=\left\{\begin{array}{ll}
5 x-4 ; & 0<x \...
0
votes
1answer
25 views
Find a function satisfy certain condition
Find function $f:\mathbb{R}\to\mathbb{R}$ such that $f(2)=2$ and
$$\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}}}{...
1
vote
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, ...
1
vote
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. ...
0
votes
0answers
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 $y''$
...
2
votes
0answers
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 \...
0
votes
0answers
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 ...
0
votes
0answers
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}-...
0
votes
0answers
20 views
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^...
1
vote
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
$$ X(t)=e^...
-1
votes
0answers
7 views
What is the maximum of the function in this feasibility?
g(x,y)=xy2+x2−4xy. The region is only feasible if y≥−1, x≤3 and y≤x.
I found the minimum in (2,2) but the other points were saddle points. Is there any maximum extrema in this region?
-2
votes
3answers
55 views
Evaluation of $\lim_{x \to 0} \frac{\sin(3x)-4\sin(2x)+5\sin x}{x^5}$ without l'hopital rule [closed]
$$\lim_{x \to 0} \frac{\sin(3x)-4\sin(2x)+5\sin x}{x^5}$$
How do you evaluate this limit without using l'hopital a tiring 5 times?
0
votes
0answers
26 views
Gradient descent with non-Lipschitz continuous gradients
In general, we know that for strongly convex functions for which we can compute the Hessian and find the Lipschitz constant $L$ of the gradient, gradient descent will converge provided that the step ...
0
votes
1answer
25 views
Is monotonicity a necessary condition for the inverse function theorem?
A textbook I was reading, Introduction To Real Analysis By Robert G. Bartle (page 169) states that the inverse theorem is defined as:
Let $I$ be an interval in $\mathbb{R}$ and let $f: I \...
0
votes
0answers
19 views
Prove that every power series centered at $a\in\textbf{R}$ whose radius of convergence is $R$ is differentiable in $(a-R,a+R)$.
Let $f:(a-R,a+R)\to\textbf{R}$ be the function
\begin{align*}
f(x) := \sum_{n=0}^{\infty}c_{n}(x-a)^{n}
\end{align*}
Then the function $f$ is differentiable on $(a-R,a+R)$, and for any $0 < r < ...
4
votes
1answer
57 views
Differentiation of an integral depending on a parameter
Let $f(t):=\int_0^{\pi/2} \arccos\frac{t-\tan^2x}{t+\tan^2x}\,dx$, for $0\leq t\leq 1$. I would like to differentiate $f$ with respect to $t$ by taking the partial of the integrand:
$$
f'(t)
= \int_0^...
0
votes
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 ...
0
votes
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 = ...
0
votes
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
\...
0
votes
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 ...
1
vote
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\...
1
vote
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!
0
votes
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 ...
0
votes
0answers
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 $...
0
votes
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 ...
1
vote
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
votes
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 ...
0
votes
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} $ $...
0
votes
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 ...
0
votes
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 ...
4
votes
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,...
0
votes
0answers
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 ...
0
votes
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 \ &(...
1
vote
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&...
0
votes
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 ...
1
vote
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 ...
0
votes
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?
0
votes
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 ...
2
votes
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 $a_0 = 0$...
0
votes
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 ...
-2
votes
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 ...
0
votes
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
votes
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 \...
