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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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Differentiating Dirac delta with product rule

I have here an equation. $$ h'(t_2) \delta(t_1 - t_2) = [h(t_2) - h(t_1)] \delta'(t_1 - t_2) $$ I checked the equality by integrating both sides with a test function. $$ \int d t_1 \phi(t_1) \ldots \...
Bio's user avatar
  • 922
0 votes
0 answers
15 views

Give an approximation of $f(x)$ given that $m\leq f'(x)-g'(x)\leq M$

Given 2 real differentiable functions on $(a,b)$ $f(x)$ and $g(x)$, give an approximation of $f(x)$ given that $m\leq f'(x)-g'(x)\leq M$ for some real numbers $m, M$ and $a>0$. Also, $g(x)$ is ...
MiguelCG's user avatar
  • 275
6 votes
0 answers
32 views

$f:\mathbb R\to [0,\infty)$ be a 3 times differentiable and $\max_{x\in\mathbb R}|f'''(x)|\le 1$. Prove that: $f''(x)+\sqrt[3]{\frac{3}{2}f(x)}\ge0 $

I have a problem in Analysis: Let $f:(-\infty,\infty)\to [0,\infty)$ be a three times differentiable and satisfy: $$\max_{x\in\mathbb R}|f'''(x)|\le 1$$ Prove that: $$f''(x)+\sqrt[3]{\frac{3}{2}f(x)}\...
Đạt Nguyễn's user avatar
3 votes
3 answers
112 views

Show that $\frac{x^3+y^3}{x^2+y^2}$ is not differentiable at $(0,0)$

Let be $f:\mathbb{R}^2\to\mathbb{R}$ where $$ f(x,y):=\begin{cases} \frac{x^3+y^3}{x^2+y^2},&(x,y)\neq (0,0)\\0,&(x,y)=(0,0).\end{cases} $$ Show that $f$ is not differentiable at $(0,0)$. My ...
Philipp's user avatar
  • 4,554
1 vote
0 answers
28 views

Taylor Approximation of Third Order

Problem: Given the function $ f(x, y) = e^{x^2} \log(1 + x + y) $ near the point $ (0, 0) $, find the third-order Taylor approximation of the function at $ (0, 0) $ using known series and verify the ...
j.primus's user avatar
-6 votes
1 answer
46 views

Let f : R → R be differentiable at x = 0. Define g(x) = f(x^2 ). Show that g is differentiable at 0 (by the chain rule). [closed]

Let $f : \mathbb{R} → \mathbb{R}$ be differentiable at $x = 0$. Define $g(x) = f(x^2 )$. Show that $g$ is differentiable at $0 $(by the chain rule).
Niyatee Mahobiya's user avatar
1 vote
1 answer
53 views

Study if the function $F(x)=\int_{0}^1D(xu)du$ is differentiable

Given the function $D(x)=\inf\{|x-n|:n\in \mathbb{N}\}$, study the differentiability of $F(x)=\int_{0}^1D(xu)du$. My try in $0$: $\lim_{h\to 0}\frac{F(h)-F(0)}{h}=\lim_{h\to 0}\frac{\int_{0}^1D(hu)du}...
MiguelCG's user avatar
  • 275
0 votes
1 answer
35 views

Solution of linear dynamical systems without Fourier transforming.

I'm trying to understand an exercise for my exam of mathematical methods for physics, in which it is asked to find the general expression of the Green function of the following system without using ...
deomanu01's user avatar
0 votes
2 answers
59 views

Taking the derivative with respect to a function of x but slightly more complicated

Let $f(x)$ and $g(x)$ be continuous functions of $x$, how would you find $$\frac{d}{dg(x)}\bigg(\frac{df(x)}{dx}\bigg)$$ just in terms of $x, f(x), g(x)$ and derivates with respect to $x$, so $f'(x)$ ...
user82832's user avatar
0 votes
0 answers
21 views

Composition of the norm with an absolutely continuous function is absolutely continuous

Let $f:[a,b]\to \mathbb{R}^n$ be an absolutely continuous function. The $g:[a,b]\to \mathbb{R}, \ g(x):=|f(x)|,$ is the composition of an absolutely continuous and of a Lipschitz function. Hence, $g$ ...
Pong's user avatar
  • 23
-3 votes
1 answer
43 views

How can I calculate this derivative using the fundamental theorem of calculus? [closed]

Let g be a continuous function for all x such that g(1)=5 and the integral of g from 0 to 1 is 2. If f(x) is it (see the image), prove f'(x) (see the picture). enter image description here Dada una ...
Edwin Zamarripa's user avatar
1 vote
0 answers
18 views

Unboundedness of Differential Operator by Fourier Transformation of Multiplication Operator $||x|| = \infty, D \mapsto k \implies ||D|| = \infty ?$

The multiplicative operator $T_g$ is defined by $f(x) \mapsto g(x)f(x) $ and is bounded in $L^2[\mathbb{R}]$ if $||g(X)||_\infty < \infty$. The operator $T_x$ is unbounded. The operator $D:=-i\...
theta_phi's user avatar
  • 115
3 votes
1 answer
52 views

What is the Euclidean norm of the vector containing all $k$-order partial derivatives of $|x|$?

Denote $|x|$ the Euclidean norm of a vector $x\in\mathbb R^N$. Also denote $D^kf$ as the vector in $\mathbb R^{N^k}$ containing all $k$-order partial derivatives of the function $f\colon\mathbb R^N\to\...
Raoní Cabral Ponciano's user avatar
-1 votes
1 answer
74 views

Find the $54$th derivative of a function at the point $x = 0$. Write the result in the form of multiplication. [closed]

Let's have the function $y = (x^2 + 2x) \cdot \sin\bigl(1.01x + \frac{\pi}{4}\bigr)$. Find the $54$th derivative of a function at the point $x = 0$. Write the result in the form of multiplication. \...
Rastislav Nemec's user avatar
0 votes
1 answer
20 views

Derivation of Continuity Equation for an Incompressible flow

Good day guys, I was playing around with the following form of the continuity equation: $$ \frac{\partial \rho}{\partial t} - \nabla \cdot (\rho \vec{v}) = 0 $$ For an incompressible fluid: $\frac{D\...
RMS's user avatar
  • 352
-1 votes
0 answers
32 views

Using the Airy Function to give a solution to a differential equation

Questions (b) to (e) attached How do I solve part (e)? It says to use the Airy Function and its derivative to give a solution to the differential equation but I am really unsure as how to proceed. ...
Chris Williams's user avatar
2 votes
1 answer
43 views

Why $\nabla f$ do not exactly coincide with $D f$ (it's its transpose)

Is there any reason (historical, or of any other kind) to why $$\nabla f= \begin{bmatrix}\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial z} \\ \end{bmatrix}...
niobium's user avatar
  • 1,221
0 votes
1 answer
54 views

Can $F(x)=g(f(x))$ be differentiable at $x=\alpha$ if f and g are not at $x=\alpha$

is $F(x)=g(f(x))$ always non-differentiable at $x=\alpha$ if: a) f is differentiable at $\alpha$ and g is not différentiable at $f(\alpha)$ b) f is not différentiable at $\alpha$ and g is ...
edster101's user avatar
0 votes
1 answer
19 views

Derivative of Convolution with Respect to Input Image

Given a discrete convolution operation for image processing like this: $$ H(x, y) = \sum_{i=-a}^{a} \sum_{j=-b}^{b} K(i, j) \cdot I(x - i, y - j) $$ It is common for CNNs to take the derivative with ...
James Li's user avatar
0 votes
1 answer
13 views

How do I find the largest possible layout with the amount of money avaliable?

I have a question from a practice exam and I was wondering how to start this. Here's the Question: On a property next to a straight road, the owner is constructing a rectangular paddock. The side next ...
AlexanderWaller's user avatar
0 votes
0 answers
33 views

How to obtain a PDE from a solution

I am trying to do something similar to this question but for a PDE. Specifically, I have these some equations and also know the answer as it's from a paper but cannot understand it or redo it. ...
Zack Fair's user avatar
  • 101
0 votes
0 answers
64 views

Applying chain rule/product rule to $\frac{d}{d\tau} \left( \frac{d}{dt} \theta(t(\tau)) \, \frac{d}{d\tau} t(\tau) \right)$

How can I apply the chain rule to following function? $$ \frac{d}{d\tau} \left( \frac{d}{dt} \theta(t(\tau)) \, \frac{d}{d\tau} t(\tau) \right) $$ The right way to rewrite the above expression is by ...
Federica Guidotti's user avatar
1 vote
0 answers
39 views

deriving differential equation from difference of PDE solutions

Consider defining $$f(x,t)=g(x,t)-h(x,t)$$ where the PDEs describing $g(x,t)$ and $h(x,t)$ are $\it{known}$ but the solutions themselves are $\it{unknown}$. It could be impractical to solve the PDEs ...
Clayton Estey's user avatar
2 votes
0 answers
86 views

Solve $\frac{dy}{dx}=\frac{2x^2-xy+y^2}{y-x}$.

This problem gives me a lot of trouble. I am asked to solve the differential equation $$\frac{dy}{dx}=\frac{2x^2-xy+y^2}{y-x}.$$Note that this is non-linear, non-homogeneous, non-exact, and ...
user108580's user avatar
0 votes
0 answers
20 views

Existence of all partial derivatives implies anything about total derivative’s existence?

Say a function $f: \mathbb{R}^n \to \mathbb{R}^m$ is $C^{\infty}$ if its partial derivatives of all orders exist. Does this imply anything about the existence of its total derivatives, or if not then ...
Abced Decba's user avatar
0 votes
0 answers
13 views

Does this modification of the derivative after integral change the result?

Consider a two-variable real function $f(x,y) = \frac{x^2}{x^2+y^2}$ who has only one singularity point (0,0) in the domain of definition with $\lim_{x\to 0}\lim_{y\to 0}f(x,y)=0,\lim_{y\to 0}\lim_{x\...
JohnWu's user avatar
  • 3
0 votes
1 answer
41 views

On how to demonstrate equivalence of two particular differentiation operations [closed]

$$ \frac{df(y')}{dy} \stackrel{?}{=} \frac{d\left( \frac{d f(y')}{d(y')} \right)}{dx} \tag{1} $$ In a recent answer on physics stackexchange (subject: calculus of variations), I asserted the above ...
Cleonis's user avatar
  • 212
1 vote
0 answers
72 views

Show that $f$ is not differentiable but partially differentiable at $(0,0)$

Let be $f:\mathbb{R}^2\to\mathbb{R}$, where $$ f(x,y):=\begin{cases} x,&x\neq y\\1,&x=y, x\neq 0,y\neq 0\\0,&x=y=0 \end{cases} $$ Show that $f$ is not differentiable but partially ...
Philipp's user avatar
  • 4,554
0 votes
0 answers
14 views

Semidifferentiability at the extremum of an interval and continuous extension of derivative

Let $f:[a,b] \to \mathbb{R}$ be differentiable on $(a,b)$ with continuous derivative $f'$. (i) Assuming that $f'$ can be continuously extended at $a$, is it true that $f$ is semidifferentiable at $a$ ...
Paolo Intuito's user avatar
0 votes
1 answer
25 views

Inconsistency on Heaviside Derivative

I've been stuck while trying to redo some calculations of a paper (https://arxiv.org/abs/2011.13267) and thought some of you might help. Essentially what I'm trying to do is calculate: \begin{equation}...
Pedro Pinho's user avatar
1 vote
1 answer
37 views

$f(x) = (\sum_{k=0}^n a_k x^k)^{\frac1n}$ is sublinear if $a_k \ge 0, x \ge 0$, i.e. $f(a + b) \le f(a) + f(b)$ for $a, b \ge 0$.

$f(x) = (\sum_{k=0}^n a_k x^k)^{\frac1n}$ is sublinear if $a_k \ge 0, x \ge 0$, i.e. $f(a + b) \le f(a) + f(b)$ for $a, b \ge 0$. I have some solution. I know that there's a very short and cute ...
Sergei Nikolaev's user avatar
0 votes
1 answer
42 views

What is the explanation of these special cases of tangents [closed]

The graph $y = |x|$ has no tangents at $x=0$ since it is not differentiable at that point, but it is possible to draw an infinite number of tangents to the graph at $x=0$, so why does it technically ...
Doodieman360's user avatar
-1 votes
0 answers
31 views

is w dot grad u = grad dot uw [closed]

In the subject of an exam about functional analysis and finite element method there is a question : Verify that $$ w \cdot \nabla u = \nabla \cdot uw $$ But I don't see why it's true since $$ w \cdot \...
Pierre-Antoine Senger's user avatar
2 votes
2 answers
85 views

Justify if there exist a differentiable function f such that $|f(x)|<2$ and $f(x)f'(x)\geq \sin(x)$

The problem states: Justify if there exist a differentiable function f such that $|f(x)|<2$ and $f(x)f'(x)\geq \sin(x)$ for every $x\in \mathbb{R}$. What I got by another question I made about a ...
MiguelCG's user avatar
  • 275
0 votes
0 answers
29 views

How to find the derivative of Q function [closed]

How to take the derivative of the given expression with respect to $x$ $$ Q \left( \frac{\sqrt L ( \ln (1+x) - R)} {\sqrt {1- \frac{1}{( 1+x)^2}}} \right) $$ where $Q(x)$ is the complementary error ...
Mina Kay Nak's user avatar
1 vote
0 answers
12 views

Mean value theorem application for a theorem on concave functions

Let $f$ be a function with continuous third derivative, with $f'$ strictly convex on $(−\infty, 0)$ and strictly concave on $(0,\infty)$. Prove that there exist only two differentiable functions $a \...
ABlack's user avatar
  • 580
1 vote
0 answers
38 views

$f \in C(\mathbb{R}^2)$ and for all $g \in C^1([0, 1], \mathbb{R}^2)$ their composition is in $C^1$. Is $f$ necessarily in $C^1$?

A function $f \in C(\mathbb{R}^2)$ is such that for any function $g \in C^1([0, 1], > \mathbb{R}^2)$ their composition $f \circ g$ belongs to $C^1([0,1])$. Prove or give a counter example that $f \...
ABlack's user avatar
  • 580
3 votes
1 answer
76 views

Justification for differentiation under integral

Given a function I seek to find its derivative $$f(x) = \int_{\frac{1}{x}}^{\frac{e^x}{x}} \frac{\cos(xt)}{t} \, dt, \quad (x>0)$$ My question is regarding the justification of the differentiation ...
Teodoras Paura's user avatar
1 vote
0 answers
31 views

Comparing two series expressions for $1/\zeta(s)$. What can be said about their complex roots?

The following two expressions involving the inverted Riemann $\zeta(s)$ functions are well known: \begin{align} \frac{1}{\zeta(s)} &= \sum_{n=1}^\infty \frac{\mu(n)}{n^s} \\ -\frac{\zeta'(s)}{\...
Agno's user avatar
  • 3,191
1 vote
2 answers
62 views

Ambiguity in solving differential equations

Suppose we want to solve the differential equation $y'=x \sqrt{y}$. Easy right? Because you can transform the equation into a separable one. However, I think that there are more than meets the eye. ...
legogubben's user avatar
0 votes
1 answer
42 views

Calculating the Derivative of a Complex Vector Function

I am trying to calculate the derivative of the function $$f(x) = x^T a (x^T a)^* = x^T a x^H a^* = x^T a a^H x^*$$, where $(.)^T$, $(.)^H$, and $(.)^*$ represent the transpose, Hermitian (conjugate ...
Omid Abasi's user avatar
2 votes
1 answer
87 views

Show that $f(x)=\int_{0}^{x}\sin(t)g(x-t)dt$ is 2 times differentiable and that $f''+f=g$ [duplicate]

Given g(x) a continuous function on $\mathbb{R}$, show that $f(x)=\int_{0}^{x}\sin(t)g(x-t)dt$ is 2 times differentiable and that $f''+f=g$. This problem reminds me another one where $$f(x)=\int_{0}^{...
MiguelCG's user avatar
  • 275
-1 votes
1 answer
76 views

Justify if there exist a differentiable function f such that $|f(x)|>2$ and $f(x)f'(x)>sin(x)$

The problem states: Justify if there exist a differentiable function f such that $|f(x)|>2$ and $f(x)f'(x)>sin(x)$ for every $x\in \mathbb{R}$. I thought about using trigonometric functions like ...
MiguelCG's user avatar
  • 275
0 votes
0 answers
34 views

Matrix derivative of matrix commutator

I'm working on some functional and I have to calculate its second derivative with respect to some matrix-variables. I'm just left with the following derivative to perform: $$ V''= -6 i \lambda \gamma ...
Fredrigo6's user avatar
0 votes
0 answers
20 views

The upper bound of a definite integral involving a derivative of an exponential function

I have a question. Is there anyone who knows how to evaluate the upper bound of the absolute value of the following integral. $$\int_{n}^{\infty} \Bigl(\frac{\exp(iax)}{x}\Bigr)^{(n)} \exp(ix) dx$$, ...
hitsu's user avatar
  • 85
-1 votes
0 answers
16 views

I can't show that for a function f:R^2-->R if df/dx=df/dy and f(x,0)>0 then f(x,y)>0. [closed]

I can't show that for a function f:R^2-->R if df/dx=df/dy, f/in C^1 and f(x,0)>0 for all x in R then f(x,y)>0.
Hamid's user avatar
  • 1
0 votes
1 answer
46 views

Existence of composite function with algebraic and exponential functions

We are supposed to test the validity of two statements: (1) There exists a differentiable function $g:R\rightarrow R$ such that $g(x^3+x^5)=e^x-100$ (2) There exists a continuous function $g:R\...
Equiposied's user avatar
0 votes
0 answers
50 views

Equivalence of Taylor Expansion and L'Hospital

Is the rule of L'Hospital equivalent to Taylor Expansion?: A analytic (or for this sake one time differentiable) function $f(x)$ can be expanened around $x_0$ with $f(x) \approx f(x_0) + f'(x_0) (x-...
theta_phi's user avatar
  • 115
0 votes
1 answer
27 views

Sign of cross-partial derivative

Consider a function $f(x,y)$. Let $x, y \geq 0$, and $\frac{\mathrm d}{\mathrm dx}f(x,y), \frac{\mathrm d}{\mathrm dy}f(x,y) >0 \ \forall x, y$. I would tend to think that the cross partial ...
maxence's user avatar
-3 votes
0 answers
31 views

How to find the variation of volume using differentials and derivatives [closed]

I've been stuck on this problem and I honestly don't know where to start. How much variation dr in the radius of a coin can be tolerated if the volume of the coin is to be within $11000$ of their ...
Jie Jo's user avatar
  • 1

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