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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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1answer
13 views

$\frac 1 {\overline {f(1/\ \overline z)}}$ is differentiable for $|z|>1$

Let $S=\{z \in \mathbb C : |z|<1\}$. Suppose that $f: S \to \mathbb C$ is differentiable everywhere in its domain. Suppose further than $f(z)$ is never zero. I want to show that $g(z):=\frac 1 {\...
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16 views

Prove this formula follows from a function being continuously differentiable?

I'm studying for an exam in an electrical engineering course (stochastic process in dynamic systems), though this section is strictly on the math. A given practice problem (with no solution given, of ...
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1answer
28 views

If $f$ is $C^1$ s. t. $\Vert f(x) - f(y) \Vert \geq k \Vert x - y \Vert$, then $f$ is a diffeomorphism of $\mathbb{R}^{n}$

Let $f:\mathbb{R}^{n} \to \mathbb{R}^{n}$ be a function of class $C^1$ and suppose that there is $k>0$ such that $$\Vert f(x) - f(y) \Vert \geq k \Vert x - y \Vert$$ for any $x,y \in \mathbb{R}^...
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2answers
41 views

Is there a function $f \in C[0,1]$ such that $f(0)=f(1)=0$, $f'(0)=0$, $f'(1)=1$ and $\|f\|_{C[0,1]} < \epsilon$, $\|f'\|_{C[0,1]} < \epsilon$

Given $\epsilon \in (0,1)$ I am looking for a twice continuously differentiable function $f\colon [0,1] \to \mathbb{R}$ satisfying the following conditions: $$ f(0)=f(1) =f'(1)=0, \quad f'(0)=1, $$ $$...
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Verification: Investigation of a linear map on surjectivity and injectivity

The linear map I'm investigating is defined like this: $l_2: P_3(\mathbb{R}) \rightarrow \mathbb{R}, l_2(p):= p'(1)$ And these are my calculations: Injectivity: $Let \,\, p,q \in P_3(\mathbb{R}), ...
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1answer
38 views

Proof integration identity $\int_{0}^{1}dx\int_{0}^{x}e^{x^2}dy=\int_{0}^{1}dy\int_{y}^{1}e^{x^2}dx$

I have to prove this identity: $$\int_{0}^{1}dx\int_{0}^{x}e^{x^2}dy=\int_{0}^{1}dy\int_{y}^{1}e^{x^2}dx$$ I've shown that: $$\int_{0}^{1}dx\int_{0}^{x}e^{x^2}dy=\int_{0}^{1}xe^{x^2}dx=\frac{1}{2}(e-1)...
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1answer
37 views

An identity on $\small{}_pF_q\left(\left.\begin{array}{c} a_1+1,a_2+1,\dots ,a_p+1\\ b_1+1,b_2+1,\dots ,b_q+1\end{array}\right| z\right)$

I stumbled upon this relation while trying to answer this post. I was trying to find a relation between the two generalized hypergeometric functions, $$A=\,_3F_2\left(\color{blue}{\tfrac12,\tfrac12},\...
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1answer
22 views

How to compute the Jacobian matrix of a multivariate function in a nonstandard matrix?

Given a function $f:R^2\rightarrow R^2$ such that $f(x,y)=(xy, \cos xy)$, I need to compute the Jacobian matrix Df with respect to the basis $\{(1,0), (1,1)\}$. Not confident in my answer though. ...
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1answer
40 views

Find $\lambda$ such that $f$ it is differentiable in zero and has a continuous derivative in zero

I am trying to solve this task Find $\lambda>0$ such that $f=\begin{cases}0& x=0\\ |x|^{\lambda}\cdot \sin\frac{1}{x} & x\neq 0 \end{cases}$ a) is differentiable in zero b) ...
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Why is $\|A-BCA\|_F^2$ non-convex regarding $B,C$?

I noticed somewhere, a function similar to $\|A-BCA\|_F^2$ was claimed to be non-convex regarding $B,C$, while a relaxation like $D=CA$ makes $\|A-BD\|_F^2$ convex respect to $B,D$. The matrices $A, B,...
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An exercise from Zorich on bounded differentiable functions

I have to prove this: $$\left\{\begin{matrix} f\in\mathcal{C}^{(n)}(-1,1) \\ \sup_{x\in (-1,1)}|f(x)|\leq 1 \\ |f'(0)|>\alpha _n \end{matrix}\right. \Rightarrow \exists x_1,...,x_n \in (-1,1) : f^...
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2answers
59 views

Matrix exponential is differentiable at $0 \in \mathbb{R}^{n,n}$

Given$$\exp : \mathbb{R}^{n,n} \mapsto \mathbb{R}^{n,n} \qquad A \mapsto \sum_{k=0}^{\infty} \frac{A^k}{k!}$$ where $ \mathbb{R}^{n,n}$ is equipped with Operator Norm. I am trying to show that $\exp$ ...
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2answers
39 views

Write a Limit to calculate $f'(0)$

Let $f(x) = \frac {2}{1+x^2} $ I need to write a limit to calculate $f'(0)$. I think I have the basic understanding. Any help would be greatly appreciated. d=delta and so far what I have is $f'(...
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1answer
29 views

When is the polynomial $P(|x+y|)$ total differentiable?

If $P \in \mathbb{R}[x]$ is a polynomial, under which sufficient condition is the function: $$f: \mathbb{R}^2 \to \mathbb{R}: f(x,y) = P(|x+y|)$$ total differentiable? So for a function to be total ...
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30 views

Derivative the exponential map $e^A$

Can the derivative the map $$ f(A) = e^A, A \in \mathbb{R}^{n \times n} $$ be defined as $$ \lim_{h\to 0} \frac{e^{A+h\Delta} -e^{A}}{h} =\lim_{h\to 0} e^{A} \frac{e^{h\Delta} - I}{h} $$ where $\...
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17 views

Spectrum of Derivative Operator

Good evening! Given the operator $A$ acting on $L^2(0,1)$ with $Au = u'$, $\mathcal{D}(A) = \{u \in H^1(0,1):u(1)=0\}$, I am trying to show that $A$'s spectrum is empty, which is stated as easy to ...
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Find a function to smoothly complete this frame

I've been trying to find equations to model a shape that looks like the curved outline of old TV monitors. This is what it looks like (please excuse the low quality): To do this, I picked random ...
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1answer
31 views

On an application of the chain rule.

Let $f: \mathbb{R} \rightarrow \mathbb{R} $ be continuously differentiable, then it's well know that \begin{align*} f(x) & = f(y) + \int_0^1 \frac{\partial f (y- t(x-y))}{\partial t} \, dt \\ ...
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1answer
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Compute the gradient of $f(x)=\|\text{diag}(x)\|$ with the chain rule

Consider the function $f:\mathbb{R}^n\to\mathbb{R}$ given by $f(x)=\|\text{diag}(x)\|$, where $\text{diag}(x)\in\mathbb{R}^{n\times{n}}$ is the diagonal matrix with diagonal entries $x_1,x_2,\dots,x_n$...
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1answer
30 views

No. of critical points

Domain: $\mathopen]1,4\mathclose[$, $f(x)= 3x^2 - 6x$ How many critical points exist? (Zero, 1, 3, 4) By diff.($x$): $f'(x)= 6x-6$ Then $x=1$ is local point of minimum, but according to the interval ...
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2answers
67 views

Does $|f^\prime|<1$ imply that $\forall_{x,y}|f(x)-f(y)|<|x-y|$?

I have a task that I think reduces to proving that $f$ is a contraction mapping. We know that $\forall_x|f^\prime(x)|<1$. Therefore if I could prove that $|f^\prime|<1\implies|f(x)-f(y)|<|x-y|...
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1answer
20 views

Study differentiability of a multi variable function

Let $f(x,y)=\sqrt{x^2+(y-1)^2}$. Study the differentiability of the function at the point $(0,1).$ I know that the derivative of a multi variable function is calculated as follows: $$\lim_{h\to0}\...
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1answer
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Let $X =\{n^3 + 3n^2 +3n | n>0\}$ and $Y = \{ n^3 -1 | n>0\}.$ Prove that $X =Y$.

Let $X=\{n^3+3n^2+3n\mid n>0\},$ and $Y=\{n^3-1\mid n>0 \}$. Prove that $X=Y$. I tried proving it by adding some values to $X$ and $Y$, but I could not make it equal.
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46 views

Confusing Lagrange multipliers question

Let $a_1,a_2, \dots, a_n$ be reals, we define a function $f: \mathbb R^n \to \mathbb R$ by $f(x) = \sum_{i=1}^{n}a_ix_i-\sum_{i=1}^{n}x_i\ln(x_i)$, in addition, we are also given that $0 \cdot \ln(0)$ ...
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2answers
21 views

Finding the slope of the line normal to the graph at a given point

My Problem: What is the slope of the line normal to the graph of $f(x) = e^x-x^e-e$ at the point where the graph crosses the $x$-axis? a. $-0.288$ b. $-0.110$ c. $3.471$ d. $9.106$ I ...
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1answer
16 views

Percentage increase/decrease

Given equation V(T^n)=C . I am trying to find percentage increase in T when V is halved and n=0.5 So, method 1: using differential dT/T=(-1/n)*(dV/V). Putting dV/V=-50% I get dT/T=100% method 2: V1T1^...
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2answers
20 views

Finding where two graphs have perpendicular tangent lines

I've been stuck on this calc problem for a while: Let $f$ be the function given by $f(x) =\ln(x+1)$ and let $g$ be the function given by $g(x) = x^{-1/2}$. At what value of $x$ do the graphs of $f$ ...
4
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1answer
55 views

On anti-derivative of functions

Let $f,g,h: \mathbb R \to \mathbb R$ be differentiable functions. (1) Does there necessarily exist a differentiable function $F: \mathbb R \to \mathbb R $ such that $F'=\max \{f' ,g' \}$ ? (2) Does ...
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1answer
42 views

Maclaurin's series for $\tan(x+x^2)$

I would like to find the Maclaurin's series for $\tan(x+x^2)$ stopped at $x_0^3$. I tried with: $$\tan(x+x^2)=1+\frac{1+2x_0}{\cos^2(x_0+x_0^2)}x+....,$$ but it seems like the derivation gets more ...
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1answer
34 views

How can I graph this derivative of a quarter of a semicircle?

For graphing the derivative of the circle, I know that the equation of a circle is $x^2+y^2 = r^2$ and in this case r = 4 With implicit differentiation I know that $y' = \frac{-x}{y}$ or $\frac{-x}{\...
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Clarification about the symbol $d/dx$ [duplicate]

As we know the definition of differentiation , $d/dx$ is a operator and it's not $d÷dx$.But, how can we write $dy=f(x)dx$ from $dy/dx=f(x)$.
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how to graph the derivatives of certain kinds of piecewise functions

For questions like these, how can I graph the derivative? For the first image, the sideways x^3 graph is the original and for the second image the v-shaped thing is the original function. For the ...
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1answer
45 views

How can I graph the derivative of 1/4th of a circle or a semicircle in a piecewise function? (Also other kinds of piecewise functions)

I'm having trouble with questions like these. In the first image, the original function is what is the two sharp lines and a semicircle in between. I understand how to find and graph the derivative of ...
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1answer
25 views

Taylor's theorem implies $f(x,y) - f(a,b) = (x-a)\frac{\partial}{\partial x}f(\bar{x},y) + (y-b)\frac{\partial}{\partial y}(fx,\bar{y})$?

Let $f: B_{r}(a,b) \subset \mathbb{R}^{2} \to \mathbb{R}$ be a differentiable function. Prove that for all $(x,y) \in B_{r}(a,b)$ there is $\bar{x} \in [a,x]$ and $\bar{y} \in [b,y]$ such that $$f(x,...
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1answer
45 views

Why can't I reduce the total differential?

I have encountered the following equation: $g: \mathbb{R}^m \rightarrow \mathbb{R}$ $u: \mathbb{R}^n \rightarrow \mathbb{R}^m$ $z = g(\mathbf{y})$, $\mathbf{y} = u(\mathbf{x})$ then using ...
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1answer
40 views

Existence of continuous derivatives of partial functions and total differentiability

We know that for a function $\mathbb{R}^m\to \mathbb{R}^n$ the existence and continuity of partial derivatives implies the differentiability of that function. Will this hold true for functions on ...
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0answers
11 views

Worked examples of Lie derivative

I'm trying to find the Lie derivative of a 2 form $\sin(\theta)d\theta d\phi$ with respect to a vector field given in a differential basis and I think the way to go here is to use Cartan's formula but ...
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1answer
38 views

Extending a function defined in $\mathbb{R}^n\setminus\{0\}$ to a continuous function defined in $\mathbb{R}^{n}$.

Let $g: \mathbb{R}^n\setminus\{0\} \to \mathbb{R}$ be a function of class $C^{1}$ and suppose that there is $M > 0$ such that $$\left|\frac{\partial}{\partial x_{i}}g(x)\right| \leq M.$$ Prove ...
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3answers
41 views

Derivative of $ f(x)=(\frac{x^2}{3x-1})^4(2x-3)^2$

I determined $\Large f'(x)=((\frac{x^2}{3x-1})^4(2x-3)^2)'$ but the solution I'm given is different than what I got and I dont understand why. Here is what I worked out: $\Large f'(x)=((\frac{x^2}{3x-...
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0answers
8 views

derivative of surface integral over sphere

let $u$ be harmonic in the domain $U \subset \mathbb{R}^n$ and $B_R(0) \subset U$ and $u(0)=0, u\neq 0$. Let $0<r<R$. Define $a(r):= \frac{1}{r^{n-1}} \int_{\partial B_r(0)} u^2dS, b(r):= \frac{...
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1answer
22 views

Characterize the critical points of $\Vert Ax - b \Vert^{2}$

Let $A$ a $m \times n$ matrix, $b$ a $m \times 1$ matrix and $x$ a $n \times 1$ matrix. Consider $f: \mathbb{R}^n \to \mathbb{R}$ defined by $$f(x) = \Vert Ax - b \Vert^2.$$ Determine a condition ...
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1answer
43 views

Can anyone solve this for me !? [on hold]

Let f(2)=3 and f'(2)=3 then the value of Lim x-2 {xf(2)-2f(x)}÷(x-2)
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1answer
19 views

Graph of a function in the neighbourhood of $x=0$

I'm new in Calculus II. Given the function $f(x)=(x^2(e^x−1))^{\frac{1}{5}}$, I would like to determine a qualitative graph of the function in the neighbourhood of x=0. I know that there might be an ...
2
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3answers
263 views

Derivative of an inverse of a function [on hold]

I'm new into Calculus. Given $f:\mathbb{R}\rightarrow \mathbb{R}$, I know that $f(0)=2$ and $f'(0)=3$. I can't understand why, given a function $g=f^{-1}$, it is true that $g'(2)=\frac{1}{3}$.
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0answers
14 views

How to determine a qualitative graph of a function in the neighbourhood of a point?

I have the function $f(x)=(x^2(e^x-1))^{\frac{1}{5}}$ and I would like to determine a qualitative graph of the function in the neighbourhood of $x=0$. I know that there might be an inflection point ...
0
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2answers
41 views

Finding extrema of $f(x) = \frac {\sqrt{x^4 - 4x^2 + 4}}{x^2 -1}$

Find all of the extreme points of the following function and sketch a graph. $f(x) = \frac {\sqrt{x^4 - 4x^2 + 4}}{x^2 -1}$ So far what I did is simplify the function to get the following: $f(x) = ...
6
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0answers
42 views

A question about converging derivatives

Suppose $f \in C^{\infty}(\mathbb{R})$ and $\forall x \in \mathbb{R} \text{ } \exists \lim_{n \to \infty} f^{(n)}(x) = g(x)$. Does this mean that $$ \exists a \in \mathbb{R} \forall x \in \mathbb{R}...
1
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2answers
61 views

How does one interpret $\dfrac{dx}{dy}$ for a function which isn't invertible?

I was just going through the proof of derivative of inverse functions. The statement reads: If $y= f(x)$ is a differentiable function of $x$ such that it's inverse $x=f^{-1}(y)$ exists, then $x$ ...
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0answers
10 views

Uniform precession motion, relative derivation, poisson vectors.

Suppose that you have a gyroscope with revolution symmetry around a perpendicular axis $\bf{e}$ such that the inertia tensor of this gyroscope can be written: $${\bf{Jc}} = {A\bf{I}} + (\Gamma - A)\...
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1answer
29 views

How to compute $u_x(\sqrt {(x^2+y^2)})=$?

This is in reference to this question $g(z) = f(|z|)$ is not holomorphic for a non constant function $f$ If $u$ is a function of single variable $x$ and if I want to differentiate $u(\sqrt {(x^2+y^...