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

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Derivative of projection

Let $\Omega \subset \mathbb{R}^n$ be a limited domain of class $C^\infty$ (open, connected) and $\varepsilon > 0$ small such that $$ \Omega_\varepsilon = \{y \in \overline{\Omega} : d(y, \partial \...
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19 views

Show that the function is infinitely differentiable on the entire real line [on hold]

Show that the function $$f(x)=\begin{cases}(1+x)^{1/x} &\mbox{if }x\neq 0,\\ e &\mbox{if } x=0 \end{cases}$$ is infinitely differentiable on the entire real line.
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1answer
22 views

Show relation and linearity related to differentiable functions

I have problems solving the following exercise: (a) Let $n\in \mathbb N$, $a\in \mathbb R$ and $f:\mathbb R^n \backslash \{ 0 \} \to \mathbb R$ $\mathbb R$-differentiable. Show that the relation $$...
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1answer
16 views

Show Each function is equivalent using 2 conditions.(Real analysis)

It is might be easy for you. The Question There are functions $ f, g, c, s $ $f,g : R \rightarrow R $ and $s,c : R \rightarrow R $ ($R$ is a set of the real number) These functions satisfy ...
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3answers
126 views

$f$ be a smooth function on real line , $f(0)=0$ , $f(x)>0, \forall x \ne 0$ and any $f^{(n)}(0)=0$ ; is $\sqrt f$ smooth?

Let $f: \mathbb R \to \mathbb R$ be an infinitely differentiable function such that $f(0)=0$ , $f(x)>0 , \forall x \ne 0$ and $f^{(n)}(0)=0$ ( the $n$-th derivative ) $, \forall n \in \mathbb N$ ...
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2answers
89 views

How to prove this statement? (Real analysis)

This might be the basic question in real analysis. A function $f$ is $ C^2 $ function on the closed interval$ [0,1]$ Also the function $ f $ is satisfying $ f(0) = f(1) =0 $ Plus, $\vert f''(x) \...
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1answer
46 views

Some doubts with the sign of a derivative

Good evening to everyone. The derivative is defined in the following order: $$ \frac{d}{dx} f(x)=\frac{{-x^2-x+11}}{\left(x+3\right)^2}e^{2-x}\:$$ for $ x < -3 $ $$\:\frac{d}{dx}f(x)=\frac{{x^2+x-...
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1answer
23 views

What is $\vec{v}(\vec{\gamma}(t))$?

If we got the curve $\vec{\gamma}:[0,1]\rightarrow\mathbb{R^3}$ $$\vec{\gamma}(t) = \left(\! \begin{array}{c} t \\ t^2+1 \\ t \end{array} \!\right) $$ And the vector field $\vec{v}:\mathbb{R^3}\...
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6answers
94 views

find ${dy}/{dx}$ if $x^y + y^x = 1$

Find ${dy}/{dx}$ if $x^y + y^x = 1$. I have no idea how to approach this problem. Can somebody please explain this to me?
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0answers
20 views

(Conceptual_Calculus) Differential Conditions v. Derivative Conditions

I have few questions regarding the reason we learn about the differential conditions in higher dimensions and dealing with multivariable calculus. In the context of optimization (e.g. finding ...
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21 views

mixed partial derivative of a function

Find second order mixed partial derivative, $\frac{\partial^{2} f}{\partial y \partial x}$ of $$\frac{x \log(y)}{ye^x}$$ I am not able approach this problem. I tried differentiating it wrt $x$ (...
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2answers
31 views

Derivative problem(I think, that is Implicit function theorem)

I have a function: $$F(x,y) = 2x^4 + 3y^3 +5xy$$ And input $x$ and output $y$ we know that this relation $F(x,y) = 10$ confirms. We know, that this happens when x = 1 and y = 1. By small change of ...
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0answers
37 views

Is finding the extreme points of a differentiable function by first derivative always correct?

I came across this question where I was asked to find the local minimum and local maximum of the function $$y=\sec x + 2\ln(|\cos x|),$$ domain of $x$ being $(0,2\pi)-\{\pi/2 , 3\pi/2\}$. I found its ...
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1answer
51 views

What is the difference between derevative w.r.t a vector and directional derivative?

Say we have a scalar-valued function $f: \mathbb R^3 \rightarrow \mathbb R$, such that: $$f(\mathbf x) = \mathbf x^T\mathbf a$$ $\mathbf x$ and $\mathbf a$ are two vectors. The derivative of $f$ ...
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2answers
40 views

Help with derivative of integral function?

How do you differentiate: $\displaystyle f(x)=\int_{a}^{b}e^{x^{2}+t^{2}}dt$ I tried writing $f(x)$ as the difference of the antiderivative of the function $\displaystyle e^{x^{2}+t^{2}}$ and I get $\...
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1answer
33 views

Derivative of narrowband signals

I just read a statement in an article that "for a narrowband signal $u(t), -\frac{d^2(u)}{dt^2} \sim u(t)$". Is this appropriate? Here, $u(t)$ is a transient displacement field and we are talking ...
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2answers
75 views

Partial derivative definition

What is the partial derivative of $$\frac{\partial x}{\partial y}$$ when $x$ and $y$ are a part of a function $f(x,y)$? Using an example of: $$f(x,y) = x+y$$ Given the definition of holding all ...
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0answers
20 views

Maxima and minima nth derivative reasoning

I found a statement somewhere in my notes that if we have a higher order function and lets say we take the nth derivative of it. If n is odd and the result turns out to be any number except zero then ...
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1answer
22 views

Differentiation product of functions in multidimensional Analysis

Define $k: \mathbb{R}^d \to \mathbb{R}^{m\times m}$ such that $ k(x)=g(x)f(x)^T$, where $f: \mathbb{R}^d \to \mathbb{R}^m, g: \mathbb{R}^d \to \mathbb{R}^m$ are differentiable functions. Prove that $k$...
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2answers
70 views

How can we say the derivative is exact if the difference quotient has a domain restriction?

I think I've finally been able to voice my confusion when it comes to derivatives and limits. Let's first look at the difference quotient for a function $f(x)=x^2$ $$\lim_{h\to0} \frac{f(x+h)-f(x)}{...
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0answers
19 views

Differential Application [on hold]

I have the following question. One hour after $x$ milligrams of a particular drug are given to a person, the change in body temperature $T$ (in degrees Fahrenheit) is given by: $$T(x)=x^2\left(1 - \...
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0answers
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Defining derivatives and integrals for hyperoperations > 2

Derivatives and Integrals are continuous generalizations of the Forward Difference and Summation additive operators respectively. We can do the same with multiplication and get multiplicative calculus ...
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2answers
81 views

Increasing function with $f'(x)=f(f(x))$ [duplicate]

Is there a strictly increasing function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x$?
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shifting integration variable and taking derivative seemingly giving problem

I am doing loop integral in quantum field theory, and an issue in shifting integration variable is giving me a problem. Let me illustrate with an example. I have an integral that looks approximately ...
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2answers
15 views

Evaluating a statement without calculating the indefinite integral

I'm cramming for a supplementary exam so you might see a ton of questions like these in the 48+ hours to come <3 The question is more of just a yes or no ; Evaluate the statement without ...
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1answer
17 views

Derivative of Incomplete Gamma Function

For the following incomplete Gamma function: \begin{equation} Γ(1+d,A-c \ln x)=\int_{A-c\ln x}^{\infty}t^{(1+d)-1}e^{-t}dt \end{equation} I am trying to calculate the derivative of $Γ$ with respect ...
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1answer
51 views

Study of differentiablity of function

Study the differentiability of the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ $f(x,y)=\begin{cases} \frac{x^3+y^3}{x^2+\left|y\right|} & (x,y)\ne(0,0) \\ 0 &(x,y)=(0,0) \\ ...
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3answers
33 views

Derivatives with different rules

I'm having trouble with this one problem that just deals with deriving. I can't seem to figure out how they got their answer. Any help would be appreciated! Thanks! $ \frac{(x+1)^2}{(x^2+1)^3} $ The ...
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1answer
30 views

Find the Derivatives of $g(x) = \sqrt{3-2x^2}$ and $h(x) = \ln {(x^2 – x)}$

I am asked to find the derivatives of $g(x) =\sqrt{3-2x^2}$ and $h(x) = \ln{(x^2 – x)} $ For: $g(x)h(x)$ and $\dfrac{h(x)}{g(x)}$ and $h^3 (x)$ First off I am not sure if my derivatives are ...
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3answers
194 views

Derivative of the magnitude of a vector. Does it exist, or not?

I have a puzzling situation involving derivatives. I want to derivate: $$ \frac{d}{dx}| \mathbf F(x)| $$ This was actually something involving physics. Lets be 2-dimensional for simplicity. Let a ...
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1answer
57 views

Can this $dx$ be taken out?

I have this expression: $$\frac{1}{f(x)}\frac{df(x)}{dx}=\frac{dg(x)}{dx}$$ Can the two $dx$ be "simplified"? Namely, to get $$\frac{df(x)}{f(x)}=dg(x)$$ Is this right? If it is, what is the ...
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2answers
43 views

Find $\frac{dy}{dx}$ if $x^3 + x^2y + xy^2 + y^3 = 81$

I need to find $\frac{dy}{dx}$ if $x^3 + x^2y + xy^2 + y^3 = 81$ I am trying to first get y in terms of x, but that is quite lengthy and feels like I am doing something wrong. How do I go about this ...
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54 views

More convenient form of derivative of $\mathrm{sinc}(x)$

$\mathrm{sinc}(x)$ is defined as $\frac{\sin(x)}{x}$ except continuous at $x=0$ (insert the removable singularity). The derivative of $\mathrm{sinc}(x)$ is usually given as the derivative of $\frac{\...
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32 views

Understanding a calculation deduced for the function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$

With my current knowledges I don't know if this is a bad question, but since I am interesting in this kind of calculations I want to ask you, if I was wrong or if if my statement is obvious. From ...
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1answer
25 views

How to calculate $∇(r^2/(2z(1+a/z^2)))$ in cylindrical coordinates

How to calculate $$∇\bigg(\frac{(ρ^2)}{2z(1+\frac{a}{z^2})}\bigg)$$ where the function is in cylindrical coordinates $$ρ^2=x^2+z^2$$ $$∇\bigg(\frac{x^2+z^2}{2z(1+\frac{a}{z^2})}\bigg)$$ Is the ...
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2answers
20 views

Sign of the derivative $ -e^{\frac{1}{2x+2}}\left(sgn\left(x\right)+\frac{1-\left|x\right|}{2\left(x+1\right)^2}\right) $

Good morning to everyone. I have a problem with finding the sign of a derivative: $$ \frac{d}{dx}f(x)=-e^{\frac{1}{2x+2}}\left(sgn\left(x\right)+\frac{1-\left|x\right|}{2\left(x+1\right)^2}\right) $$ ...
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6answers
34 views

Critical points of a cubic function

There is a function $x^3 - 6x^2 + 9x + 1$. Its critical points are $1$ and $3$. I am very confused, if these points are maximum and minimum points respectively or are both inflection points. Can ...
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1answer
18 views

Derivative of dot product with transposed function

According to this post Derivative of dot product I have a similar task: $$\langle f(x),g(x) \rangle = f(x)g(x)^T=j(x)$$ I have to show: $j'(x)=g'(x)f(x)^T+g(x)^Tf'(x)$ I know how to ...
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0answers
11 views

Mean continuity of gradient

Let $f:\mathbb R^n\longrightarrow R$ be a differentiable function, and suppose $\nabla f$ is bounded. Prove that $$\lim_{r\to 0}\frac{1}{\omega_n r^n}\int_{B_r(x)}[\nabla f(y)-\nabla f(x)] dy=0.$$ ...
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1answer
24 views

How can I prove that $(0,0)$ saddle-type inflection point of $x|y|+y|x|$?

How can I prove that (0,0) is saddle-type inflection point? the function is: $f(x,y)=x|y|+y|x|$ How can I find the second derivative by $x$ (for the hessian matrix)
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1answer
80 views

Why can this differential equation be written in $3$ different ways?

Suppose we have the following differential equation using operator notation: $$(D-x)(D+x)y=0\tag{1}$$ where $$D=\frac{d}{dx}$$ Now I could rewrite $(1)$ as $$\begin{align}\require{enclose}(D-x)(D+x)y&...
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2answers
108 views

Examples of (not) uniformly continuous, non-differentiable, non-periodic functions

Let $I\subseteq\mathbb{R}$ and $f:I\to\mathbb{R}.$ $(0)$ If $f$ is discontinuous on $I$, then it is not uniformly continuous. $(1)$ Suppose $I$ is open and bounded. If $f$ is unbounded on $I$,...
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1answer
51 views

function is not differentiable on $\mathbb R\setminus\{0\}$

I need to prove that the given function $f$ is not differentiable on $\mathbb R \setminus\{0\}$. $$ f(x) = \begin{cases} x^2, \ x \in \mathbb{Q}\\ 0, \ x \in \mathbb{R}-\mathbb{Q} \end{cases} $$ ...
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36 views

Derivatives that are tangent to the original function

I was recently studying parabolas $ f(x) = ax^2 + bx + c $ whose derivative $f'(x) = 2ax + b$ is tangent to itself -- one example would be $f(x) = x^2 -6x +10;$ it is easy to see that if $c = a + \...
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2answers
41 views

Intuition behind the derivative of are of a square? How to properly use the derivative ?

If I derive the formula $$S=16t^2$$, where S denotes the distance and t denotes time I get $$ds/dt= 32t$$. This in return give me a formula for the speed of the object at any time t. However if we ...
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1answer
43 views

Derivative of $\arcsin \frac{x-1}{x+1}$

I was looking at a question that asks for the derivative of $\arcsin (\frac {x+1}{x-1}) $. The solution starts by saying $y = \frac{x+1}{x-1}$, so $1-y^2= \frac{4x}{(x+1)^2}$ and $\frac{1}{\sqrt{1-y^...
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1answer
47 views

How many solutions does $a^x=2016x$ for $a > 0$ have?

How many solutions does $a^x=2016x$ for $a > 0$ and $x \in \mathbb{R}$ have? Note that for $x < 0$ we have $a^x > 0$ and $2016x < 0$ so we can consider only $x \ge 0$. Let $f(x) = 2016x - ...
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0answers
12 views

Apply Laplace Operator with two functions

Proof that $\Delta u(x,y):=\partial_1\partial_1u(x,y)+\partial_2\partial_2u(x,y)=0$ is valid for $\log(x^2+y^2)$ and $\arctan\left(\frac{x}{y}\right)$. Is it enough to differentiate the functions in ...
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1answer
24 views

Derivative of dot product of Residual Sum Square in matrix notation

I am trying to derive the following expression w.r.t. $\beta$: \begin{equation} RSS(\beta) = (\mathbf{y} - \mathbf{X} \beta)^T (\mathbf{y} - \mathbf{X} \beta) \end{equation} I know that the ...