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

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1answer
14 views

How is the class related to derivability?

Good evening to everyone. I have a question where they require me to find the derivability. After I read the answer sheet I saw that the function has the class $ C^1 $. How is the class related to ...
3
votes
1answer
28 views

Why $\frac{d}{dt}f(x+t(y−x))<0$ if $x < y, f(y) < f(x)$

Here excerpt from a book: Аssume that $f$ satisfies $\nabla f(x) \ge 0$ for all $x$, but is not nondecreasing, i.e., there exist $x,y$ with $x < y$ and $f(y) < f(x)$. By ...
3
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2answers
92 views

$f: \Bbb R^2 \to \Bbb R$ whose partials exist. Show: $\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$

Let $f: \Bbb R^2 \to \Bbb R$ be a function whose partial derivatives exist. Now i have to show: $$\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$$ Any tipps on how ...
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2answers
51 views

How to prove differetiability in $\Bbb K^2$?

I have to investigate differentiability in all points of the following function: $$f: \Bbb {R}^2 \to \Bbb R \: \: \: \: \: \: \: f(x,y):=\begin{cases} y-x &\mbox{if } y\ge x^2 \\ 0 & \mbox{if }...
2
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1answer
104 views

Proof from Calculus 1

Last days, from going into a website of the university of Pisa, I found an exercise given in the previous exams, in 1999. The problem was like: Given a continuous function $f$ in $\mathbb R$, and ...
0
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1answer
38 views

How this integral is evaluated $\frac{\partial }{\partial x}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$?

How this integral is evaluated? $$\frac{\partial }{\partial y}\left(\int _y^x\cos \left(-5t^2-2t-4\right)\:dt\right)$$ And in general, are there general methods for partial differentiation ...
0
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2answers
56 views

Where have I gone wrong in finding the derivative of $2^{\sin x}$?

I was finding the derivative of $2^{\sin x}$. My attempt $(1)$- $$y=2^{\sin x}$$ $$\implies\ln y=\ln2^{\sin x}$$ $$\implies\log_ey=\ln2^{\sin x}$$ $$\displaystyle\implies e^{\ln2^{\sin x}}=y$$ $$...
2
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1answer
37 views

Using the chain rule for cos and sin functions

I am having issues with derivatives containing chain rules. I know there is multiple threads already but after reading a few, I still find myself confused. I also checked the actual answer following a ...
0
votes
1answer
51 views

Definition of derivative to calculate $x\sqrt{|x|}$ at $x=0$

So the question says use the definition of the derivative to calculate the derivative of $x\sqrt{|x|}$ at $x=0$. I understand the definition of derivative but have no idea where to go from there to ...
1
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2answers
31 views

Why is it true that $S'(t)/S(t) = d log(S(t)) / dt$?

I came across this identity in derivation of the hazard rate in survival analysis.
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1answer
27 views

Derivative of a characteristic polynomial at an eigenvalue

Let $p(\lambda)$ be the characteristic polynomial of an $n\times n$ matrix $A$. We know that the roots of $p(\lambda)$ are the eigenvalues of $A$, hence the sum of the roots of the polynomial (taking ...
4
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0answers
33 views

How would I find the nth derivative of a function, where n is imaginary? What about where n is not a constant? [duplicate]

Forgive me if this question has already been asked. I was unable to find anything relevant to this question. The $n$th derivative of a function, $f^n(x)$ is well-defined for $n\in\mathbb{Z}^+$. As ...
0
votes
1answer
30 views

What should i conclude from the following workout?

We know that for any value of $x$ other than $0$, $a^x\ne e^x$ where $a>e, a\in R^+$ but we do know that for some value of $p$, $$pa^x=e^x\ldots(1)$$ you see $p$ is a positive number because of ...
0
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0answers
34 views

Exercise at Differential Equations: derivatives of boundary conditions

$ Let\quad U\quad be \quad a \quad smooth \quad solution \quad of \quad the \quad following\quad boundary \quad value\quad problem: $ $ -cU'+ (F(U))'=εU''\qquad U(-\infty)= A \quad and \quad U(+\...
0
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0answers
26 views

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 \...
1
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0answers
17 views

Polynom subspace of continuously differentiable Functions

Let $n\in \mathbb{N}$ and $a\in \mathbb{R}$. Then $\mathcal{C}^n(\mathbb{R})=:V$ and $$\langle f,g\rangle :=\sum_{k=0}^n {f^{(k)}(a)g^{(k)}(a)}$$ is a positive semidefinite Bilinear Form for all $f,g\...
0
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1answer
76 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 $$...
0
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1answer
20 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
147 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 ) $, \forall n \in \mathbb N$ ...
7
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2answers
96 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) \...
1
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1answer
55 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
26 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}\...
2
votes
6answers
109 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?
2
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0answers
24 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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0answers
22 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 $x$ (...
0
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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 ...
0
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0answers
38 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 ...
1
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1answer
52 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 $f$ ...
2
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2answers
44 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 $\...
1
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1answer
38 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 ...
6
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2answers
79 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 ...
0
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0answers
22 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 ...
0
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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 $k$...
0
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2answers
74 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)}{...
3
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0answers
23 views

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 ...
5
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2answers
98 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 $x$?
2
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1answer
53 views

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 ...
1
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2answers
17 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 ...
1
vote
1answer
19 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 ...
3
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1answer
53 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 ...
1
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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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4answers
207 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 ...
0
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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 ...
0
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2answers
61 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 ...
4
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0answers
60 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{\...
0
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0answers
34 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 ...
1
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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 ...
1
vote
2answers
28 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: $$ \frac{d}{dx}f(x)=-e^{\frac{1}{2x+2}}\left(sgn\left(x\right)+\frac{1-\left|x\right|}{2\left(x+1\right)^2}\right) $$ ...
0
votes
6answers
46 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 ...