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

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13 views

Different results on doing $\frac{\partial}{\partial y}\left(\int_r^y \frac{1}{\sqrt{y^2-s^2}} ds \right)$ in different ways

I have a confusion when trying to get the result of the expression below, $$ I = \frac{\partial}{\partial y}\left(\int_r^y \frac{1}{\sqrt{y^2-s^2}} ds \right). $$ All variables are real and $y>r$. ...
-5
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0answers
57 views

A not very easy problem… [on hold]

I leave a challenge, a derivative problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - \alpha x \sin x = \mathcal{O}(x^4), \text{ as } x\to 0 $$
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1answer
22 views

Speed at which hands of clock approaching one another.

A clock's hour hand's length is $1$, and its minute hand's length is $r$. First I had to find the distance between the tips of the hands at 4:00. I did this using the law of cosines. This gives me ...
0
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1answer
26 views

Find the number of points of where this function isn't differentiable.

I want to find the number of points of where this function isn't differentiable: $$f(x) = \max\{4,1+x²,x²-1\} $$ I tried drawing a graph but it didn't help.
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3answers
82 views

Differentiation under the integral sign: Where is my mistake?

So I'm trying to find $\int_0^\infty \sin(x^2)\,dx$ by the method of differentiation under the integral sign. The idea is to use differentiation with respect to t on A(t) -- defined below -- and then ...
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2answers
25 views

$f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$

I think that I understand what the question wants me to do: $f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$ I worked out the partial derivatives: ...
5
votes
4answers
38 views

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

Find the derivative of the function $$f(x) = \frac{\cos{(x^2 - 1)}}{2x}$$ This is my step-by-step solution: $$f'(x) = \frac{-\sin{(x^2 - 1)}2x - 2\cos{(x^2 -1)}}{4x^2} = \frac{2x\sin{(1 - x^2)} - ...
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1answer
37 views

Differentiability question using the definition

Could anyone please help me with where to start with this question? (c)
2
votes
0answers
33 views

Having difficulty calculating a derivative using the definition and limit question

My first question is how do we justify that the limit of $\sin (x)\sin \frac{1}{x}$ as $x$ tends to $0$ is $0$? Would we say $-\sin(x)\leq \sin (x)\sin (\frac{1}{x})\leq \sin (x)$ and then use the ...
2
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0answers
38 views

Need example for derivative in relation to a product life cycle, please.

I am looking for help in explaining how one would use derivatives in relation to a product life cycle, with a mathematical example. Any help is greatly appreciated.
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0answers
17 views

Condition for all derivatives to be L-Lipschitz

Let $f:\mathbb{R}\to\mathbb{R}$ be a function with infinitely many derivatives and let us use the notation $$ f^{(n)}(x)=\frac{\mathrm{d}^nf(x)}{\mathrm{d}x^n}. $$ Assume that $f^{(n)}$ is ...
5
votes
2answers
72 views

Proving that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$ using derivatives

Let $a,b,c\in\mathbb{R}^+$ and $abc=1$. Prove that $$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$$ This isn't hard problem. I have already solved it in following way: Let ...
0
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1answer
24 views

Calculus Velocity and Acceleration

Here's the question: I know that: $a(t) = -9.8$ So I integrated the acceleration function to find the velocity: $v(t) = -9.8t + c$ And because $v(0) = -5$, I can determine that $c = -5$, thus: ...
3
votes
2answers
41 views

Asymmetric second difference quotient?

I need to find (approximate) the second derivative of a discrete function. Usually I would approximate the second derivative with $$f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}\tag{1}$$ In my case, ...
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1answer
17 views

Calculus: Velocity and Acceleration Question [on hold]

So I'm a little stuck on this question: I'd really appreciate it if you could explain how I'd go about working this out so I know the process, rather than just the answer. Thanks!
-1
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1answer
21 views

Given $g \in C^2(\mathbb{R})$ Study derivability of $f(x) = \frac{g(x)}{x}$ [duplicate]

Given $g \in C^2(\mathbb{R})$ ($g$ is a function twice derivable), with $g(0)=0$. consider the function $f: \mathbb{R}\rightarrow\mathbb{R}$ defined by $$ f(x) = \frac{g(x)}{x}\:\:\:\:\forall x \in ...
1
vote
4answers
51 views

Difficult Calculus Question (Differentiation)

I stumbled across this question in my maths textbook and have no idea how to solve it. I have looked at the answers and don't understand how to get it from the question. It's in the chapter titled ...
0
votes
2answers
41 views

solve diferential equation difficulties

I'm studying math and I've founded this equation: $\frac{dp}{dt}=0.5p-450$. I write it so: $p'=0.5p-450$. Derivating the two sides: $p''=0.5p' \Rightarrow p''-0.5p=0$ General solution: $m^2-0.5m=0 ...
1
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1answer
26 views

Calculate the (variational) derivative of the following equation;

Consider $ E[u]= \int^1_0 \big(u'(x)\big)^2+\big(u(x)\big)^2-2f(x)u(x) dx.$ Calculate the variational derivation for a function $v$; in other words, calculate $\frac{d}{d\epsilon}E[u+\epsilon v]$ at ...
0
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1answer
16 views

How to prove a function derivability in a interval? [on hold]

How can I see if a function is derivable on a specific interval. What method do you use?
-1
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3answers
46 views

find the derivative of the integral

Prove that the following integral $F(x)$ is differentiable for every $x \in \mathbb{R}$ and calculate its derivative. $$F(x) = \int\limits_0^1 e^{|x-y|} \mathrm{d}y$$ I don't know how to get rid of ...
7
votes
1answer
84 views

Does the inverse function theorem hold over $\mathbb{Q}$?

Let $f:\mathbb{Q}^n \longrightarrow \mathbb{Q}^n$. We can define what it means for such $f$ to be differentiable: (The differential will be a linear transformation $\mathbb{Q}^n \longrightarrow ...
2
votes
2answers
35 views

How can I determine the sequence which has this generating function?

In a discrete mathematics past paper, I must find the first eight terms of the sequence whose generating function is $$\frac{x^2}{(1-x)(1-2x)}.$$ I have looked at both of the following posts: How ...
4
votes
6answers
154 views

Are the any **non-trivial** functions where $f(x)=f'(x)$ not of the form $Ae^x$

This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is ...
0
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1answer
28 views

Q/ evaluate (d/dx)ln |tan x| .

Q/ evaluate (d/dx)ln |tan x| . If we apply this theorem ( ) then we will have just = ( 1/tan x ) But in the book they solve it like this ( ) So, what i'm asking about it is, why they but more ( ...
1
vote
1answer
36 views

Derivative of matrix exponential wrt to each element of Matrix

I have $x=\exp(At)$ where $A$ is a matrix. I would like to find derivative of $x$ with respect to each element of $A$. Could anyone help with this problem?
2
votes
2answers
68 views

L'Hôpital's rule exercise with natural log function

I'm looking for some advice on the following exercise: $$\lim_{x \to 0^+}{\ln{(\frac{1}{x}})}^x$$ This is my work so far: $$\lim_{x \to 0^+}{\ln{(\frac{1}{x}})}^x = \lim_{x \to ...
0
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2answers
36 views

Total Differential / Ito dynamics

I found this process in a scientific paper: $M_t = \int_{0}^t e^{-(t-u)} \frac{dS_u}{S_u}$ where $dS_t = S_t (\phi M_t + (1-\phi)\mu_t) dt + \sigma S_t dW_t$ and I want to compute the ...
7
votes
2answers
134 views

Polynomial Functional Equation.

Let $f(x)$ be a one-one, polynomial function such that $f(x)f(y)+2=f(x)+f(y)+f(xy) \ \forall \ x,y \in \mathbb R - \{0\}$, $f(1) \neq 1$, $f'(1)=3$. Find $f(x)$. I tried to find the degree of ...
2
votes
1answer
29 views

differentiable and uniform continuity of f and F

Given $f: \Bbb R \to \Bbb R$. define new function: $F(x) =\frac{f(x)-f(a)}{x-a}$ for $x\neq a$. Prove that $f$ is differentiable at $a$ if and only if $F$ is uniformly continuous in some punctured ...
0
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0answers
23 views

Given $g \in C^2(\mathbb{R})$ Study Derivability of $f(x) = \frac{g(x)}{x} $

Given $g \in C^2(\mathbb{R})$ ($g$ is a function twice derivable), consider the function $f: \mathbb{R}\rightarrow\mathbb{R}$ defined by $$ f(x) = \frac{g(x)}{x}\:\:\:\:\forall x \in \mathbb{R}^*, ...
0
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1answer
39 views

Proof for Mean of Geometric Distribution

I am studying the proof for the mean of the Geometric Distribution http://www.math.uah.edu/stat/bernoulli/Geometric.html (The first arrow on Point No. 8 on the first page). It seems to be an arithco ...
0
votes
1answer
44 views

Tangent map of orthogonal projection

Given a tangent bundle $TM$ and its natural projection $\pi:TM\to M$, I want to compute tangent map $T\pi:TTM\to TM$. Here is my method. Suppose a curve $c:\mathbb{R}\to TM$ with $c(0)=(x,y)$ and ...
-2
votes
2answers
42 views

Cauchy-Riemann equations Complex Numbers [on hold]

I have used the theorem if f'(z) = 0 then f(z) is a constant. I have proved it by using Cauchy Riemann's theorem. b
3
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2answers
34 views

Complex Differentiation

Can anyone give a hint to how to approach this question?
-2
votes
2answers
86 views

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$ [on hold]

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$... if $x > 0$ and $cos(x)$ $> 0$
0
votes
1answer
18 views

Maximum volume of a cuboid with constraints

Find maximum volume of cuboid for which sum of three dimensions (x,y,z) is not greater then 108. I am looking for the most straightforward approach to the question. Thus the volume will be $xyz$ and ...
0
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2answers
18 views

Using substitution to make an equation into a separable differentiable equation

I have the question: By making the substitution $y = t^nz$ and making a cunning choice of n, show that the following equations can be reduced to separable equations and solve them. $$\dfrac{dy}{dt} = ...
2
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4answers
68 views

What is the result of $\frac{d\dot x}{dx}$?

I have a problem with a step in assignment. What is the result of $\frac{d\dot x}{dx}$ ? $x$ is the displacement, and $\dot x$ is the speed. I'm not sure if this equation itself is right. Thank you! ...
0
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1answer
19 views

Computing the Frechet derivative

How does a computation involving the Frechet derivative typically proceed? The definition, $$\lim_{h \to 0} \frac{\| f(x + h) - f(x) - f'(h)\|}{\|h\|} = 0$$ seems somewhat cumbersome to use in order ...
3
votes
3answers
36 views

$\sqrt{y}+\sqrt{x}=\sqrt{A}$ … prove that x-intercept + y-intercept of any tangent = constant [on hold]

This is equation of a curve $\sqrt{y}+\sqrt{x}=\sqrt{A}$ $A$ is constant $T$ is a tangent of the curve from any point on it $B$ is y-intercept of $T$ $C$ is x-intercept of $T$ ...
2
votes
4answers
43 views

Gradient of a curve $y=\ln \sqrt{x+y}$

Find the gradient of the curve $y=\ln \sqrt{x+y}$ at the point when its y-coordinate is 1. My attempt, I differentiated and I got $\frac{dy}{dx}=\frac{1}{2x+2y-1}$. But I've problem in finding the ...
1
vote
1answer
39 views

Computing Fréchet derivative

I am reading Methods in Nonlinear Analysis by Kung-Ching Chang and having trouble in obataining a Fréchet derivative in the text. For those who has the book, it is on page 37, which concern Euler ...
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2answers
40 views

Is there has a smart way to compute the 1order derivative of the circle equation? [on hold]

I have encountered a compute problem. This exercise has given the circle equation and a para-curve equation with unknown parameters, the para-curve and circle has the same radius of curvature, and ...
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3answers
33 views

Logarithmic differentiation of $y=(x^2(7x-14)^{1/3})/(1+x^2)^4$

Honestly I have no clue how to rewrite then start it. I know you have to Ln both sides but how would you Ln the right side?
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0answers
13 views

Derivative of convoluted 2D image w.r.t. to its coefficients

I am creating an image with the following variables with the following dimensions: $A: (1,i)\\ X_a: (i,x,y)\\ B: (1,j)\\ X_b: (j,x,y)\\ Image=A\cdot X_a\odot B\cdot X_b $ Where $\odot$ stands for ...
0
votes
1answer
23 views

Calculate angle of inclination in NE direction [on hold]

A man is on the hill in a point $(-100, -100, 430)$, the hill is given by an equation $z=500-0.003x^2-0.004y^2$. What is the angle of inclination in NE direction? (i guess ne direction on the ...
0
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6answers
95 views

Why $\int 2 \cdot \frac{\ln(x)}{x} dx$ is $\ln(x)^2 + C$?

Why the integral of $2 \cdot \frac{\ln(x)}{x}$ is $\ln(x)^2 + C$ (where $C$ is of course a constant) ? After some years of my high school math classes, I am again doing derivatives and integrals, but ...
0
votes
2answers
18 views

Derivative of: $x\log_2(x)$

Can someone please help me with the derivative of this function: $$y = x\log_2(x)$$ This is the answer: $$1+\ln(x)\over\ln2$$ When I try to solve it I get stuck here: $$\log_2(x) + {x\over x\ln2} ...
0
votes
0answers
22 views

Differentiate $g\circ f$ transformation

Differentiate $g \circ f$ of the following functions: $$f: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$f(x,y)=(x-y,x+y)$$ $$g: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$g(x_1,x_2)=(e^{x_1} \cos ...