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

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4answers
100 views

Finding the $n$-th derivative of $f(x)=e^{x}\sin(x)$

I am trying to find the general form for the $n$-th derivative of $f(x)=e^{x}\sin(x)$. I have calculated the derivatives up to $5$, but I am having trouble coming up with a general rule. Here is my ...
1
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1answer
60 views

The uniqueness of solution to $1+2^{\log_3x}=x$

I have this equation: $$1+2^{\log_3x}=x \text{ where } x \in \mathbb{R}$$ Anyone can immediately see the solution, $x=3$, but the remaining problem is to prove that $x$ is the unique solution. We can ...
3
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4answers
40 views

A particle moves along the x-axis find t when acceleration of the particle equals 0

A particle moves along the x-axis, its position at time t is given by $x(t)= \frac{3t}{6+8t^2}$, $t≥0$, where t is measured in seconds and x is in meters. Find time at which acceleration equals 0. ...
0
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0answers
34 views

Prove that $f(z(t)), f(w(t))$ are perpendicular at $t=0$

I have the following problem but I'm not sure if my proof is correct: Let $f(z)$ be a holomorphic function. Let $z(t)=a(t)+ib(t)$ and $w(t)=c(t)+id(t)$ be perpendicular at $t=0$. We have shown in ...
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0answers
21 views

From current plot $y=f(x)$ get plot $dx/dy$ vs $y$

I have a plot $y = f(x)$ where $y$ is voltage and $x$ is capacity. Now I want get from this graph the $dx/dy$ vs $y$ plot. How can I get this new graph?
3
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1answer
48 views

What is the derivative of $\int_{-10}^{-3} e^{\tan(t)} \,dt$ with respect to x?

We were learning about the Fundamental Theorem of Calculus today in my high school and the above integral came up as an example of an integral with a "constant" value. At first I accepted that the ...
0
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0answers
15 views

(beginner question) How to find points where a series stops being flat, or becomes flat?

I have a series of distributions that fall into three classes: series is flat(tish), then falls, then becomes flat(tish) again series is flat and remains flat series is flat(tish), then falls, and ...
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1answer
37 views

Proof on Differentiation in Banach spaces

Prove that f: $\Bbb R^2$-> $\Bbb R$, (x,y)$\mapsto$ x$^2$+ 2xy$^2$ +5y$^3$ is differentiable at (2,1) with DF(2,1)=[6,3]. Now I know that the partial derivatives 1) $\partial f/\partial x (2,1)=2x + ...
2
votes
3answers
281 views

Prove with use of derivative [closed]

How to prove this inequality using derivative ? For each $x>4$ , $$\displaystyle \sqrt[3]{x} - \sqrt[3]{4} < \sqrt[3]{x-4} $$
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1answer
28 views

What is a good resource for a more intuitive/flexible understanding of optimization

Take the following example of optimization: $$cost = 10*x + 20*y$$ Where x = cans of soup, y = cans of juice It is easy to see in this scenario what we need to do in order to minimize cost. Just ...
1
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2answers
63 views

Calculus optimisation with the speed formula

For a ship travelling at ${x}$ km/h the running cost in £ is ${(x^2 + {13500\over x})}$ per hour. Find the speed that minimises the cost of a 300km journey. The speed formula is ${speed = {distance\...
0
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2answers
44 views

Finding a two-variable function that is distinct from another on every open disk, with specifics.

Consider the two-variable function $$f(x, y) = \sin(x) + \cos(x) + y^2.$$ Find a two-variable function $g(x, y)$ that is distinct from $f(x, y)$ on every open disk which contains the point $(1, 2)$ ...
2
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0answers
38 views

Derivative of the area of a circle - Unsure why my answer is incorrect

The initial radius of a circle is $3$cm, but it grows at a rate of $\frac{1\text{cm}}{\text{second}}$ The problem is taken from this Khan Academy video I work out my answer in a similar way to his ...
0
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0answers
31 views

Is the following function continuously differentiable?

I am given a piecewise function, $f(x,y)=(xy,\frac{x^4}{x^2+y^2})$ if $(x,y) \neq 0$ and $f(x,y)=(0,0)$ if $(x,y)=0$. Thus $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am asked if this is ...
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1answer
44 views

How do you find the derivative of the following?

My main issue is that I do not know specifically, anything about the subject of differentiation. Let $m,n \in \mathbb{N}$. Let $A$ be an $n \times m$ matrix and $F_A:\mathbb{R}^m \rightarrow \...
2
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1answer
25 views

Distance between the nullpoints of the series of derivatives of ln(x)/x

I plotted a function $f(x) = \frac{ln(x)}{x}$, and continued with $f'(x)$, $f''(x)$, $f'''(x)$. I noticed how the intersections between the functions and the x-axis seemed to be roughly equally ...
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0answers
16 views

A question about notation in derivatives inn $\partial_{\bar{A}}L^I$

In this paper http://arxiv.org/abs/1210.2332, when the authors say eq (2.17): $$\partial_{\mu}L^I=\partial_{\bar{A}}L^I\partial_{\mu}\bar{z}^A+\partial_AL\partial_{\mu}z^A$$, does they mean by $\...
1
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1answer
101 views

prove that the following function is decreasing?

I am trying to prove that the following function is decreasing. \begin{align}&f(t)=\frac{1-g(t)}{\sqrt{1+e^t}}\cdot\exp\left(-\frac{te^t}{2(1-e^t)}\right)&t<0\end{align}where $ g(t)=\dfrac{(...
2
votes
1answer
110 views

Motion of particle on parabola

A particle moves along the parabola $y=x^2$ and has an acceleration vector directed toward the focus point $(0,\frac{1}{4})$. As the particle moves rightward through the origin, its speed is equal to ...
-1
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1answer
102 views

Calculation of Rise in Height of water in a Frustum of Right Circular Cone

A volume of frustum of right circular cone is calculated as follows. With known h, R & r of a container with the shape shown below, how to find out the rise in height for each time $7m^3$ of water ...
1
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1answer
142 views

Functional difference between d(total) and partial

edit 2(final answer in terms of python) ...
0
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1answer
58 views

Particle motion along a circle

There exists a particle which moves with constant speed 5 unit/sec along a circular path of radius 3 units which is centered at the origin in the plane given by the equation 2x+2y+z = 0. Calculate the ...
-1
votes
2answers
67 views

Integral of $\int\frac{1}{1+2e^x}dx$

It seems there are two ways to find the integral of this function $f(x) = \frac{1}{1+2e^x}$. In both paths I only do operations that I know are true, but for some reason one of them gives me the right ...
0
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1answer
42 views

Find $F(x)=\int_0^xf(t)\,{\rm d}t$ where $f(x)= 1/(x-5)^2 + x^3$ [closed]

Given the function $f(x)= 1/(x-5)^2 + x^3$, find $F(x)=\int_0^xf(t)\,{\rm d}t$. I'm not sure how to go about this problem since my function is in terms of $x$ and not $t$.
8
votes
2answers
102 views

Is $f(x)=x|x|$ differentiable everywhere?

When $f$ is a function $\mathbf{R}$ to $R$. I know $\lim_{x \to 0+}\frac{f(x)-f(0)}{x-0}= \lim_{x \to 0+}\frac{x^2}{x}=0$ and $\lim_{x \to 0-}\frac{f(x)-f(0)}{x-0}= \lim_{x \to 0-}\frac{-x^2}{x}=0$, ...
0
votes
0answers
19 views

Local Maxima in two variable function

Given the following function: f(x,y) = 1,000,000*y/(x+y)-y How do I find a local maxima of the function? I understand that I should calc dx=0 and dy=0 and then ...
0
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0answers
21 views

Mean value theorem proof problem

Suppose f is defined and differential for every $x>0$ and $f'(x)\rightarrow 0$ while $x\rightarrow \infty$ $g(x)=f(x+1)-f(x)$ Prove that $g(x) \rightarrow 0$ as $x\rightarrow \infty$ I am curious ...
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0answers
16 views

inclusion $C^r(U,C^r(\mathbb{R}^m,Y)\subset C^r(U\times\mathbb{R}^m,Y)$

Let $U\subset\mathbb{R}^n$ be open, $Y$ Banach space, $r\in\mathbb{N}$. Define a map $C^r(U,C^r(\mathbb{R}^m,Y)\to C^r(U\times\mathbb{R}^m,Y)$ by $f\mapsto ((x,y)\mapsto f(x)(y))$ Question: Is ...
1
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4answers
57 views

Integration by parts - hint

I'm stuck on a passage on my textbook: $$ \int \frac{1}{(1+t^2)^3} dt = \frac{t}{4(t^2+1)^2}+\frac{3}{4} \int \frac{1}{(t^2+1)^2} dt$$ I know that it should be easy but I just can't figure out what ...
2
votes
2answers
42 views

Find the $\Delta y$ of $f(x)={1 \over x^2}$; $x=2; \Delta x = 0.01$

Find the $\Delta y$ of $f(x)={1 \over x^2}$; $x=2; \Delta x = 0.01$ when $\Delta y = f(x+ \Delta x) - f(x)$ So here's what I did: $$\Delta y = f(x+ \Delta x) - f(x) \\ \Delta y = {1 \over (x+ \...
0
votes
1answer
22 views

Given that $f_1$ and $f_2$ are differentiable, find $Df(x_o)$ in terms of $f_1'(x_o)$ and $f_2'(x_0)$. [closed]

this is my first time asking a question on here but I am completely stuck. to answer this question, we need to use linear approximation and I'm just confused Let $f: \mathbb{R} \to \mathbb{R}^2$, $...
0
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0answers
13 views

Behavior of $J/I$ w.r.t $m_1$, $I=\int_{m_1k}^{\infty} t^{N-k}e^{-t} dt$ and $J=(m_1k)^{N-k} e^{-m_1k}-\int_{m_1k}^{\infty} \log(t) t^{N-k}e^{-t} dt$

Let us define $I=\int_{m_1k}^{\infty} t^{N-k}e^{-t} dt$ and $J=(m_1k)^{N-k} e^{-m_1k}-\int_{m_1k}^{\infty} \log(t) t^{N-k}e^{-t} dt$. We assume that $m_1 \ge 0$, $k \ge 0$ and $k \le N$. Using the ...
0
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3answers
75 views

Does differentiation of $f(x)=\log(x)$ yield two different results?

The two different results are :$\frac{1}{x}$ and $\frac{-1}{x}$. I read in my book that: $$\frac{d(\log x)}{dx}=\frac{1}{x}$$ where $x>0$ And: $$\frac{d(\log(-x)}{dx}=\frac{1}{x}$$ where $x<...
1
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3answers
55 views

Total derivative notation help

consider the function $$f = f(x(t),y(t))$$ I know that its total derivative wrt t is $$\frac {df}{dt} = \frac {\partial f} {\partial x} \frac {dx}{dt} + \frac {\partial f}{\partial y} \frac {dy}{dt}...
0
votes
1answer
48 views

Find the derivative of $y = x^{1/2}$ by using differentiation from first principle. [duplicate]

For this question, I tried to apply the derivative limit formula on it but I have a problem with the square root there: $$\lim_{\Delta x \rightarrow 0}\frac{\sqrt{x+\Delta x}-\sqrt x}{\Delta x}$$ If I ...
8
votes
7answers
398 views

How do I simplify and evaluate the limit of $(\sqrt x - 1)/(\sqrt[3] x - 1)$ as $x\to 1$?

Consider this limit: $$ \lim_{x \to 1} \frac{\sqrt x - 1}{ \sqrt[3] x - 1} $$ The answer is given to be 2 in the textbook. Our math professor skipped this question telling us it is not in our ...
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2answers
37 views

How to differentiate $ y=\sin^2(2x)\cos(x) $?

I was solving some A Level past papers and I came across this question. We have the equation of the line $ y=\sin^2(2x)\cos(x) $ for $ 0\leq x \leq \frac{\pi}{2} $ and there is a maximum point M. We ...
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1answer
20 views

Taking the derivative of a function of a convex combination of vectors, $f((1-t)x + t\cdot y)$

Let $f$ be a differentiable function, $x\not = y$ and vectors (say in $\mathbb{R}^n)$, and define $g:(0,1] \to \mathbb{R}$ by $$ g(t) = f((1-t)x + t\cdot y) $$ How would I differentiate this with ...
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2answers
34 views

Need clarifying on basic derivatives of natural log/e

So here's the question: Find the derivative: $ y= e^{\cos(x)}$ Hint: This is a combination of the chain rule and the natural log. The derivative is $(\ln a)(a^{f(x)}) * f'(x)$ So ...
0
votes
5answers
87 views

What is the derivative of $2x + \sin 2x$

I can't figure out how to differentiate $2x + \sin 2x$. I'm not sure if I should multiply the $2 + \cos 2x$ by $2$. Basically I want to know what is the correct way to differentiate $2x + \sin 2x$.
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1answer
39 views

General clarification for derivative notation

I am a bit confused on the different notations of derivatives, could you help me clear it up? The following can be interpreted as: the total derivative of f wrt x, or equivalently, the derivative ...
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2answers
25 views

Sufficient conditions for applying Taylor theorem

Consider a real-valued function $f:\mathbb{R}\rightarrow \mathbb{R}$. Is assuming $f(.)$ twice differentiable at $a \in \mathbb{R}$ enough to apply the Taylor Theorem stating $$ f(x)=f(a)+f'(a)(x-a)+\...
1
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2answers
54 views

Find if $\sqrt[4]{x^4+y^4}, \sqrt{x^4+y^4}$ are differentiable in $(0,0)$

Find if $$f(x,y)=\sqrt[4]{x^4+y^4}$$ $$g(x,y)=(f(x,y))^2$$ are differentiable in $(0,0)$. well, $g(x)$ is clearly $\sqrt{x^4+y^4}$, so I guess the answer will be similar to $f(x)$. $f_x=x^3/(...
0
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1answer
77 views

Implications of bounded second derivative

Consider a real-valued function $f: \mathbb{R}\rightarrow \mathbb{R}$. Suppose we are said that the second derivative exists and is bounded in a neighbourhood of $x\in \mathbb{R}$. Does it imply that ...
0
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1answer
38 views

A partial derivative problem related with elasticity of substitution in Advanced Micro

Exe 3.8 Sorry, it is a problem that appears in Jehle and Reny Advanced Microeconomic Theory (3rd ed) exercise 3.8. But I think it's a partial derivative question. Letting $f_i(\mathbf{x})=\partial f(\...
4
votes
2answers
50 views

First derivative meaning in this case

If we have a function: $$f(x)=\frac{x}{2}+\arcsin{\frac{2x}{1+x^2}}$$ And it's first derivative is calculated as: $$f'(x)=\frac{1}{2}+\frac{1}{\sqrt{1-\big(\frac{2x}{1+x^2}\big)^2}}\frac{2+2x^2-4x^2}{(...
1
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4answers
40 views

Differentiabillity and continuity

If I have a function like $f(x)= \left\{\begin{array}{lr} 2, & \text{for } x>0\\ -2, & \text{for } x\leq0 \end{array}\right\}$ it is obviously not continuous in $x=0$...
1
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1answer
154 views

Prove that this function is differentiable

I came across this problem while I was studying for a preliminary exam and now I've devoted quite some time to it and can't figure it out. Any help would be greatly appreciated! Let $f : \mathbb R \...
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0answers
30 views

Explanation of Template Matching formula

Can someone please explain the formula f.) on OpenCV template matching Formula: Suppose template image is 3x4 and source image is 15x20 how would the mathematical operations follow...
3
votes
1answer
40 views

$f \in C^1$ defined on a compact set $K$ is Lipschitz?

Let $f: \Omega \subseteq \mathbb{R}^N \to \mathbb{R}^M$ be $C^1$, and $K \subseteq \Omega$. Prove that $f \mid_K$ is Lipschitz. Letting $x,y \in K$, I know that $f$ is loccaly Lipschitz, I thought ...