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

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0
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1answer
37 views

Lipschitz Continuity of a Function of a Matrix

Define $f(A): \mathbb{R}^{p\times m} \to \mathbb{R}$ as follows: $$ f(A) = \frac{1}{2}\|Y-XAB\|_F^2 = \frac{1}{2}\text{tr}\{(Y-XAB)^T(Y-XAB)\}, $$ where matrices $Y\in\mathbb{R}^{n\times q}, X\in\...
0
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0answers
6 views

Signing *change* of probability that one random variable is lower than another

Let $\tilde{z}_L \in [0,1]$ and $\tilde{z}_H \in [0,1]$ denote two random variables, with $F_L(z|\theta) := \Pr\{\tilde{z}_L \leq z|\theta\}$ and $F_H(z|\theta) := \Pr\{\tilde{z}_H \leq z|\theta\}$. ...
1
vote
2answers
32 views

Differentiability of a function and its square root

Consider a function $f:\Theta \subseteq \mathbb{R}^l \rightarrow [0,\infty) $. Let (1) $\sqrt{f(\theta)}$ is differentiable at $\theta_0 \in \Theta$ (2) $f(\theta)$ is differentiable at $\theta_0\in ...
0
votes
1answer
19 views

Derivative of a summation.

If a function $E={1\over2}\sum_{n=1}^N(y_k-t_k)^2$ And if $a_k = y_k$ then how ${\partial E \over {\partial a_k}} =y_k - t_k$ Can anyone please tell me how final answer was obtained using partial ...
0
votes
1answer
54 views

Is there better way to solve this derivative: $((5\tan 5x - 3\cot 5x)\arcsin(\frac{x+3}{x-1}))'$?

I've done $$(5\tan 5x- 3\cot 5x)'\arcsin\frac{x+3}{x-1} - (5\tan 5x- 3\cot5x)(\arcsin\frac{x+3}{x-1})'$$ And I've gotten $$5\left(\frac{5}{\cos^25x}+\frac{3}{\sin^25x}\right)\arcsin\frac{x+3}{x-1} ...
1
vote
0answers
51 views

Understanding derivative notation in those equations

I am given the following set of equations from a physics course, which is about longitudinal waves in rods. My questions are: On the second line you have $ (\frac{\partial \Delta}{\partial x})dx $...
0
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1answer
30 views

Question on one-sided derivatives

Assume we have a function $f$, say on $\mathbb{R}$, such that $f$ is continuously differentiable in all $x$ smaller than some given $x_0 \in \mathbb{R}$. I am a bit confused about the connections of ...
0
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1answer
50 views

Determine where $f(x)=\sin(2x)$ is increasing and decreasing and find absolute extrema on $(0,2\pi)$

So this is the problem: determine where $f(x)=\sin(2x)$ is increasing and decreasing and find absolute extrema on $(0,2\pi)$. I took the derivative and found it to be $f'(x)=2\cos(2x)$. When setting ...
2
votes
3answers
102 views

Finding the $n$-th derivative of $f(x)=\log\left(\frac{1+x}{1-x}\right)$

I am trying to find the general form for the $n$-th derivative of $f(x)=\log\left(\frac{1+x}{1-x}\right)$. I have rewritten the original formula as: $\log(1+x)-\log(1-x)$ for my calculations. I ...
1
vote
2answers
66 views

How to find derivative of $\sin\sqrt{x}$ using difference quotient?

The definition of derivative of a function $f(x)$ is $$\lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$ Using this definition, the derivative of $\sin\sqrt{x}$ will be: $$\lim_{h\to0} \frac{\sin\sqrt{x+h}-\sin\...
1
vote
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
vote
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
votes
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
33 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 ...
1
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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
votes
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 ...
0
votes
1answer
36 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
271 views

Prove with use of derivative [closed]

How to prove this inequality using derivative ? For each $x>4$ , $$\displaystyle \sqrt[3]{x} - \sqrt[3]{4} < \sqrt[3]{x-4} $$
0
votes
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
59 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
votes
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
votes
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
30 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 ...
0
votes
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
votes
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 ...
1
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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
vote
1answer
100 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
votes
1answer
93 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
vote
1answer
137 views

Functional difference between d(total) and partial

edit 2(final answer in terms of python) ...
0
votes
1answer
55 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
votes
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
101 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
votes
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 ...
0
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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
vote
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
votes
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
vote
3answers
54 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
397 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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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
85 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 $2x + \sin 2x$.