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

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2
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1answer
92 views

Rearranging an equation to form the limit definition of derivative

I am following a proof which starts with the following inequalities: $$S_{i}(v) \geq S_{i}(v+dv) + (-dv)P_{i}(v+dv)$$ $$S_{i}(v+dv) \geq S_{i}(v) + (dv)P_{i}(v)$$ From this, we rearrange to form: ...
1
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3answers
36 views

Differentiable approximation $f(x) = x$ for $x>0$ and $0$ otherwise.

I would like to find a twice continuously differentiable approximation of $$f(x)= \begin{cases} 0 & x\leq 0 \\ x & x>0. \\ \end{cases}$$ Are there any approximations ...
0
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4answers
47 views

Differentiate the following expression: $f(x)=2x^2 + 7x - \ln(x^2+1)$

Differentiate the following expression: $f(x)=2x^2 + 7x - \ln(x^2+1)$ I changed the $\ln(x^2+1)$ to $\frac{1}{x^2+1}$, but I don't know how to solve this.
0
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0answers
29 views

To find maximum number of solutions to $f(x) = y$ in (0,1)

Let f : (0,1) to R be a continously differentiable function such that f' has finitely many zeroes in (0,1) and f' changes its sign at exactly two of these points .Then for any y , element of R ,the ...
2
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0answers
45 views

Truth of an inequality involving differentials

Is the following inequality true? $$ s\frac{\partial \frac{\partial f(s,t)}{\partial s}}{\partial t}-\frac{\partial f(s,t)}{\partial t}>0 $$ Given that $f(s,t)$ is a monotonically-decreasing ...
3
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4answers
110 views

Find derivative of $x^{x^x}$

Trying to find the derivative of: $$ x^{x^x} $$ I have a solution but cannot understand the third transition:
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2answers
41 views

Differentiable $N$ times, $f'(x_{0})=\cdots=f^{(N-1)}(x_0)=0$, $f^{(N)}(x_0)>0\implies f$ increasing on $x_0$?

Let $I\subseteq \mathbb {R}$ be an open interval and $f:I\rightarrow \mathbb {R}$ is differentiable $N$ times in $x_0\in I$. It's given that: $$f'(x_{0})=\cdots=f^{(N-1)}(x_0)=0, \qquad ...
1
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0answers
55 views

Need help to find maximum of this summation equation

I tried to solve secretary problem myself, and I found its bruteforce equation. I am proud that I understood what it is and I can also solve its variations by changing equation. However I don't have ...
0
votes
2answers
132 views

Intuitive meaning of second, third and fourth derivatives at a point.

Can someone explain me the intuitive meaning of second, third and fourth derivatives of a function say, $f(x)$ at a point (say, $a$)? I know it's hard to explain to someone novice like me! But an ...
3
votes
3answers
85 views

How to simplify $y = \frac{\sin\rho + \sin2\rho}{\cos\rho + \cos2\rho}$

How can I simplify this function before I differentiate it? $$y = \frac{\sin\rho + \sin2\rho}{\cos\rho + \cos2\rho}$$ Of course you could immediately start off by using the quotient rule, but that ...
0
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0answers
20 views

System of derivatives

System: $$H\to ^{\alpha I} I \to^{\beta}$$ $$\therefore \frac {dH}{dt}=-\alpha I H$$ and$$\frac {dI}{dt}=\alpha I H-\beta I$$ $$\alpha=0.01 \;and \;\beta=0.2$$ Initial conditions: $$H(0)=800 \; and ...
3
votes
4answers
73 views

Finding the largest constant $C$ such that $|\ln x−\ln y| \geq C|x−y|$ for all $x, y \in (0, 1]$

Find the greatest value of C such that $|\ln x−\ln y|≥C|x−y|$ for any $x,y∈(0,1]$. What should my approach be? I can't think of much options.
3
votes
2answers
76 views

Find the derivative of $\arccos\frac{b+a\cos x}{a+b\cos x}$

Find the derivative of $\arccos\dfrac{b+a\cos x}{a+b\cos x}$ is there a smart way to find this derivative i tried by the conventional chain rule way, and it got very complicated
0
votes
1answer
103 views

For what values of $x$ in $[0,2π]$ does the graph of $y=\cos{x}/(\sqrt{3}+\sin{x})$ have a horizontal tangent?

For what values of $x$ in $[0,2π]$ does the graph of $y=\cos{x}/(\sqrt{3}+\sin{x})$ have a horizontal tangent? List the values of $x$ below. I solved the problem and at last it came out like ...
1
vote
1answer
50 views

shift taylor series coefficient

Let say I have analytic function $f(z)$ with taylor series $\sum a_nZ^n $ I want to find function $g(z)$ that It's taylor is $\sum a_{n+1} Z^n $ I need that for every $n>1$ : $g ^{(n-1)}(z)$ = ...
1
vote
2answers
413 views

Differentiable approximation of the absolute value function

Are there any good approximations of the absolute value function which are $C^2$ or at least $C^1$? I've thought about working with exponentials and then adding in more terms to keep the function from ...
2
votes
0answers
63 views

Is $f(x,y)=xy/(x^{2}+y^{2}) $ differentiable or continuous?

I'm taking a course in Analysis of several variables and the text we're following is Analysis on Manifolds - Munkres. I'm having issues to interpret properly the results I'm getting in my exercises. ...
3
votes
1answer
81 views

When does differentiability of $g\circ f$ and $f$ resp. $g$ imply differentiablity of $g$ resp. $f$?

To me the following seems intuitively true: If $f$ is differentiable at $x$ with surjective derivative then $g$ is differentiable at $f(x)$ iff $g\circ f$ is differentiable at $x$. On the other ...
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0answers
22 views

Is $\left\lVert x \right\rVert*x$ a function of class $C^{\infty}$

I just liked to know if that function ($\left\lVert x \right\rVert*x$) is of class $C^{\infty}$ in $x=0$. I think it is'nt, and I proved that, but my teacher said me to prove that this function is of ...
7
votes
1answer
208 views

Condition for increase in the optimum of a general function

For a function $f(x,y)$ with the following properties: $f(x,y)$ is strictly increasing as a function of $x$ $f(x,y)$ is strictly decreasing as a function of $y$ $\lim_{x\to\infty}\frac{\partial ...
0
votes
2answers
40 views

Delta notation in Thermodynamics

Assume we want to calculate the finite Enthalpy change for a process. $$H=U+pV$$ $$\Delta H=\Delta U + \Delta(pV) $$ Everything clear so far, but I do not understand how my teacher consecutively ...
1
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1answer
39 views

How to solve this differential equation??

Good morning (or evening) to everybody. I would like to know how may I work to solve this differential equation: $$\dot{R}^2 = \alpha\dot{r}^2 - \beta\dot{r}^4$$ Where $R$ is $R(t)$ and $r$ is also ...
2
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0answers
20 views

Unitary derivative for plane curves

Let $\mathbf{r}:I\to\mathbb{R}^2$, where $I\subseteq\mathbb{R}$ is an open interval, be a continuous function that is not constant on any subinterval $J\subseteq I$ such that at each point $t\in I$ ...
2
votes
0answers
25 views

A lateral differentiable function and a countable set

Let $f:I\subset\mathbb{R}\to\mathbb{R}$, where $I$ is an open iterval, be a function that admits lateral derivatives at each point in $I$. Is the following set countable: $J=\{x\in I\ |\ ...
1
vote
2answers
112 views

What does $\frac{dg}{dx}$ mean?

What does $\frac{dg}{dx}$ mean? Specifically, I'm trying to solve$$ \frac{1}{3}\frac{dg}{dx}\frac{1}{1+g^2} $$ where $$ g(x) = \frac{3x\left(1-x^2\right)}{x^4-4x^2+1} $$ I know $\frac{d}{dx}$ ...
0
votes
1answer
186 views

Show that the path followed by the boat is the graph of the function.

The problem I am trying to figure out is as follows: A man initially standing at the point O walks along a pier pulling a rowboat by a rope of length L. The man keeps the rope straight and taut. The ...
1
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1answer
47 views

Derivative not defined, but does exist?

Consider the function f: $\ (x,y) = y \sqrt{x^2 + y^2} $. The derivative at the origin is zero. However, if I calculate the partial derivative with respect to x, for example, I get the following: $ ...
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3answers
56 views

For what values of $k$ is $g(x)=x^3+kx^2+x$ one-to-one?

I need to find for what values of $k$ $g(x)=x^3+kx^2+x$ is one-to-one. I tried finding for what values it is strictly increasing and got the derivative to be $3x^2+2kx+1>0$, but I'm not really sure ...
0
votes
1answer
50 views

Updated: $\frac{d}{dx} [\frac{1}{3}arctan(\frac{3x(1-x^2)}{x^4-4x^2+1})+C]=\frac{x^4+1}{x^6+1}>0.$

For all C show that $\frac{d}{dx} [\frac{1}{3}arctan(\frac{3x(1-x^2)}{x^4-4x^2+1})+C]=\frac{x^4+1}{x^6+1}>0.$ So from the first answer below by Chinny84, $ ...
4
votes
2answers
70 views

Show that $f:\mathbb{R}^2\to\mathbb{R}$ is constant.

Let $f:\mathbb{R}^2\to\mathbb{R}$ such that $\left|f(x)-f(y)\right| \le \|x-y\|^2$ for every $x,y\in\mathbb{R}^2$. Show that $f$ is constant. So I think that if $f$ is constant then $\nabla f ...
0
votes
1answer
32 views

Gradient of this function

I am to solve an optimization problem as described below: $$ \min f(x) = \frac{1}{2}\left\lVert x - x_{b} \right\rVert^{2}+ \frac{1}{2}\left\lVert \epsilon \right\rVert^{2}$$ with $$ Hx -y = \epsilon ...
1
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0answers
68 views

Mean value theorem and differentiability in endpoint

The mean value theorem says that for $f: [a,b], \rightarrow \mathbb{R}$ cont on [a,b] and diff on $(a,b)$, there exists a $\xi$ so $f'(\xi) = \frac{f(b) - f(a)}{ x - a}$. Is it possible to replace ...
1
vote
1answer
46 views

If $f^2$ and $f^3$ are smooth, does it follow that $f$ is smooth?

Let $f: \mathbb{R} \to \mathbb{R}$ be given. Assume that the square and cube of $f$ are smooth. Is $f$ smooth? That is if $f \cdot f \in C^{\infty}$ and $f \cdot f \cdot f \in C^{\infty}$, does it ...
0
votes
1answer
71 views

tricky derivative second set of eyes…

Here is the original function: I was wondering if I could get an assist with figuring out the derivative and to see if I was missing any crucial steps in the process. I have attached an image of ...
-1
votes
3answers
71 views

How to find $y(x)$ from $y'+y=\cos x$?

How to find $y(x)$ from this equation? $$y'+y=\cos x$$ Would I just subtract $y$ and integrate?
2
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0answers
126 views

Sketch a function in the interval [-2,3] that has the following characteristics

Sketch a function in the interval [-2,3] that has the following characteristics: Here is what I got: Is this correct?
1
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0answers
21 views

How is quasiconcavity and the sign of cross partial related?

Suppose $f(x,y)$ is differentiable. If $f(x,y)$ is quasiconcave, is the cross partial nonnegative? Or, if $f(x,y)$ is strictly quasiconcave, is the cross partial strictly positive? My initial ...
1
vote
1answer
89 views

Uniform limit in definition of second order directional derivatives

If $f:E\rightarrow F$ is twice differentiable at $x\in E$, do we then have $$\lim_{h,k\rightarrow 0}\frac{A_x(h,k)-f''(x)(h)(k)}{\|h\|\|k\|}=0$$ where $A_x(h,k):=f(x+h+k)-f(x+h)-f(x+k)+f(x)$? This is ...
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2answers
158 views

Word Question Involving the Definition of the Derivative

A skydiver jumps out of a plane from a height of $2200$m. The skydiver's height, $h$ meters above the ground, can be modeled by the function $$s = 2200 - 4.9t^2$$ How fast is the skydiver ...
1
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1answer
55 views

How to best simplify a chain/product rule with lots of trig functions?

I've found the derivative of the following: $$g(x) = \sec(8x)\tan(5x^9)$$ to be $$g'(x) = 8\sec(8x)\tan(8x)\tan(5x^9) + 45x^8 \sec(8x)(\sec(5x^9))^2$$ I'm aware that the trig identities are ...
0
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1answer
38 views

Where does this function have a derivative of zero? Where is it not differentiable?

Use the following graph to answer the questions. Did I answer these questions correctly? I sketched the derivative roughly on the same grid, then noted (estimated) where the derivative was ...
2
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0answers
53 views

Graphing the derivative from a graphed function

For the graphs of $f(x)$ make a sketch of the graph $f'(x)$ Here are the sketches I have so far. The graphs on the top row are the original functions. Did I graph these correctly?
1
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2answers
22 views

Finding the derivative Maxwell Boltmann distribution

I need to basically show that the slope is $0$ when $\epsilon = kT$ $f(\epsilon)=\left(\frac{8 \pi}{m}\right)\left(\frac{m}{2 \pi kT}\right)^{\frac{3}{2}}\epsilon e^{\frac{-\epsilon}{kT}}$ So I ...
0
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1answer
31 views

Continunity of a two variable function in Apostol's analysis

In discussing how the concept of differentiability implying continuity cannot be applied to functions of several variables, Apostol proceeds to give an example to demonstrate why. The function he uses ...
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3answers
281 views

what is the relation between Limit and Derivative?

We have that: $$\begin{align} \lim_{x \to 2} x^2 &= 4\\ D_{x = 2}\left(x^2\right) &= 4 \end{align}$$ In both cases the result is $4$. So limit and derivative are always the same? If not, what ...
0
votes
1answer
49 views

Partial derivative of a CDF

How to calculate this partial derivative? $$\frac{\partial}{\partial a}\int_{-\infty}^a(a-x)f(x)\text{d}x$$where $f(x)$ is a pdf. Since I'm on the midway of a proof. So it will be great if the result ...
4
votes
1answer
109 views

Fundamental proof of Taylor's theorem using little-o notations

Is there a fundamental proof of Taylor's theorem using little-o notation? I assume $f:E\rightarrow F$ as a mapping between Banach spaces and write $(h^i)$ for $(h,\ldots,h)$ ($i$ times iterated). ...
0
votes
1answer
36 views

Prove that for $f: \mathbb{R} \to \mathbb{R} f'(x_{0}) \leftrightarrow \exists_{a \in \mathbb{R}} f(x_{0}+h) = f(x_{0})+a*h+R(h)$

Prove that for $f: \mathbb{R} \to \mathbb{R} \enspace f'(x_{0})$ exists and is finite $\Longleftrightarrow \exists_{a \in \mathbb{R}} \enspace f(x_{0}+h) = f(x_{0})+ah+R(h) \enspace $where ...
0
votes
2answers
45 views

Integration Under the integral sign on indefinite integrals

Is it possible to perform the differentiation under the integral sign for an indefinite integral (anti-derivative)? that is, if $f(s) = \int F(s,t) dt $ then, is $f'(s) = \int (d/ds(F(s,t)))dt$ ...
2
votes
1answer
78 views

absolute extrema problem…

My calc skills are kind of rusty. Was wondering if I could get an assist on this one perhaps? I am looking for the absolute extrema if they exist and all the x values they occur at in the domain. ...