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

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-1
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1answer
18 views

Check differentiablity of $f$

Consider a function \begin{equation*} f(x)=|\cos x|+|\sin (2-x)|. \end{equation*} At which of the following points f is not differentiable? a)$\{(2n+1)\frac\pi2:n\in \Bbb Z\}$ b)$\{n\pi:n\in \Bbb ...
-1
votes
0answers
16 views

A “prove or disprove question” on absolutely continuous functions

Let $f:\left[a,b\right]\rightarrow\mathbb{R}_{+}$ be an absolutely continuous function. ($a<b$). Prove or disprove that the right ( respectively left) derivative of f exists at each point of the ...
2
votes
1answer
27 views

Are the extrema of this function global or local?

Last question about this function, I promise. The function $f: \mathbb R \rightarrow \mathbb R$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; x < -3 \\ 0 & ...
0
votes
1answer
18 views

Derivatives of real part of function

My physics textbook gives me the complex form equation of simple harmonic motion as: $$z = Ae^{i(\omega _{o}t+\phi )}$$ and then defines $$ x = Re (z) $$ From there they argue that $$ \frac{\partial ...
1
vote
1answer
43 views

Chain rule and a function of a function of a function

Suppose we have a composite function: $f(g(h(x)))$, and we want $\frac{\partial f}{\partial h}$. By the chain rule $\frac{\partial f}{\partial h} = \frac{\partial f}{\partial g}\frac{\partial ...
1
vote
3answers
49 views

Derivative of a Rational function $f(x)=\sqrt{2x-5\over3x+1}$

I'm trying to find the derivative of, $$f(x)=\sqrt{2x-5\over3x+1}$$ I think I can change this into $$f(x)= \left({2x-5 \over 3x+1}\right)^{1\over2} \\ =[(2x-5)(3x+1)^{-1}]^{1 \over 2}$$ Am I not ...
0
votes
0answers
19 views

Can someone please help me to prove if a matrix is non-negative?

Let $\textbf{r}_{1}$ and $\textbf{r}_{2}$ be two symmetric, diagonal dominate, Metlzer matrices. Let $\textbf{F}(m)= (\textbf{I}-e^{m\textbf{r}_1})(\textbf{I}-e^{m(\textbf{r}_1+\textbf{r}_2)})^{-1}$. ...
1
vote
1answer
25 views

Deciding whether $(ax+b)/(cx+d)$ is increasing

In order of studying usual function such us : 1) $f(x)=ax^2$ 2) $g(x)=\frac{a}{x}$ 3) $h(x)=\frac{ax+b}{cx+d}$ 4) $p(x)=ax^2+bx+c$ To know if a function is increasing or decreasing we can ...
1
vote
1answer
73 views

Feynman Integration Problem

$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 ...
1
vote
1answer
33 views

Critical Point of $\mathbb{R}^2 \to \mathbb{R}^2$ function

Given a function $f:\mathbb{R}^2 \to \mathbb{R}$ I can find critical points by finding the $1\times 2$ Jacobian matrix, setting each partial derivative equal to zero and solving the equations. I can ...
6
votes
1answer
61 views

If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$

Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that $f(0)=f(1)=f(2)=0$ Prove that $\forall x\in[0,2], \exists c\in[0,2], f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ This problem got me stuck. I ...
1
vote
1answer
25 views

Can someone please help me in proving this?

Let $k_{2}>k_{1}>0$, prove that for any $x>0$, $f(x)$ is a monotonically increasing function. $$ f(x)=\frac{1-e^{-k_{1} x}}{1-e^{-k_{2} x}}. $$ We can have ...
0
votes
2answers
27 views

With $f(x)= 32 \cosh(x) \sinh(2x) $, determine the slope of its tangent at $( \ln 2 , \, 75)$

With $f(x)= 32 \cosh(x) \sinh(2x)$, determine the slope of its tangent at $( \ln 2 ,\, 75)$. My work $$\sinh x \cosh y = \frac{1}{2}(\sinh (x + y) + \sinh (x - y))$$ $$\cosh(x) \sinh(2x)= ...
0
votes
2answers
25 views

$y = 3\sin^{-1}(\sqrt{x})/x$ find $y'(1/4)$

$y = 3\sin^{-1}(\sqrt{x})/x$ find $y'(1/4)$ my work is that y'= (x*$ ('\sin^{-1}(\sqrt{x}))+\sin^{-1}(\sqrt{x}) *1$)/(x)^2 my problem how to Derivative $ \sin^{-1}(\sqrt{x})$ ...
1
vote
2answers
20 views

Derivatives problem

Given the equation $f(x)=\frac{2x+4}{\sqrt{x}}$, evaluate $f(0.5)$ and $f'(0.5)$. I am having a problem understanding the problem. The first part is straight forward, but it's the second part I'm ...
1
vote
0answers
30 views

Prove that $g$ is differentiable and that $g'$ is not differentiable at $0$

For part ($c$) I assume I have to show that the limits from above and below are equal? I am having trouble doing this though... I get limit as $h$ tends to $0$ of ( $(4-h^2)^{0.5}$ - ($4^{0.5}$) ...
1
vote
3answers
38 views

A really basic integration question concerning differentials

I'm really, really confused with this. Please, please help me. $$$$ My Calculus teacher taught me that the integral symbol and the differential with respect to which we are integrating are like ...
0
votes
1answer
32 views

if g is not constant zero, $f\circ g$ has a local minimum at zero

Consider $f:\mathbb{R}^2\to\mathbb{R}\; f(x,y)=(x^2-y)(x^2-3y)$ and a linear function $g:\mathbb{R}\to\mathbb{R}^2,\; x\mapsto \begin{pmatrix} g_x(x)\\ g_y(x) \end{pmatrix} $. The claim is: If $g$ ...
0
votes
1answer
63 views

How to prove this limit of derivative to zero [on hold]

This is a test question in real analysis and I need help to prove it. Let $f:\mathbb{R}\to\mathbb{R} \in C^{\infty}$ be periodic of period $1$ and nonnegative. Show that $$ ...
0
votes
0answers
15 views

Can we calculate the derivative of a distribution function with respect to its parameters?

I am asking a very basic question. Can we calculate the derivative of a density function with respect to its parameters, mean and variance? Can we calculate the derivative of a distribution function ...
1
vote
1answer
42 views

Alternate formula for $Γ(t + 1)$

I believe that: $$\frac{d^{n}}{dx^{n}}[x^{n}] \equiv Γ(n + 1) \equiv n!$$ Would this have any application, if it has not already been discovered, which I am almost certain that it has?
0
votes
1answer
26 views

Differentiating expression involving summation

My problem seemed very simple at glance but I keep missing one term from the answer. Any suggestions? This is the problem: We have $$x_i^* + \xi_i + \frac{\alpha_i}{p_i} \left[ y - \sum_{j=1}^n ...
-4
votes
0answers
23 views

Call spread derivative [on hold]

delete this tbh lads, this is not the site for finance questions apparently Using the notation $V(E)$ to mean the value of a European call option with strike $E$, what can you say about ...
0
votes
1answer
21 views

derivative of infinite sum of terms

If we have $\sum_{i=1}^\infty f_i(x)$ and assume this is a convergent sum and asumme all the $f_i$ are differentiable in every point. Is the derivative of the infinite sum equal to the sum of the ...
0
votes
1answer
27 views

Differentiating a sum involving logs

I was doing the problem provided in the picture but I do not understand how do they obtain the answer. I am not sure how to differentiate the sum. I end up getting: alpha - 1 - 1/K. I believe I need ...
0
votes
2answers
27 views

The Fourier transform of functions with compact support is differentiable.

1) How can I prove that if $f(x)$ is a continous function with compact support (let's say $f(x)=0$ $\forall x\in B(0,R)^c$), then its Fourier transform $\hat{f}(\xi)$ is differentiable? 2) Is there ...
2
votes
1answer
39 views

Can critical point that $f''$ has different sign in its every neighborhood be a local extreme point?

Suppose that $f$ is a second order derivable function on $[0,1)$, and $f'(0)=0$. It is true that: If there exits $\delta>0$ such that $f''(x)\geq0$ for all $x\in[0,\delta)$, then $0$ is a local ...
2
votes
1answer
39 views

Can a function be differentiable at the end points of its (closed interval) domain?

Assume $f$ has a domain of $[a,b]$. Is it possible that $f$ is differentiable on the closed interval $[a,b]$, or must the maximal domain for $f'$ be $(a,b)$?
4
votes
3answers
59 views

Tangent of a Straight Line

I just got back a math test in my EF Calc class, and I disagree with my teacher on this one problem. We are using derivatives to determine equations of lines tangent to a given equation. The equation ...
0
votes
0answers
18 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 $y>r$. ...
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votes
0answers
63 views

A not very easy problem… [on hold]

I leave a challenge, a derivative problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - \alpha x \sin x = \mathcal{O}(x^4), \text{ as } x\to 0 $$
0
votes
1answer
23 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
votes
1answer
29 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.
2
votes
3answers
88 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 ...
0
votes
2answers
27 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
39 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)} - ...
0
votes
1answer
37 views

Differentiability question using the definition

Could anyone please help me with where to start with this question? (c)
2
votes
0answers
39 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
votes
0answers
45 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.
0
votes
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
votes
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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votes
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
votes
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
54 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
45 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
vote
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
votes
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
votes
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 ...