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

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Real Analysis Question- Differentiability on an interval

So this is the question I am trying to answer... At $(-2,0)$ and $(0,2)$ we just differentiate using normal rules of calculus, yes? Here is my attempt for at $x=0$. Is this correct? For d)I think ...
3
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3answers
103 views

Easy question : $\int (xdy+ydx)$

I am ashamed to ask such an easy question but, well: Lets say I got a function $$ f(x,y)=xy $$ Now let's compute the total differential of the function $$ d(f(x,y))=xdy+ydx $$ Now if I do $$ \int ...
1
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1answer
26 views

Find derivative using the definition of the derivative as a limit

$$f(x) = \frac{1} {\sqrt{x}}$$ find $f'(x)$ using the definition of the derivative as a limit. I know that $$ f'(x) = \frac{(x + \delta)^{-1/2} - (x)^{-1/2}}{\delta} $$ as $\delta$ goes to $0$. ...
2
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4answers
54 views

Real Analysis - differentiable

$f:[0,\infty]\rightarrow \mathbb{R}$ is twice differentiable. If $f''$ is bounded and exists the limit of $f(x)$ at infinity, then $\lim_{x\rightarrow \infty}f'(x)=0$. I tried to use the Taylor's ...
4
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7answers
369 views

Understanding derivatives

I don't know if this is written somewhere else. I've looked all over the internet so apologies if this has already been covered. I'm doing Year 12 Maths in Australia for what it's worth. In our ...
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0answers
12 views

selecting points on a domain which represent the derivative of a function

I'm working on some algorithm part of which entails me to subdivide a domain based on the derivative of a function. Let's just consider the 1D case with a closed and bounded domain $[a,b]$ for a ...
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2answers
73 views

derivative of $\ln(4)$

what is the derivative of $\ln(4)$? I am trying to find the derivative of this equation: $h(x)=\ln(\frac{x^3\cdot e^x}{4})$ by rules of logs I simplified the $h(x)$ to the following: ...
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0answers
17 views

Generalized “softmax”

I'm looking for a function $f$ from $\mathbb{R}^n$ to $[0,1]^n$, similar to softmax in the sense that is satisfies these properties: $\sum_i f(x)_i = c$, where $c$ is a chosen constant (i.e., c=1 ...
6
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2answers
86 views

Limit of $\frac{f\left(x+g\left(x\right)\right)-f\left(g\left(x\right)\right)}{x}$ as $x\to 0$

I'm trying to answer the following question: Let $f$ be continuously differentiable in all of $\mathbb{R}$ and let $g:\mathbb{R}\to\mathbb{R}$ be a function satisfying ...
1
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2answers
32 views

Using the same limit for a second derivative

I've been trying to answer the same question answered here: Second derivative "formula derivation" And I'm stuck in a step that is not addressed both in the answer and in the comments of ...
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0answers
15 views

diffeomorphism on intervals on R

I came across a line in a proof which involved a diffeomorphism $f:I_1 \rightarrow I_2$ (with $f$ a homeomorphism, $f,f^{-1}\in C^{\infty}$) mapping open intervals in R, which claimed that ...
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1answer
35 views

Solving second order nonhomogeneous linear equation

So i have the equation $$\frac{d^2y}{dt^2} + y = \sin(t)$$ I know the first step is to find the corresponding homogeneous equation, which i think would be: $$r^2+1=0$$ giving real roots and therefore ...
-1
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1answer
22 views

Check differentiablity of $f$ [on hold]

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 ...
-2
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0answers
24 views

A “prove or disprove question” on absolutely continuous functions [on hold]

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
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1answer
37 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 & ...
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1answer
20 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 ...
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1answer
54 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 ...
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3answers
51 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 ...
1
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1answer
27 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
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1answer
81 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
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1answer
35 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 ...
7
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1answer
74 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 ...
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1answer
27 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 ...
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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
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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})$ ...
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2answers
22 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 ...
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0answers
31 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 ( $(4-h^2)^{0.5}$ - ($4^{0.5}$) ...
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3answers
41 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
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1answer
33 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
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1answer
67 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 $$ ...
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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 ...
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1answer
44 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?
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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 ...
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0answers
24 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
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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
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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 ...
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2answers
28 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
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1answer
40 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
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1answer
40 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 $(a,b)$?
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3answers
60 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 ...
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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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0answers
64 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
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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
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1answer
31 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
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 ...
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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: ...
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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
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1answer
44 views

Differentiability question using the definition

Could anyone please help me with where to start with this question? (c)
2
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0answers
42 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
47 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.