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

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Summation of a function with the variable both in the function amd in the upper limit

E is defined as : E = c1 ( a$\rho$ + b$\rho ^{2}$ ) + c2 $\rho$ ( c + d $\sum_{j=0}^{n} (\log{ \frac{R\rho}{j} } ) $ ) + c3 $\rho ^{2}$ a, b, c, d, c1, c2, c3, R are known constants. $\rho$ is the ...
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1answer
36 views

General formula for $\dfrac{\partial^k}{\partial x^k} \left(\frac{f(x)}{g(x)}\right)$

I would like to know the general formula for expressing $\dfrac{\partial^k}{\partial x^k} \left(\dfrac{f(x)}{g(x)}\right)$ in terms of derivatives of $f(x)$ and $g(x)$. I am stuck when trying to ...
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3answers
37 views

Calculating $\displaystyle \lim_{x \to 0+} \frac{\log(\cos(x))}{x}$ where the domain of the quotient is $(0, \pi/2)$

Calculating: $$\displaystyle \lim_{x \to 0+} \frac{\log(\cos(x))}{x}$$ where the domain of the quotient is $(0, \pi/2)$ The fist step is setting $f(x)=\log(\cos(x))$ and $g(x)=x$, and verifying ...
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2answers
38 views

If $f$ is increasing on an open interval and continuous at endpoints, it's increasing on the closed interval.

Prove that if $f$ is increasing on $(a,b)$ and continuous at $a$ and $b$, then $f$ is increasing on $[a,b]$. The question then clarifies: "In particular, if $f$ is continuous on $[a,b]$ and $f'>0$ ...
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2answers
36 views

differentiation of tan(-x)

I've just started high school calculus. To differentiate trig functions the rule is $(f \circ g)' = g'(x) \cdot f'(g(x))$ So for $\tan(-x)$ would this not be $-\sec^2(-x)$? The answer says ...
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1answer
36 views

How exactly do I 'see' the function I need to make for optimization?

Optimization problems in Calculus seems to be my white whale. I always seem to struggle with it. I know that once I find the function I need to manipulate with it's pretty much smooth sailing from ...
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13 views

Second Frechet derivative: is there a mistake?

Let $x = (x_1, x_2, \ldots) \in l_2$ and $J(x) = \sum_{i = 2}^{+\infty} x_{i- 1}x_{i + 1}$ Calculate the first and second Frechet derivatives. Attempted solution First, let's notice that we can ...
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0answers
18 views

Hessian of a non-linear Matrix function

Apologies if this is a silly question, but I am really confused. I am trying to find the Hessian of a non-linear function $f$. I understand that the Hessian of $f$ with respect to $A$ is the Jacobian ...
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2answers
119 views

First derivative of multiplied powers

Wolfram Alfa shows $\frac{d}{dx}e^{4y} = 4e^{4y}$ but I do not understand how to get to that answer I have $e^{4y} = (e^4)^y$ So by the chain rule is it not the case that \begin{align} ...
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28 views

What is the equation for a tangent to the graph of $y=\arcsin(x/2)$ at the origin?

I believe arc sin is the same as inverse sin but then I don't know how to deal with taking the derivative of that.
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1answer
22 views

Problem about partial derivatives

I have two problems: Let $$f : \mathbb{R}^2 \to \mathbb{R}, f \in C^2$$. Problem 1 Find all functions such that $$\frac{\partial^2f}{\partial x \partial y} = 0$$ Problem 2 Find all functions ...
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0answers
12 views

Derivative of the Total Variation of a digital image?

Thanks for your time reading my thread. I am trying to calculate the derivative of the Total Variation (TV) of a digital image with respect to its gray-scale intensity. Say, there is an image: ...
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0answers
37 views

Books explaining differentiation under the integral sign

I've heard that this is a great tool to have in you math toolkit, but I cannot comprehend this method just from the wiki entry and 2 page pdf files. I'm looking for a book which has problems ...
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1answer
45 views

proving differentiability of functions in $\mathbb{R}^2$

Define $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ by $$f(x,y):= \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} &\text{when} \, (x,y) \neq (0,0) \\ 0 &\text{when} \, (x,y)=(0,0)\end{cases}$$ Prove ...
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22 views

Inner product space on $\mathbb{R}^n$, prove norm is differentiable

Let $\langle \cdot , \cdot \rangle$ be an inner product on $\mathbb{R}^n$ and define $N(X)=\langle x,x \rangle ^{1/2}$. Prove that $N$ is differentiable at every point except $x=0$ and $$(DN)(x)= ...
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2answers
55 views

Prove Derivative is sum of determinants

Given $n^2$ functions $f_{ij}$, each differentiable on an interval (a,b), define $F(x) = det[f_{ij}(x)]$ for each $x$ in $(a,b)$. Prove that the derivative $F'(x)$ is the sum of the determinants, $$ ...
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1answer
24 views

Showing a function is not monotonic.

I need help with what this question is asking. Define $f$ by: $$f(x) = \begin{cases}x^2\sin\frac{1}{x}, & \mbox{if }x\neq 0 \\ 0 & \mbox{if }x=0\end{cases}$$ Let $g(x) = x + 2f(x)$. Show ...
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1answer
42 views

Differentiable function bounded between constant multiples

I honestly have no idea how to do this. Can anyone help?
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2answers
43 views

An exponential/polynomial inequality

Prove that there is at least $1$ real number $a>0$ with the property $$a^x\ge x^a $$ for any $x>0$.
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Finding 'closest' function subject to constraints on derivatives

Suppose I have a real-valued function $f(t)$ for $t\in[0,T]$ s.t. $f'''(t)$ is defined as piecewise constant values: $$ f'''(t) = \begin{cases} k_0, & 0 < t \le t_0 \\ k_1, & t_0 < t ...
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1answer
28 views

Hard time with Derivatives of Inverse Functions

I'm having a really hard time with this question I keep googling for advice but can't find anything solid that's similar! Please help. I'm not sure if I should derive first or find the inverse first? ...
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3answers
35 views

Finding the absolute min and max [closed]

Find the absolute maximum and absolute minimum values of f on the given interval. $$f(x)=4-x^2;\;\; x\in [-3,1]$$
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2answers
44 views

Proofs for $g(x)=e^{-1/x^2}$ when $x\neq0$, and $g(x)=0$ when $x=0$

Sorry for the non-descriptive title - the question is a bit long. I have $g(x)$ as in the title, and we proved previously that $g'(0)=0$ using L'Hôpital's rule. Now I must show by induction that ...
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0answers
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Question about integrals of an unkown function and then differentiating to it

In my thesis I encountered the following problem: I have an unknown function y(x) and I need to calculate the following combination of integrating and differentiating: ...
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1answer
21 views

Gradient Question-Linear Regression

When discussing linear regression, we discuss the error of the out of sample data prediction. That is, $$ E_{\operatorname{out}} = \frac{1}{N} \sum_{n=1}^{N} ...
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40 views

Calculus Optimization - Finding the minimum cost

In oil pipeline construction, the cost of pipe to go underwater is 60% more than the cost of pipe used in dry-land situations. A pipeline comes to a river that is 1 km wide at point A and must be ...
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1answer
29 views

Fundamental theorem of calculus sin^3

How do I derivate the following integral: $$F(x)=\int_{-x}^{x} \sin(t^3)\,dt$$ I have used the fundamental theorem of calculus, and I get: $$\sin(x^3)(1)-\sin(-x^3)(-1)=\sin(x^3)-\sin(x^3)=0$$ ...
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1answer
19 views

Continuity of a partial derivative

I have the function $$f(x,y)=\begin{cases} x^2ysin(\frac1x) & \text{if $x$ is not 0} \\ 0 & \text{if $x=0$}\end{cases}$$ And I need to find the derivative and the ...
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1answer
63 views

Set-theoretic notion of differentiation and integration?

Since set-theory is said to be one of the foundations of mathematics, I would think that every mathematical concept is definable in set-theoretic terms, right? How would you define differentiation ...
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uniqueness of antiderivative up to constants

Statement: Let $[a.b]$ be a compact interval of positive length, and let $f,g:[a,b]\rightarrow \mathbb{R}$ be differentiable functions. Show that $f'(x) = g'(x)$ for every $x\in [a,b]$ if and only if ...
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1answer
26 views

If $\omega$ is compactly supported form then so is $d\omega$?

If $\omega$ is a compactly supported differential form then so is $d\omega$. Is it true?
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23 views

How to minimize values in equations?

If you have the equation $-a \leq \cos(45+d) \leq a$ where $a=\sqrt{\frac{(a+b)^2}{2} + c^2}$ and $(a,b,c)$ is a unit vector. for some $d$, how can you minimize $|d|$ so that the above equation is ...
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2answers
22 views

derivation using derivative rules

what is the derivative ln|x^2/2| my answer: we have two function ln|| and X^2. we use derivative of function outside multiply by derivative of function inside so = 1/(X^2/2). X^2/2 . x ANS = X. ...
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0answers
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How to find the solution in $x$ when the differential equation is in $\partial_y$ such that $\partial_y = \partial_x + f(x)$

For my problem, the differential equation can be simplified by defining a new operator like $\partial_y = \partial_x + f(x)$. But at the end I need a solution in terms $x$. How can I do that? As an ...
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3answers
40 views

series calculation involving factorial

How would one calculate following $$\sum_{k=2}^\infty \frac{k^2+3k}{k!}$$ I searched youtube for tutorials (patricJMT and other sources) where I usually find answers for my math problems, I think I ...
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0answers
32 views

How to show a function is identically equal to zero?

Let $g:\mathbb{R}^{2}\rightarrow{\mathbb{R}}$ be an arbitrary integrable function such that $$ \int_a^b \int_x^b {g(x,y)(y-x)^{n}}dydx=0 $$ for all $a,b\in{\mathbb{R}}$ with $a<b$. Then how do we ...
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2answers
27 views

Need help on finding the derivative of a function.

My problem is; find the derivative of $$y= \frac{\cos(\pi x)}{\sin(\pi x) + \cos(\pi x)}$$ Can someone please explain to me how to do the process in detail. I get the fact that you can use the ...
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3answers
104 views

Taylor Theorem inequality

Prove that for all $f\in C^2([0,1])$ with $f(0)=f(1)=0$ and $|f''(x)| \le 1$ $$|f(x)| \le \frac{1}{2}x(1-x)$$ $\forall x \in [0,1]$.
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1answer
32 views

second order nonhomogeneous differential equation help? (easy)

finn the general solution to the nonhomogeneous differential equation $$y''+ 2y'-3y = 5e^{-3x}$$ and I have to use undetermined coefficients? ok so what I did was found out that the homogeneous ...
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1answer
64 views

Complex Analysis - Definition of Singular Point

I have been reviewing Dennis Zill's Complex Analysis text and he defines a singular point as a point $z$ at which a function $f$ fails to be analytic. Now he goes on to talk about isolated ...
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3answers
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If Limit of function and derivative exist, then limit of derivative is 0 [duplicate]

Any hints for this question , My attempt; Say $f(x):0$$\rightarrow$$\mathbb{R}$ The by MVT, there exists a $c$$\in$$(0,\infty)$ , such that; $f'(c)=$$\frac{f(x)-f(0)}{x-0}$ but im not sure about this ...
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Using an appropriate form of the chain rule, find all (partial) derivaitves of the 1st order of …

... $f(g(x,y))$, where $f(z)=\ln(1+z)$ and $g(x,y)=\sqrt{x^2+y^2}$. Please help, very confused about this question. Thanks.
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Directional Derivative of a function containing an Indicator function.

I'm trying to understand a passage in Koenker's Quantile regression book (p.33). It says: (note that y,x, are vectors and w is the direction vector) With the first part of the outcome no problem: ...
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1answer
49 views

Differentiability conditions for a piecewise function

So this is an analysis class, and we just started the unit on differentiability -- however I missed the class. Can someone start me off with a good real analysis definition for differentiability of ...
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1answer
35 views

One-sided total derivative

Given a function from half space into euclidean space: $f:\mathbb{H}^m\to\mathbb{R}^n$ Suppose its one-sided limit exists at a specific point: $\lim_{\mathbb{H}^m\owns v\to 0}\frac{1}{\lVert ...
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1answer
35 views

Why is that a risk averse consumer buys the optimum insurance when there is actuarially fair insurance?

I've asked the same question at the Quantitative Finance StackExchange. Consider the following example: "As a risk-averse consumer, you would want to choose a value of x so as to maximize expected ...
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1answer
47 views

Very basic questions on chain rules and product rules

I have serious gaps in maths and would like to ask some basic questions. I know there is the following chain rule for the first derivative: $$ Dh(x) = Dg(f(x))Df(x)\quad\quad (1) $$ where $h(x) = ...
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1answer
27 views

slope of curve represented by discrete points

I have data which are visualized in this chart: I need to compute slope of increasing / decreasing parts of the curve. I can't use any 2 points because of noise in data. Maybe numerical derivative ...
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3answers
75 views

Is $f(x)=x^2 \sin\left(\frac 1x\right)$ continuously differentiable?

$f(x)=x^2 \sin\left(\frac 1x\right)$ on $(-\infty, 0) \cup (0, \infty)$ and $f(0)=0$ Show that this function is not continuously differentiable in $\mathbb R$. I don't know how to show ...
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0answers
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A vector analysis question requiring multivariable calculus

The question is taken from a Vector-Analysis worksheet as an extension exercise, here is the first part: Let $\underline{v}$ be a vector field $\mathbb{R}^{2} \backslash (0,0)$ of the form ...