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

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1answer
39 views

Simplification of the derivative

I have this equation $y=x^{5x^3}$ by doing a log transformation we get, $log (y) = 5x^3 log (x)$ upon doing a differentiation w.r.t $(x)$, we get $$\frac{1}{y}\frac{dy}{dx} = 5x^3.\frac{1}{x} + ...
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1answer
33 views

Composite function derivative/ Chain Rule

If $F(x) = f(g(x))$, $g(2) = 4$, $g'(2) = 3$, $f'(4) = 5$, what is $F'(2)$? Please explain how you got the answer as well.
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1answer
40 views

Need help to simplify the derivative

Can someone tell me what would be the output of this equation? $$\frac{d}{dx}[\cos^4(x)\cdot\cos (x^4)] = -4x^3\cdot\cos^4(x)\cdot\sin (x^4)+4\cos(x^4)\cdot\cos^3(x)\cdot\sin(x)$$ But am not getting ...
3
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1answer
36 views

need help to understand the differential equation

In one of the books, it was mentioned $\frac{d}{dx}(x^3 \tan x)= (x^2\sec^3x+3x^2\tan x)$, but i think it should be $(x^3\sec^2x+3x^2\tan x)$. I feel its a printing mistake. Just wanted to be sure, ...
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2answers
52 views

find the derivative

$$\cfrac{(x-6)(x^2+4x)}{x^3}$$ and $$\left(\frac{q^6+4}{2q}\right)\left(\frac{q^8+ 6}q\right)$$ Okay so Im reviewing for a test I have tomorrow and these two derivative question come up ive been ...
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1answer
38 views

Show that the linear function f(x)=Ax is differentiable at a

How to show that the linear function f: $R^{n}$ $\rightarrow$ $R^{n}$, defined by f(x) = Ax, is differentiable at a generic point a, where A is a n $\times$ n matrix, and what is Df(a)? from the ...
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1answer
28 views

“Global surjective theorem”

So in Multivariate analysis we are doing mapping Theorems and we had one homework problem that I have not been able to solve for over a day now. $ g: \mathbb{R^p \to R^p}\,\text{belongs to class ...
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2answers
58 views

derivative /algebra/some help [closed]

i want some help please thank you
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1answer
27 views

Looking for verification of related rates logic!

I'm working through a really interesting problem about related rates here and I'm pretty sure I've got it figured out. But I'd like to get an expert's opinion on my method here: "Sand is falling ...
0
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1answer
35 views

Need help with logarithmic differentiation

I need to use logarithmic differentiation to get f(x)=x$\sqrt{(x+1)(x+2)(x+3)(x+4)}$. I've been working on it for a while and could use some help. Thanks!
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3answers
59 views

Critical numbers of the function: $x\sqrt{5-x}$

Let f(x) = $$\displaystyle f(x) = x\sqrt{5-x} $$ On the interval: [-6,4] Critical numbers are the the values of x in the domain of f for which f'(x) = 0 or f'(x) is undefined. Derivative of the ...
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0answers
25 views

Need help in differentiating following problem

I need to find the theta, so the x (distance) is largest. I tried solving it by using differentiation, finding when slope is zero, which should give me the answer. ...
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1answer
28 views

differentiability of partial derivatives

Prove that if f a function of n variables is continuously differentiable in an open subset U of $R^n$ then the partial derivatives of f are continuously differentiable. I used the definition of f ...
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2answers
78 views

Prove that if $\lim \limits_{n \to \infty} f(n)=2$ and $\lim \limits_{x \to \infty} f'(x)=0$, then $\lim \limits_{x \to \infty} f(x)=2$.

Let $f: \Bbb{R}\rightarrow\Bbb{R}$ be a differentiable function and let $\lim \limits_{n \to \infty} f(n)=2$ ($n$ runs for Natural numbers), and $\lim \limits_{x \to \infty} f^\prime(x)=0$. Show ...
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1answer
20 views

finding the values of a such that an implicit function g(y)=x has max,min,saddle points along y=0

I've got $$f(x,y)= a\exp(1+xy) + a^2 \sin(x) +1$$ for which I've shown that there exists an implicit function $x=g(y). ( df/dx \neq 0)$ and $df/dx = a y \exp(1+xy) + a^2 \cos x$ now in the ...
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0answers
28 views

Continuous second derivative over the support of a Daubechies4 wavelet

I can not entirely follow the proof from section 3.1.1 from the book "A primer on Wavelets" by Walker. After the first part (listed below), I can grasp the rest so if you could help I would greatly ...
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0answers
28 views

Why doesn't the $L_2$ norm differentiable at $x=0$?

Why doesn't the $L_2$ norm differentiable at $x=0$? Let's define $N(x)$ as the norm function. I know that for every $x\ne 0$: $$\frac{\partial N}{\partial x_i}(x) = \frac{x_i}{\|x\|}$$ What ...
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1answer
65 views

Derivative of a summation function in order to minimize the function

I'm asked to minimize this function $$f\left(x\right)= \sum_{k=1}^K \left(g\left(w\left(k\right)+\alpha\right)-t\left(k\right)\right)^2$$ with respect only to $\alpha$. Function ...
2
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1answer
43 views

Show that $F(x) = f(\|x\|)$ is differentiable on $\mathbb{R}^n$. [duplicate]

Let an even function $f:\mathbb{R}\to\mathbb{R}$ which is even and differentiable. We define $F:\mathbb{R}^n\to\mathbb{R}$ as $F(x) = f(\|x\|)$. Show that $F(x)$ is differentiable on ...
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1answer
30 views

Show that a function is a contraction in the metric d(x,y) = |lnx - lny|.

We have a function $f: \: (0,\infty) \rightarrow (0,\infty)$, and there is a constant $0<k<1$ s.t. $$x|f'(x)| \leq kf(x).$$ I want to show that $f$ is a contraction. Solving the differantial ...
2
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1answer
53 views

Finding all real solutions to the equation $3^x+4^x=5^x$

Find all real solutions to the equation $$3^x+4^x=5^x.$$ My attempt: It is evident that $x=2$ is a solution. However, I think that there are no other solutions. So, I define a function ...
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2answers
37 views

Showing that if $xf(x)=\log x$ for $x>0$ then $f^{(n)}(1)=(-1)^{n+1}n!\bigg(1+\dfrac12+\dfrac13+\ldots+\dfrac{1}{n}\bigg)$

Let $f(x)$ be a function satisfying $$xf(x)=\log x$$ for $x>0$. Show that $$f^{(n)}(1)=(-1)^{n+1}n!\bigg(1+\dfrac12+\dfrac13+\ldots+\dfrac{1}{n}\bigg),$$ where $f^{(n)}(x)$ denotes the $n$th ...
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3answers
44 views

How to do the derivative when an exponent has an exponent

I am trying to solve an equation that is in the form of $y(x) = (c + x^2)^{x^2}$. Note $c =$ constant My initial thoughts are I need to look into using ln and e to solve this. However what I am ...
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3answers
104 views

How to solve an Inverse differentiation problem

If f is a one-to-one function where $f(3)=2$ and $f'(3)=6$, what is the value of $(f^{-1})'(2)$? I am not even sure where to start with this question. I was hoping someone can help $f$ of $3 =2$ and ...
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1answer
145 views

Related Rates and Angle of elevation

I have been trying to wrap my head around related rates, which are super interesting but very difficult for me personally. Would anyone care to verify if my logic is correct here? "A balloon rises ...
2
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1answer
55 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: ...
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3answers
35 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 ...
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4answers
43 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.
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0answers
22 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 ...
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0answers
43 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 ...
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4answers
103 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
38 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 ...
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0answers
38 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 ...
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2answers
59 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
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3answers
83 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 ...
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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 ...
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4answers
65 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
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2answers
72 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
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1answer
43 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 ...
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1answer
23 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 $n>1$ : $g ^{(n-1)}(z)$ = ...
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2answers
112 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 ...
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0answers
59 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
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1answer
76 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
20 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 ...
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1answer
203 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 ...
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2answers
33 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 ...
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1answer
38 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
18 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
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0answers
23 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\ |\ ...
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2answers
96 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 $\frac{d}{dx}$ ...