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

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Munkres Analysis on Manifolds Differentiation Question

Below is a problem from Munkre's Analysis on Manifolds book. I'm unsure of how to approach this; it seems to me to apply the defintion of the derivitative, but I cannot seem to get that to work out. ...
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2answers
73 views

18th derivative of $\arctan(x^2)$ at point $x=0$

$$\frac{\mathrm d^{18}}{\mathrm dx^{18}} \arctan(x^2)$$ Without using Taylor. I relay don't have any idea how to use General Leibniz rule or any other idea how to get result.
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2answers
94 views

Prove that $x^y < y^x$

Assuming that $e<y<x$, prove that $ x^y < y^x$. I think this must be easy, but I can't work it out. Thanks in advance for any kind of help.
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1answer
64 views

Minimize the Area

The lower corner of a page is to folded to reach the opposite inner edge. We have to find the width of the folded part if the Area of the folded part is minimum. Now how I proceeded: Let the width ...
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2answers
64 views

Numerical Differentiation Matlab

I am trying to estimate the second derivative of $\sin(x)$ at $0.4$ for $h=10^{-k}, \ \ k=1, 2, ..., 20$ using: $$\frac{f(x+h)-2f(x)+f(x-h)}{h^2}\tag{1}$$ And then plot the error as a function of ...
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2answers
52 views

How to find derivative?

How can I get from step $3$ to step $4$ in the problem? I've tried this: $6x^2 - 9x + 8x -12 + 2\Delta x - 6x^2 -8x +9x + 12 + 3\Delta x$ It doesn't cancel out to $17\Delta x$, though. I've also ...
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1answer
39 views

Confused on what variable this refers to in Calculus I (Related Rates)

A spotlight on the ground shines on a wall $12 \mbox{ m}$ away. If a man $2 \mbox{ m}$ tall walks from the spotlight toward the building at a speed of $2.2\mbox{ m/s}$, how fast is the length of his ...
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0answers
42 views

Is it possible to differentiate the Gamma function with product rule?

I want to differentiate the Gamma function using product rule: $$\Gamma(x+1)=x\Gamma(x)$$ $$\frac{d}{dx}\Gamma(x+1)=\frac{d}{dx}x\Gamma(x)$$ $$=x\left(\frac{d}{dx}\Gamma(x)\right)+\Gamma(x)$$ If ...
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1answer
36 views

how to solve an nth derivative for the equation $\ln((1+x)/(1-x))$

I'm trying to find the $n$th derivative of this function. I've got that the first term is: $$ \frac{2(n!)x^{n-1}}{(x^2-1)^n} $$ Any improvement on this would be very helpful.
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1answer
19 views

Directional derivative into the begining…

Hey guys another try to do a Directional derivative, i want to do a Directional derivative of $z= x^2*e^{2x+3y}$ at $(2,-1)$ to the begining $$(0,0)$$ This is what ive done: so the vector will be ...
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1answer
103 views

Am i doing correct the Directional derivative?

im trying to find Directional derivative,I done few examples and im not sure if im doing it correct, so please tell me if im right... A. I need to find the Directional derivative of $$ ...
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3answers
56 views

How to calculate this integral without any integration techniques?

Differentiate $f(x) = (5x+2)\ln(2x+1)$ with respect to $x$. Hence, find $\int \ln(2x+1)^3dx$. Because of the word "Hence" I'm assuming that the question doesn't allow integration techniques such ...
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2answers
46 views

f is a differentiable function. Which of the limits is equal to f′(a)?

The options are shown in the image Answer is the last option. But i am not able to understand this at all. I dont know how to approach this question. can someone explain this thoroughly. Thanks in ...
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1answer
24 views

Parametric derivatives

Let $f(x) = \dfrac{2\sqrt{1+x^2}-5\sqrt{1-x^2}}{5\sqrt{1+x^2}+2\sqrt{1-x^2}}$. Hence, find $\frac{dy}{dz}$ when $y=\cot^{-1}(f(x))$ with respect to $z=\cos^{-1}{\sqrt{1-x^4}}$. To get this into a ...
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0answers
33 views

Find the derivative. y = arctan(9 tanh(x))

Find the derivative. y = arctan(9 tanh(x)) My attempt: $\frac{dy}{dx}=\frac{1}{1+(9\tanh x)^2}\frac{d}{dx}(9\tanh x)\;\;$ is this right way to solve this problem
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0answers
50 views

Complex differentiable function is identically zero

Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is a differentiable function such that $f(\frac{1}{n})=0$ for all $n\in \mathbb{N}$ then $f=0$.. I know one uniqueness result namely if $f=0$ on an ...
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1answer
31 views

Quick clarification on the definition of vector field

I am having a class on differential geometry and another on ODEs In the ODE class, if we were given something of the type $$\dot x = x^2$$ The professor refers to $x^2$ as the vector field. In ...
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1answer
58 views

Help with implicit differentiation problem [duplicate]

Here is the problem: A ladder 15 metres high is propped up against a high wall. The bottom of the ladder slides away from the wall at a rate of $1\ {m/s}$. How fast is the top of the ladder ...
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1answer
94 views

Finding the $n^{th}$ derivative of $x^r$

I'm looking for a non-piecewise function -- $g(n,x)$ -- that satisfies this equation: $g(n,x)=\large\frac{d^{n}}{dx^{n}}x^{r}$ Where $n\in\Bbb{Z}$ and is the $n^{th}$ derivitive of $f(x)$ I ...
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2answers
51 views

Proving a function is differentiable if $|f(x)| \leq \lVert{x}\rVert^2$

I have to prove that a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is differentiable in $0$ knowing that $|f(x)| \leq \lVert{x}\rVert^2$. \ This is what I have: $ 0 \leq |f(0)| ...
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1answer
82 views

Diffrenece of exponential functions

Prove that the function $f: \Bbb R \to \Bbb R$, $f(x)=2016^x-2015^x+x$ is strictly increasing. I tried to find the derivative, but it didn't help me.
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28 views

Multi-derivative of standard normal CDF

I am trying to solve the following $m$th-derivative of standard normal cdf, $$\frac{\text{d}^m}{\text{d}a^m}\Phi \left(\frac{a+\mu u}{\sqrt{u}}\right),$$ where $m> 0$ is an integer , ...
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0answers
16 views

How to calculate the Bouligand derivative (B-derivative)

Let $H(x)=\min (f(x),h(x))$ where $f$ and $h$ are continuously differentiable functions from $\mathbf{R}^n$ to $\mathbf{R}^1$. The Bouligand derivative (B-derivative) $BH(z)$ at $z$ of $H$ is given ...
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1answer
19 views

Equality of first-order partial derivatives

Let $f(u,v)$ be a "sufficiently good" function of two variables. I need to find sufficient conditions on $f$ such that $$ \frac{\partial f(u,v)}{\partial u}=\frac{\partial f(u,v)}{\partial ...
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4answers
63 views

Are there any pairs of functions where $g(n,x)=f^{(n)}(x)$?

Are there any non-piecewise pairs of functions that satisfy this quality? $g(n,x)=f^{(n)}(x)$ Where $n\in \Bbb{Z}$ and is the $n^{th}$ derivitive of $f(x)$ This is a long shot but I'm just ...
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2answers
47 views

If $f'(x) = 0$ for every $x \in D$, then $f(x) = k$ for all $x \in D$, even when $D$ is not an interval.

Either give a proof or a counterexample to the following statement: If $f : D \to R$ is a differentiable function and $f'(x) = 0$ for every $x \in D$, then $f(x) = k$ for all $x \in D$, even when $D$ ...
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2answers
34 views

using L'Hospital solve $\lim_{x \to \infty} x - x^{2}\ln(1 + \frac{1}{x})$

I can't get this to $ = \frac{0}{0}$ form so I can use l'Hospital rule $$\lim_{x \to \infty} x - x^{2}\ln\left(1 + \frac{1}{x}\right)$$ tips? [EDIT] $$\lim_{x \to 0} \frac{1}{x} - \frac{\ln(1 + ...
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72 views

Shouldn't l'Hopital's rule work for every limit, not just indeterminate forms?

Why does taking the ratio of $f'(x)$ to $g'(x)$ as $x \to a$ give you the correct limit when $f(a)$ and $g(a)$ $= 0, \infty, -\infty$ , but not for other values of $a$? If the rationale for using ...
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2answers
39 views

Computer Vision Models 4.3 - Derivative of Summation

I am reading through the Computer Vision: Models, Learning, and Inference book to get an understanding of computer vision. The author describes the high-level steps taken to arrive at one of the ...
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1answer
26 views

If $f$ is differentiable at point, then error term of linear approximation is continuous in neighbourhood around that point

In this post it is said that if $f : \mathbb R \to \mathbb R$ is differentiable at $a$ then there exists a continuous function $\varphi$ defined on an interval $[-\epsilon, \epsilon]$ such that ...
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1answer
34 views

Applications of Derivatives problem

$$f(x) = x^3 + ax^2 + bx + 5\sin^2x $$ is an increasing function on the set $R$. Then $a$ and $b$ satisfy: $a^2 - 3b - 15 > 0$ $a^2 - 3b + 15 > 0$ $a^2 - 3b + 15 < 0$ $ a> 0$ and $b ...
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2answers
29 views

Differentiability class of Matern function (based on Modified Bessel Function of second kind)

I am working on some techniques using the Matérn covariance function: $h(r) = \frac{2^{1-\nu}}{\Gamma(\nu)}\Bigg(\sqrt{2\nu}\frac{r}{\rho}\Bigg)^\nu K_\nu\Bigg(\sqrt{2\nu}\frac{r}{\rho}\Bigg)$ with ...
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1answer
16 views

Finding function for capital interest

Haven't fully grasped derivatives and I believe this question really holds the gist of it Your bank account has a continuous capital interest rate of 7%. The formula for this is $$\frac{dB}{dt} = ...
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0answers
16 views

Conditions for weak differentiability of composition of $C^1$ real function with weakly time-differentiable $H^1$-valued function

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain and $\mathbb{R} \ni T > 0$. I will abbreviate $X=H^1(\Omega)$ and write $X'$ for its topological dual. Given $$u\in L^2\left(0,T;X \right)$$ ...
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1answer
60 views

Vector function derivatives for discrete adjoint equation

I'm in the process of deriving a discrete adjoint equation. I'm trying to find the derivative of the vector $\textbf{X}_{1}$ with respect to the vector $\textbf{X}_0$ but I am not able to, ...
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2answers
83 views

100th derivative of $(1-2x)^{2/3}$ at point $x=0$

$$\frac{\mathrm d^{100}}{\mathrm dx^{100}} (1-2x)^{2/3}$$ Without Taylor. I relay don't have any idea how to use General Leibniz rule in this case.
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2answers
44 views

100st derivative $(\sinh(x)*\cosh(x))^2$ at point $x=0$

$$\frac{\mathrm d^{100}}{\mathrm dx^{100}}(\sinh(x)*\cosh(x))^2$$ Without Taylor I try this :$\sinh(x)'=\cosh(x)'$ but that didn't help in using General Leibniz rule.
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1answer
61 views

$n^{th}$ derivative of $\cot x$

What is the $n^{th}$ derivative of $\cot(x)$? I tried to differentiate it may times: I can't see a pattern forming. Please help.
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1answer
23 views

Continuity of the directional derivatives implies continuity at the point ?

This might be a trivial question. Consider a function $f:\mathbb{R^2}\rightarrow \mathbb{R}$ and consider some point $(a,b)\in \mathbb{R^2}$. Suppose we know that all the directional derivatives ...
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1answer
30 views

Show that $f$ is differentiable at $x=1$.

Let $f$ be a real valued continuous function defined on $[0,2]$ such that $f$ is differentiable at all point except possibly at $1$. Suppose that $\lim_{x\to 1}f^{'}(x)=5.$ Show that $f$ is ...
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2answers
78 views

Is the composition of the differentiating operator commutative?

First of all, can I check that $d\over dx$ can be considered an operator, or function (as it says in the title)? Is the composition of the differentiating operator commutative? In other words, if ...
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2answers
75 views

100th derivative of $\frac{1+x^2}{1+\tan^2(x)}$ at point 0

$$\frac{\mathrm d^{100}}{\mathrm dx^{100}}\frac{1+x^2}{1+\tan^2(x)}$$ Without Taylor Is there a way to solve this problem by using General Leibniz rule. I tried but numerator make problem.
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1answer
96 views

Symmetric matrix - Langrange Multiplier

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix. Let $S=\{x \in \mathbb{R}^n \mid \|x\|_2=1\}$ be the unit sphere in $\mathbb{R}^n$ with respect to the $2$-norm ...
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0answers
18 views

Differentiability of multivariable functions.

I have the following two questions: (1) What are the most important techniques to show if a multivariable function is differentiable. (2) I know how to show that a multivariable function is not ...
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1answer
43 views

Does differentiability on a set imply continuous differentiability on the set? Counterexample?

Of course, differentiability implies continuity, but for a function to be differentiable on a set, say $[a,b]$, then, for the limit to exist, would we not need it to be defined on the set? I hear ...
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13 views

differential operator of composition functions

I got confused about the following derivative: $$d(df\circ g)(p)h(p)= d^2f(p)(h(p),g(p))+df(p)\,dg(p)h(p)$$ I can't see the relation between the left and right hand sides!
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44 views

Vector Calculus in Curvilinear Coordinates and Index Notation

I am trying to understand how would I use the index notation in curvilinear coordinates. Checking out this reference, I got until this point $$\vec{\nabla} = \sum_a \vec{e}_ah_a^{-1}\partial_a $$ ...
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2answers
33 views

The Critical Value of $f(x)= x^{6\over 5}-12x^{1 \over 5}$

I need to find the critical number of $$f(x)= x^{6\over 5}-12x^{1 \over 5}$$ Here's what I've tried. $$f'(x) = {6 \over 5}x^{1\over 5} - {12\over 5}x^{-{4 \over 5}}\\ = {6x^{1\over 5} \over 5} - {12 ...
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3answers
28 views

differentiation of $\operatorname{erfc}(\sqrt{ax})$

I need your help to figure out the derivative of $\operatorname{erfc}(\sqrt{ax})$ with respect to $x$. Based on my knowledge on Wolfram references, they cite that: $$\frac{d ...
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1answer
25 views

Directional derivative of determinant at the identity is the trace of the matrix?

Let $f:A\mapsto \rm{det}(A)$, Prove that $\left(Df\right)_{{\rm id}}\left(H\right)={\rm tr}\left(H\right)$ for all $H\in\mathcal{L}\left(\mathbb{R}^{n}\to\mathbb{R}^{n}\right)$. The question ...