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

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36 views

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

here are the options: $$\begin{align*} \lim_{h \to 0} \frac{f(a+8h)-f(a-5h)}{-2h} \end{align*}$$ $$\begin{align*} \lim_{h \to 0} \frac{f(a+1h)-f(a-9h)}{-4h} \end{align*}$$ $$\begin{align*} \lim_{h \...
2
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0answers
34 views

Just curious if this can be integrated easily

So, I have the following equation: $$ \frac{\ddot{x}}{\dot{x}}=-\frac{6}{t} $$ Where the dot notation is the derivative with respect to the time variable, $t$. How could one integrate this easily? ...
1
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1answer
44 views

Calculating the minima and maxima of equations involving the trig functions

This is the function given: $$S(x) = \sin^2 x$$ First I calculate the first and second derivatives: \begin{align} \frac{dS}{dx}\; & = 2\cos(x)\sin(x) \\ & = \sin(2x) \\ \frac{d^2S}{dx^2}\; &...
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0answers
23 views

Prove differentiability and find derivative

Let $f:\mathbb R^2 \rightarrow \mathbb R^2$ given by $$f (x,y)=(e^x\cos y, e^x\sin y)$$ explain why $f$ is differentiable at every $(a,b)$ and find $f'(a,b)$. I know I need to use the definition ...
1
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1answer
23 views

Bilinear, inner product, transpose and derivative

Let $A$ be an $n\times n$ matrix, let $G: \mathbb{R}^n\to \mathbb{R}$ given by $$G(x)= \langle Ax,x\rangle.$$ Show that $D G(a)(h)=\langle(A+A^T)h,a\rangle$. I proved before that $Dg(a)(h)=f(a,h)+f(...
0
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0answers
20 views

How to derive this formula for numerical differentiation? [duplicate]

I've read from a book that numerical differentiation for a point $x_i$ can be obtained by taking a linear combination of function values of other points $f(x_j)$. I don't have a clue how the ...
-2
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1answer
68 views

How was this formula for differentiation derived?

[![enter image description here][1]][1]Please tell me how this formula for numerical differentiation derived. I think it has something to do with Vandermonde Matrices but I am not quite sure how to go ...
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1answer
38 views

Solving a differential equation confusion

How do I solve this - $P'-7P+12y=0$ Where P is $dy/dx$. P.S-I tried to attempt using clairauts Differential equation But it does not match that format.
2
votes
1answer
87 views

Is there a function that $f^{-1}(x)=f'(x) $?

$$f^{-1}(x)=f'(x) \forall x \in R$$ $R$ is a set of real numbers. I try to find it or disprove that fact but I didn't make it.
0
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1answer
38 views

Derivative of arctan [closed]

What is the derivative of $\arctan (x^2)$? I got this problem while doing an integration; the integration was as follows: $$\int\frac{x \arctan x^2}{1+x^4}dx$$
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2answers
54 views

Prove $ \dfrac{2\ln(\cos x)}{x^2}<\dfrac{x^2}{12}-1$

Prove: $$\dfrac{2\ln(\cos x)}{x^2}<\dfrac{x^2}{12}-1$$ for $$x \in (0,\frac{\pi}{2})$$ I tried regular derivative methods to prove this. I thought a while about using Taylor series, but without ...
7
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2answers
156 views

Prove that $ \lim_{x \to \infty} f^{n}(x) = 0$

Let $n$ be a positive integer. Assume $ \lim_{x \to \infty} f(x)$ and $ \lim_{x \to \infty} f^{(n)}(x)$ are both real numbers. Prove that $$ \lim_{x \to \infty} f^{n}(x) = 0$$ We have that $$\lim_{...
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vote
2answers
39 views

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. ...
3
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2answers
77 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
96 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.
8
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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 ...
4
votes
2answers
73 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 $h$...
3
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2answers
53 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 ...
0
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1answer
42 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 ...
2
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0answers
44 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 $\...
1
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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.
0
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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 $$u=...
0
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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 $$ u=arcsin(\frac{...
3
votes
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 as ...
0
votes
2answers
50 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 ...
0
votes
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
34 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 ...
2
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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 ...
1
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1answer
60 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 ...
0
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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 ...
0
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2answers
53 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)| \leq\lVert{0}...
1
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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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0answers
29 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 , $\mu>0$,...
0
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0answers
18 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 ...
0
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1answer
20 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 v},\...
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votes
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 ...
1
vote
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 $D$ ...
2
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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 + x)}{...
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2answers
76 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
41 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 ...
1
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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 $\...
3
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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 $...
3
votes
1answer
18 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} = 0....
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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 $$u\in L^2\left(0,T;X \right)$$ ...
0
votes
1answer
63 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, $\textbf{...
4
votes
2answers
84 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.
0
votes
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.
7
votes
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.