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

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38 views

Let $f$ such that $\lim_{\varepsilon\rightarrow 0^+}\frac{f(x+\varepsilon y)-f(x)}{\varepsilon}=b+a\cdot y$ $\forall y$. Show that $b=0$.

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a function such that $$\lim\limits_{\varepsilon\rightarrow 0^+}\dfrac{f(x+\varepsilon y)-f(x)}{\varepsilon}=b+a\cdot y$$ $\forall y\in\mathbb{R}^n$. Show ...
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25 views

If all the critical values are shown in the chart below…

This is a strange question: If all the critical values of f(x) are shown below in the chart, which of the following could be values for f'(x) at x=1.5,2.5 and 3.5? CHART: x: 1, 2, 3, 4 f'(x): 0.5, ...
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1answer
21 views

Finding/approximating possible antiderivative given d/dx at multiple points

Suppose I am given multiple x values and the derivative of f(x) at each point. Ex: d/d(0) = 0, d/d(2) = 2, d/d(3) = 3. How do I find a function with these derivatives?
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26 views

Why does this equality stand?

We have that $$\frac{\partial}{\partial{t}}J=\begin{vmatrix} \frac{\partial}{\partial{t}}\frac{\partial{\xi}}{\partial{x}}& \frac{\partial{\eta}}{\partial{x}} & ...
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0answers
96 views

Total differential proof , need help understanding. Integration factor.

Now we're trying to find a solution for: $$ \mu(t,x):\qquad(*) \frac{\partial \mu}{\partial x}P- \frac{\partial \mu}{\partial t}Q + \mu\left(\frac{\partial P}{\partial t} - \frac{\partial Q}{\partial ...
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1answer
22 views

Prove that $\frac{f(x)}{x^n}=\frac{f^{(n)}(\theta x)}{n!},0<\theta <1$ if $f^{'}(0)=…=f^{(n-1)}(0)=0$ using Cauchy's mean value theorem

I don't know how to apply theorem on the problem. By this theorem, if two functions $f$ and $g$ are defined on $[a,b]$ continuous on $[a,b]$, differentiable on $(a,b)$ and $g^{'}(x)\neq 0$ for every ...
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2answers
75 views

What does it mean if the derivative of a function is a constant?

I was doing a homework problem to find the derivative of an equation and got "7" as the answer. I was trying to think about what it means if a derivative is a constant like that, is it just that the ...
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1answer
40 views

Evaluate $n$th derivative of a function

Is there some algorithm that is useful for finding the $n$th derivative of a function without the need to recognize the pattern?
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2answers
138 views

Finding the derivative to nth order [closed]

How to find $$\frac{d^ny}{dx^n}$$ of $$y=\frac{x}{lnx-1}$$ Appreciated advance
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1answer
41 views

Show $f(x) = x^2\sin{\frac{1}{x}} + \frac{x}{2}$ is not increasing on any open interval containing $0$

$f(x) = x^2\sin{\frac{1}{x}} + \frac{x}{2}$ for $x\not= 0$ and $f(0) = 0$. Show $f$ is not increasing on any open interval containing $0$. At first glance, we notice $f'(x) \le 0$ for some $x \in I$ ...
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2answers
85 views

Differentiation always easy?

There are many examples of real functions admitting antiderivatives (since e.g. continuous), but where computing a concrete antiderivative is a seriously hard problem even if an elementary one exists. ...
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0answers
37 views

Question about the coefficient of operator

Note that the "coefficient of" operator is an operator that takes the coefficient of the power series. We start with the following: $$ \frac{1}{f(x)+z} - \frac{1}{f(x)} = \sum_{k=0}^\infty ...
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1answer
21 views

differential equation, partial derivatives: Why is the following true?

$$x_1'=f_1(t,x_1,...,x_n) \\ x_2'=f_2(t,x_1,...,x_n) \\ ....... \\x_n'=f_n(t,x_1,...,x_n)$$ This a system of equations, now the text book says let's differentiate the first equation of the system ...
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2answers
33 views

confusion in using Lebiniz integral rule

I was trying this question - Let $$f: (0,\infty )\rightarrow \mathbb{R}$$ and $$F(x) = \int_{0}^{x}tf(t)dt$$ If $F(x^2)= x^{4} + x^{5} $, then the value of $\sum_{r=1}^{12}f(r^{2})$ is I applied ...
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1answer
36 views

Differentiability implies continuity - A question about the proof

I have a question, to aid my understanding, about the proof that differentiabiility implies continutity. Differentiability Definition When we say a function is differentiable at $x_0$, we mean that ...
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1answer
15 views

Differential of a function between two normed spaces

I have a question about the differential of a function between two normed spaces. It is a simple question about the definition. In my textbook from my university, the definition is as follows: Let ...
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0answers
10 views

Composition of functions class $C^n$

Suppose I have 2 functions class $C^n$ and I consider their composition. Would that still be a $C^n$ function? If so why (I demand a proof). It seems logical to me that it is true but I can't find a ...
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1answer
26 views

Gradient of a scalar function: path taken by a particle

Just the last part, I have no idea where to start.
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3answers
69 views

How to apply fundamental theorem of calculus to multiple integrals

I have the following problem at hand: $$\lim_{\epsilon \to 0}\dfrac{1}{4\epsilon^2}\int_{z-\epsilon}^{z+\epsilon}f(e_1,y) \int_{\frac{z+y}{2}-\epsilon}^{\frac{z+y}{2}+\epsilon}g(e_2,x) dx dy$$ $f$ ...
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1answer
23 views

derivative free optimaization method

Currently I am working project on the derivative of free optimization methods. however, I want find practical problem that solved using this method. So, how can I get solve practical examples using ...
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1answer
361 views

What does an apostrophe mean in a function?

In a workbook, I saw the function $f(x)=x^2$. Then, there was the same function with an apostrophe $f'(x)$. It was stated that $f'(x)=2x$. What is the apostrophe, and why does it change the function? ...
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1answer
65 views

Gradient of an absolute value [closed]

What is the gradient of $|\vec{x}|^2$? Is it simply $2\vec{x}$, or does the answer get expressed using absolute value notation?
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1answer
34 views

Trigonometric Differentiation. Height of a wave.

The movement of the crest of a wave is modelled with the equation $h(t)=0.3\cos 3t+0.4\sin 3t$. Find the maximum height of the wave and the time at which it occurs. I have come up until here. please ...
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1answer
63 views

Deriving a function that includes the absolute value of a complex number

I have the function $$f:\mathbb{R}^3\to\mathbb{R},f(x)= |1 - F(x)|,$$ with $F:\mathbb{R}^3\to\mathbb{C}$. Is there a way to express the gradient $\frac{\partial f}{\partial x}$ in terms of ...
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1answer
24 views

Expressing multiplication as percentage changes.

I have seen many times that for example when we have a formula: $$A=\frac{B\cdot C}{D}$$ where $A, B, C, D$ are some variables; we can express it in the following way as 'growth rates':$%\Delta%$ ...
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1answer
50 views

Show $\frac{-1-\beta^{2}+2\beta\sqrt{1+\beta^{2}}}{-1+\beta^{2}}\geqslant -1+\sqrt{2}$ when $\beta>1$.

I was wondering if there is a quick way to prove (or disprove) the following inequality: \begin{eqnarray} \frac{-1-\beta^{2}+2\beta\sqrt{1+\beta^{2}}}{-1+\beta^{2}}\geqslant -1+\sqrt{2}, ...
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2answers
34 views

Continuity and Differentiability of f(x)

$$f(x) = \begin{cases} x^2 + 3x + 2 & \quad \text{if } x \leq 0\\ x^2 - 3x + 2 & \quad \text{if } x > 0\\ \end{cases} $$ Prove that f is continuous at $x = 0$ and not ...
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23 views

Compute the directional derivative

Compute the directional derivative. Be sure that you use a unit vector. $q(x,y,z)=4x^2-3y^3+2z^2$ at $(0, 1, 2),$ $t=2i-3j+k$ Answer given is $35/(14^{0.5})$
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13 views

Differentiating integral by substituting inverse function

I have the following cost function that I wish to minimize with respect to $\alpha$: ...
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1answer
35 views

For which $n$, $x^{n - 1} \sin{\dfrac{1}{x}}$ is differentiable for all $x$?

For a whole number $n$, if $f(x) = x^{n - 1} \sin{\left(\dfrac{1}{x}\right)}\qquad x \ne 0 \qquad \& \qquad f(0) = 0 $, then in order that $n$ is differentiable at all $x$, the value of $n$ ...
3
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1answer
48 views

What am I doing wrong with this derivative - Differential calculus (brush up)

today I felt like doing some maths and I thought to myself that practicing some derivatives would be neat-o. I sat myself the following question. ...
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1answer
31 views

f continuous and differentiable?

Consider the function $$f:\mathbb{R}^2\to\mathbb{R}\; (x,y)\mapsto \begin{cases} \frac{x^ay^b}{(x^2+y^2)^c}, & (x,y)\not=(0,0)\text{,}\\ 0, & \text{else } \end{cases}$$ I am trying to ...
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1answer
73 views

How to prove this tedious (but easy) derivative theorem

I'm reading Fulton's algebraic curves book (page 3) and I'm having problems with (4), (5) and (6) part of this theorem. This proof seems really easy to demonstrate, but there are a lot of ...
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1answer
16 views

Relation between roots of a function and roots of its derivative, IVP

I am troubled with this question of my book: I do know that f (a) = f '(a) = 0 if the multiplicity of root 'a' is greater than 2 but how that fact is exploited here or is there something more ...
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1answer
18 views

Computing wavenumbers for discrete Fourier transform

I'm trying to implement a Fortran program to compute the derivative of a function using the FFT. To begin with, just to test my installation of fftpack, I computed the Fourier transform of ...
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3answers
62 views

Is the derivative of a quadratic related to the second difference of that quadratic?

Please do not judge me too harshly for my lack of knowledge, but at school we have gone over Quadratic functions recently. Now, these types of functions are not new to me, however when we viewed a ...
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1answer
39 views

Integrating a linear-map valued function

My textbook for an Analysis course I am taking presents the Mean Value Equality theorem as Suppose $\mathbb{X}, \mathbb{Y}$ are Banach Spaces. Let $U\subseteq \mathbb{X}$ be an open set, and let ...
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1answer
27 views

derivative of sign() as active function in backpropagation

I've got the task that I need to implement the backpropagation algorithm for a neural network. My activation function is just the sign(.). $w^{\prime} = w + \space$learning rate$\space \times \delta ...
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2answers
36 views

Why $F(z) = |z|^2$ is holomorphic nowhere?

I am self-studying basic complex analysis, and am slightly confused as to how to show that $F(z) = |z|^2$ is holomorphic nowhere. A necessary and sufficient condition for the holomorphism of $F(z)$ is ...
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0answers
14 views

Maximizing the weighted sum of two CDFs subject to a constraint on the expected value.

I encountered this problem in a proof and would like to have your help: Consider the maximization problem: \begin{eqnarray} \max_{x,y}b_x\Phi(x)+b_y\Phi(y),s.t\\ ...
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2answers
46 views

In order to show that a function is C^1 is it enough to show that the 1. partial derivatives exists?

Hello I am having some issues with the following exercise: Let $\textbf{h}: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $\large \textbf{h}(u,v) = u^2 + (v-1)^2 - 5 + e^{u-2}$ (i) Show that ...
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55 views

What is $dx/dF(x)$ where $F(.)$ is a continuous, increasing function.

I was wondering if it is possible to find $dx/dF(x)$, that is, the derivative of $x$ with respect to $F(x)$, which is an increasing, continuous function. Does it involve finding the derivative of the ...
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5answers
66 views

for each $x>1 , \frac{x-1}{x}\ < \ln x < x-1$

I tried to prove this with differentiation: when $x >$ 1, all 3 functions are positive and when $x = 1$, all 3 reaches zero. And the derivatives are varying like ...
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1answer
36 views

n-th derivative with respect to $\frac{1}{x}$

Is it any easy way to calculate : $\frac{d^n x}{d\left(\frac{1}{x}\right)^n}$ for arbitrary $n\in\mathbb{N}$ ? (for $n=1$ it is obvious, but for $n>1$ the formula for $n$-th derivative of ...
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3answers
63 views

$n \text{th}$ derivative of $f(x)$?

Let $ \ f(x) \ = \ x^4 e^x \ $ . Determine the nth derivative of $ \ f \ $ at $ \ x \ = \ 0 \ $. I know by working it out that the first, second, and third derivative will be 1. The fourth, fifth, ...
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1answer
43 views

Derivation of the dirac delta

I have the following function: $$\int^{a+\epsilon}_{a-\epsilon}f(x)\dfrac{1}{\epsilon} dx$$ When I take the limit $\epsilon \to 0$, I want to show that this becomes equivalent to the behavior of the ...
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2answers
68 views

The third derivative of the first principles definition of of a derivative

So the $\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}$ this is what I learned to find the first derivative and by taking this concept and trying to find the second derivative using this method I came up ...
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4answers
95 views

l'Hospital's rule with trigonometric functions

$$\lim_{x\to0^+}\frac{1-\cos(x)}{x^2\sin(x)}$$ I keep running in circles using the L'Hospital rule. After the third time applying it I got 0 but this isnt true from the graph. I can see it goes to ...
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0answers
42 views

Chebyshev spectral derivation with 16 nodes for $\,f(x)=e^{\,\text{sin}^{2}\,(x)+\cos(x)}\,$ defined in $\,[0,2\,\pi].\,$

I'm making the following exercise in Matlab, and I'm having trouble expresing my result in $x\in[0,2\pi]$ not in $x\in[-1,1]$. I first done this (as shown below) in Gauss-Lobatto points, but I don't ...
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1answer
18 views

Gradient of a function defined on a curve

Given: $f=f(\vec{r})$ and $\vec{r}=\vec{r}(s)$, therefore $f = f(\vec{r}(s)) = f(s)$, $f$ is defined only on the curve $\vec{r}(s)$. How then does one express the gradient of $f$ $\nabla ...