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

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4
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1answer
43 views

Covariant Derivative Clarification

In my notes I have the following when taking the divergence, $\partial_\mu$ of $\partial_\alpha\varphi^\alpha g^{\mu\nu}$ $$ \partial_\mu \partial_\alpha \varphi^\alpha g^{\mu\nu} = \partial_\nu \...
0
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1answer
42 views

Fritz John PDE book chapter 1 exercise: Prove that $u$ vanishes identically if $au_x+bu_y=-u$

I was trying out this question: Let $u$ be a solution in $C^1$ of the PDE $$ a(x,y)u_x + b(x,y)u_y = -u $$ on the closed unit disc $\Omega$ in the xy-plane. Let $a(x,y)x + b(x,y)y > 0$ on the ...
1
vote
1answer
26 views

Optimization of Rectangular Beam

A rectangular beam has breadth $b$ mm and depth $d$ mm. Its strength is proportional to both the square root of its breadth and its depth cubed. What are the dimensions (in mm) of the strongest beam ...
2
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1answer
28 views

If $f\in C(\Bbb{R}^n) \cap C^1(\Bbb{R}^n\setminus\{0\})$ and $\nabla f(x) \to L$ as $x\to 0$, then $f\in C^1(\Bbb{R}^n)$

Let $f:\Bbb{R}^n\to \Bbb{R}$ be continuous on $\Bbb{R}^n$ and continuously differentiable on $\Bbb{R}^n\setminus\{0\}$. Moreover, $\nabla f(x)\to L$ as $x\to 0$. Show $f$ is $C^1$ on $\Bbb{R}^n$ I ...
-1
votes
2answers
66 views

Pth derivative of $2^x$

I know that the first derivative of $2^x$ is $2x\ln2$, but what is the pattern of it? Like for its pth derivative, what will it be? Thanks!
0
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0answers
28 views

Maximizing a product of vectors and matrices

I have a function of the form $$ f(\xi) = v^T \left( S + \xi^2 D \right)^{-1} u $$ I would like to find the critical points of this function (if there are any) on the domain $\xi \in [0,\infty)$. $...
1
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2answers
67 views

Why is derivative of $x$ with respect to $x$ equal to $1$?

I just started learning the derivatives of inverse function. The first example is based on the fact that $\frac{\mathsf{d}x}{\mathsf{d}x} = 1$ and it is stated that I should know this already. However ...
0
votes
1answer
16 views

existence of function and derivative

Given a function $f(x)=1/(x-1)$, $f(1)$ is undefined. $f'(1)$ is also undefined. Thus my question is, is it always the case for any function if a point is undefined, the derivative of the function ...
0
votes
1answer
28 views

How to interpret b in $y=x^{e^{bz}}$ in nonlinear regression?

What is the correct way to interpret b in this nonlinear equation $y=x^{e^{bz}}$? I've estimated the model and b seems to be the percent change in y with a unit change in z, but I am unsure how to ...
2
votes
1answer
32 views

Uniform convergence towards continuous derivative?

Hi I was having trouble with the following question: A user called Fischer has said that "The convergence is uniform on every compact subset of $\mathbb{R}$, however", without providing a proof. (...
1
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0answers
31 views

Defining the differentiability of a multivariable function (if/then)

I'm trying to understand differentiability for multivariable functions and am thoroughly confused by the introduction (and the direction of implications in a certain definition) I'm given the ...
0
votes
1answer
42 views

differentiate $\ln(x+\sqrt{x^2-1})$

$$\frac{d}{dx}ln(x+\sqrt{x^2-1})$$ $$\frac{d}{dx}ln(x+\sqrt{x^2-1})=\frac{1-\frac{x}{\sqrt{x^2-1}}}{x+\sqrt{x^2-1}}$$ $t=\sqrt{x^2-1}$ $$\frac{1-\frac{x}{t}}{x+t}=\frac{1}{x+t}-\frac{x}{t(x+t)}=\...
2
votes
1answer
37 views

Are there any theorems linking periodic functions to the number of times they are differentiable?

I was working through some Fourier series questions and I was wondering if the periodicity of a function has anything to do with the number of times it's differentiable. For instance, the elementary ...
1
vote
1answer
44 views

How to understand basics of vector deravative and vector field

Hello I am having trouble understand the following in my notes; It was an example; Q: Is $$v=K(-yi+xj)$$ a conservative vector field? A: If it was, then $$K(-yi+xj)=\nabla \psi$$ Here is where I ...
0
votes
1answer
43 views

matrix calculus: $\frac {\partial \vec{x}}{\partial \vec{x}^{T}}$

I'm getting confused by notation conventions. In matrix calculus, it makes sense that: $$\frac {\partial \vec{x}}{\partial \vec{x}} = I$$ where I is the identity matrix. Is it true that: $$\frac {\...
0
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1answer
20 views

$f(x,y,z) = \log(x^3+y^3+z^3-3xyz)$. Find $(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z})^2 f$.

Let $f(x,y,z) = \log(x^3+y^3+z^3-3xyz)$ then to find the value of $\displaystyle \left(\frac{\partial}{\partial x} + \frac{\partial}{\partial y}+\frac{\partial}{\partial z}\right)^2 f$. We can do it ...
0
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1answer
28 views

How can I derive the second degree equation for a curve if I know the slope at two points and the y-intercept = 0?

How can I derive the second degree equation for a curve if I know the slope at two points and the x and y-intercepts = 0?
0
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1answer
22 views

can I imply this as the meaning of differentiation?

most of my textbooks and teachers told me that dy/dx is the slope of a function at some point of x but I observed it such that: dy/dx= Change of y for every 1x am I wrong?
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2answers
33 views

Derivative of $4x^5 \tan(\frac{-1}{x})$

I can't seem to be able to get the correct signs for this derivative. $\frac{d}{dx}(4x^5\tan(\frac{-1}{x}))$ Here's my work: $= (4x^5)'\tan(\frac{-1}{x}) + 4x^5(\tan(\frac{-1}{x}))'$ $= 20x^4\tan(\...
0
votes
0answers
22 views

Finding the solution using Lagrange Multiplier

For a matrix $\mathbf{D} \in \mathbb{R}^{d\times n}$ and a vector $\mathbf{a}_i \in \mathbb{R}^{d\times 1}$, I need to solve the following problem for variable $\mathbf{d}_{:,j}$ using Lagrange ...
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0answers
27 views

Application of derivatives, proof check.

Part a) I know answer is the second one, because S is a function of three variables and therefore partial derivatives are taken, and further x,y,z are functions of t only, so total derivative is ...
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0answers
59 views

basic matrix calculus identity

I'm working out very basic matrix calculus identities. I'm using Andrew Ng's CS 229 Lecture 1 notes, pg. 9, equation 2: http://cs229.stanford.edu/notes/cs229-notes1.pdf : $$\nabla_{A^{T}}f(A) = (\...
0
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1answer
29 views

How to find an inverse of this function

So I am aware of the general rules to follow to find an inverse of a function, but it seems like I'd need something different for this one: $$f(x) = -2x^3-7x+5$$ if I try what I'm use it I end up ...
0
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1answer
28 views

What is $d(p^{1-\gamma } T^{\gamma})$?

Why does $d(p^{1-\gamma}T^\gamma$) give $\frac{dp}{p} = \frac{\gamma}{\gamma - 1} \frac{dT}{T}$? If $(p^{1-\gamma}T^\gamma) = \text{constant}$? where $p$ is pressure, $T$ is temperature and $\gamma = ...
3
votes
2answers
140 views

Differenciating a function with respect to another function confusion

I am having problem solving the following question- Differenciate $\tan^{(-1)}{(\sqrt{1-x^2}/x)}$ with respect to $\cos^{(-1)}{(2x\sqrt {1-x^2})}$, where $x$ is not $0$. My attempt - I took the tan ...
0
votes
2answers
26 views

Solving $0 = -7\csc(x)\cot(x)$

$$f(x) = 7\csc(x)$$ How do I solve this derivative when I set it equal to zero? $$0 = -7\csc(x)\cot(x)$$
2
votes
2answers
74 views

Implications of differentiability and Taylor expansion

Consider a function $\phi: \Theta \subseteq \mathbb{R}^l \rightarrow \mathbb{R}$. Fix $\theta_0 \in \Theta$. Assume: (1) $\phi(\cdot)$ differentiable at $\theta_0$ (2) The gradient at $\theta_0$, $...
1
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1answer
60 views

Is it correct to teach the derivative as the slope of the tangent line?

In introductory calculus, the derivative of a differentiable function $f$ at some point is often taught as being the slope of the tangent line to the graph of the function at that point. My question ...
3
votes
4answers
83 views

If $f'(2) = 7$ , calculate the $\lim_{h\to 0} \frac{f(2+3h)-f(2-5h)}{h}$

If $f'(2)=7$ then calculate the limit: $\displaystyle \lim_{h\to 0}\frac{f(2+3h)-f(2-5h)}{h}$ . Okay, so I know what the definition of derivative is, but how do I use the fact that $f'(2) = 7$ to ...
-1
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1answer
29 views

for what values of x will f (x).f'(x) < 0 [duplicate]

Given f (x) = x^3 - 4x^2 - 3x + 18 For which values of x will: f (x) . f'(x) < 0 How would you read the answer straight of the graph without any calculations? The answers should be in this form: ...
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0answers
27 views

Differentiation with Matrix Coefficients and scalar variable.

I know how to do this for vector variables, and want to make sure it is the same for scalar variables. Take for example $$f(t)=\frac{\underline{x}^T\textbf{A}\underline{x}}{t}$$ where $\underline{x}\...
0
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0answers
15 views

$\frac{d^2}{dx^2} = \lim_{\delta x-> 0} [f(x-\delta x) - 2f(x) + f(x+\delta x))/\delta x^2]$ find a matrix

Given that $\frac{d^2}{dx^2} = \lim_{\delta x-> 0} [f(x-\delta x) - 2f(x) + f(x+\delta x))/\delta x^2]$ find an appropriate matrix that could represent such a derivative operator, in a form ...
1
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1answer
53 views

Prove that $ \lim\limits_{n \to \infty } \sum\limits_{k=1}^n f \left( \frac{k}{n^2} \right) = \frac 12 f'_d(0). $

Let $I \subset \mathbb{R}$ be an open interval with $0 \in I$ and $f:I \to \mathbb{R}$ a continuous function, with $f(0)=0$, right-differentiable in $0$. Then: $$ \lim\limits_{n \to \infty } \sum\...
1
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2answers
88 views

What are some examples where it is analytically easier to compute integrals than derivatives?

I know that in general, there exist more functions which are integrable than there are functions which are differentiable (nowhere differentiable to be exact), at least in $C([0,1])$, by the Baire ...
2
votes
2answers
54 views

Find derivative using the limit definition for the function $f(x) = e^{2(x+1)}$

Use the limit definition of derivative to find $\frac{\mathrm d}{\mathrm dx}f(x)$ for the function $f(x)= e^{2(x+1)}$. I know it's going to be $2e^{2x+2}$. I can solve it normally, I just don't know ...
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0answers
40 views

Derivatives of a simple function

I have asked this seemingly very simple question here a while ago but did not get any answer. Consider the function $f(x)=x^b(1−x)^{1-b}$ defined on $[0,1]$, with $0<b<1$. How can we prove that ...
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0answers
25 views

Derivative of expresion with matrix in exponent

I am looking at a proof which states the following results: $$\frac d {dt}\left(A \frac{t^2}{2!} \right)=A^2t$$ where $A$ is a matrix. Why is it that the matrix gets squared, and for the derivative ...
1
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1answer
140 views

Partial derivative of weighted matrix factorization?

$C \in \mathbb{R}^{m \times n}, X \in \mathbb{R}^{m \times n}, W \in \mathbb{R}^{m \times k}, H \in \mathbb{R}^{n \times k}$ $W_{i.}$ is the $i$th row of $W$ $H_{j.}$ is the $j$th row of $H$ $$f=...
3
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3answers
57 views

Showing that a derivable function $f$ (satisfying some conditions) is null.

Enunciate: Let $f:\mathbb{R}\to\mathbb{R}$ derivable, such that $f(0)=0$ and, for all $x\in\mathbb{R}$, satisfy $f'(x)=(f(x))^2$. Prove that $f\equiv 0$. How can I prove it? I'm trying to ...
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0answers
37 views

Equality of integrals and derivatives

Probably it's a very basic question but I am confused.. Is it a true that: $$ \int\limits_{0}^{\infty} \int\limits_{0}^{\infty} \frac{\partial^2}{\partial x_1\partial x_2}f(x_1,x_2) dx_1 dx_2 = \int\...
3
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0answers
19 views

Let $f : R → R$ be a differentiable function and $f(x) = 0$ for $|x| ≥ 10$. Let $g(x) = \sum_{k∈Z} f(x + k)$. [duplicate]

Let $f : R → R$ be a differentiable function and $f(x) = 0$ for $|x| ≥ 10$. Let $g(x) = \sum_{k∈Z} f(x + k)$. Then one of the following is true: (a) $g$ is differentiable and $g'$ has infinitely ...
0
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1answer
61 views

Given a convex function $f(x)$, is $xf'(x)$ also convex?

Given a convex function $f(x)$, I'm trying to proof that $g(x) = xf'(x)$ is also convex. I have found neither a proof nor a counterexample so far. A function $g(x)$ is convex iff $g''(x) \ge 0$. ...
5
votes
3answers
86 views

$n$-th derivative of $\sin^k(x)$

I would like to know if there is a general formula to calculate the $n$-th derivative of $\sin^k(x)$ evaluated at $x=0$, that is, $$\left.\frac{d^n}{d x^n} (\sin^k(x))\right|_{x=0}$$ with $0\leq k \...
2
votes
1answer
50 views

Frechet Derivative of a direct product of functions

Given two functions $f: U \to \mathbb{Y}$ and $g: U \to \mathbb{Y}$ (where $U\subset \mathbb{X}$ is open) that are Frechet differentiable at $x$. Also, $||(x,y)||_{\mathbb{X} \times \mathbb{X}}=||x||_{...
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1answer
26 views

Global minimum of a parameteric function

Let $q:[1, + \infty) \subset \mathbb{R} \longrightarrow \mathbb{R}$ be a function defined as $ \qquad \qquad \qquad \qquad \qquad \quad q(x) = \left \{ \begin{array}{lcl} \delta_{1} & \text{ if } ...
0
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3answers
360 views

What points is the norm Frechet differentiable at

I know the definition of Frechet derivatives - there exists a bounded linear map... Maybe someone could show me a similar example on how to approach questions like these.
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0answers
14 views

Find $h : U \to \mathbb{R}^n$- Precision on a question?

Let $U \subset \mathbb{R}^n$ an open set, $f : U \to \mathbb{R}$ and $\hat{x} \in U$. Show that $f$ is differentiable at the point $\hat{x}$ if and only if there exists a continuous function $h : ...
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vote
2answers
137 views

Another $1=2$ proof [duplicate]

So a friend shows me this : $x^4= x^2+x^2+ \cdots +x^2 $ ( i.e. $x^2$ added $x^2$ times) Now take the derivative of both side; $4x^3 = 2x + 2x + \cdots + 2x $; So $4x^3 = 2x^3 \cdots $(1) And ...
5
votes
1answer
136 views

The function $\mathrm{Li}_2(x)=\int_2^x\frac{dt}{\log^2t}$, its inverse and summation

I am reading the more understandable mathematics in the section Preliminary Results of a paper in which the authors give a explanation of facts for the logarithmic integral and its inverse. In this ...
0
votes
2answers
43 views

Derivate of Function

I have to find the derivative of $\displaystyle g(z) = 1 + \sqrt{4-z}$. I wrote $\displaystyle g(z)=1+(4-z)^{\frac{1}{2}}$. Deriving - $\displaystyle g'(z)=\frac{1}{2}(4 - z)^{\frac{1}{2}-1}$ which ...