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

learn more… | top users | synonyms (2)

0
votes
0answers
54 views

5 Parameter Affine Transformation

I am working on computing affine transformation using Gradient Ascent Method, so the Inverse compositional algorithm. However, I am stuck in one probably simple step but I fail to understand them. ...
1
vote
1answer
39 views

Prove that $\frac{\pi}{2}-x<\tan^{-1}(x)<\frac{\pi}{2}-x+\frac{x^3}{3}$

Prove that for every $x>0$, it is true: $$\frac{\pi}{2}-x<\tan^{-1}(x)<\frac{\pi}{2}-x+\frac{x^3}{3}$$ We can split it into two statements: $\frac{\pi}{2}-x<\tan^{-1}(x)$ ...
8
votes
3answers
635 views

What does the derivative of area with respect to length signify?

Suppose that we have a square sheet of edge length $L$. Its area $A=L^2$. Differentiating $A$ w.r.t. L, we get $$\dfrac{dA}{dL}=2L$$ I do understand what it means to differentiate, ...
1
vote
2answers
51 views

Is a bounded continuous function defined on $\Bbb R$ differentiable?

Is a bounded continuous function defined on $\Bbb R$ differentiable? Why so? The query is fueled by the following question: Let $f : \Bbb R \rightarrow \Bbb R$ be a bounded continuous function. ...
0
votes
2answers
25 views

Find the units of measurement of constant from formula

$$m\frac{dv}{dt}=mg-kv^2$$ $v=\ms^{-1} $m=kg$ $g=ms^{-2}$ $v^2=(ms^{-1})^2 I re-arrange the formula to isolate K $$K=-\frac{m\frac{dv}{dt}}{v^2}+\frac{mg}{v^2}$$ Sub in the units ...
1
vote
1answer
46 views

How to calculate the log of a sum

I'm going over some basic statistical physics and I need to compute $$\dfrac{\partial}{\partial \beta}\ln\sum_{i}e^{-\beta E_i}$$ Theres probably some simple trick i'm missing but i'm really ...
0
votes
1answer
64 views

Show that a metric space is complete

I have the following problem: Let $X$ denote the collection of all differentiable continuous functions $f : [0, 1] \rightarrow \Bbb R$ such that $f(0) = 0$ and $f'$ is continuous. For $f, g \in X$, ...
1
vote
1answer
44 views

The chain rule, partial derivatives and general functions

I am revising for my first year Calculus examination. The following question is on a past paper and I am given the solution, however I am struggling to make sense of it: Let $V(x,y)$ be a ...
0
votes
0answers
65 views

Derivative of quadratic form w.r.t. matrix (product)

I need to show that some quadratic from: 1' A C A 1 is increasing in matrix C , where 1 is a (Kx1) vector of ones, and A and C are both (KxK) positive definite. Can I reason like this: 1) ...
1
vote
1answer
66 views

Does uniform convergance of $f$ imply convergance of derivatives?

Let $X$ denote the collection of all differentiable functions $f : [0, 1] \rightarrow \Bbb R$, such that $f(0)=0$ and $f'$ is continuous. Let $\{f_n\}$ be a Cauchy sequence. By Cauchy criterion for ...
2
votes
0answers
90 views

About differentiation under the integral sign

I would like to ask something related to the application of the differentiation under the integral sign (Leibniz rule) given by ...
1
vote
1answer
46 views

Evaluating stationary points with null Hessian determinant

Given the function $f(x,y)=x^2y^3$ I'm asked to evaluate all the stationary point. My work: I started calculating the derivatives: $f_x=2xy^3$ and $f_y=3x^2y^2$ then I looked for the point such that ...
1
vote
1answer
25 views

Unsure of meaning of question

A glass of water is rotating about its axis at constant angular velocity $\omega$. Let $y=f(x)$ denote the equation of the curve obtained by cutting the surface of the liquid with a plane passing ...
1
vote
1answer
32 views

description of the function whose graph corresponds to Figure

Consider f be a real continuous function , $f(0) = 0$ , and whose graph has the form shown in the figure: a) How can a give description of the function whose graph corresponds to Figure. b) Sketch ...
1
vote
0answers
34 views

equivalence of two definitions of differentiability

I have two definitions and I want to understand why they are equivalent. 1. Def.: Consider $U\subseteq \mathbb{R}^n$ open, $f:U\to \mathbb{R}$ a function. f is called differentiable of class $C^k$, ...
0
votes
1answer
62 views

How to find if and where $f(x)$ is continuous and/or differentiable for a given piecewise function? [closed]

What approach would be ideal in finding if and where $f(x)$ is continuous and/or differentiable when $f$ is a piecewise defined function? A concrete example is below, but I'm interested in general ...
0
votes
1answer
41 views

What Law prohibits me from doing this?

I am working on a differentiation question and my answer to simplify is $2x\ln \left(2x-1\right)+\frac{2}{2x-1}x^2$ Why cant I cancel out a x when multiplying to get $\frac{2x}{2-1}$= $2x$ ? So ...
0
votes
1answer
23 views

Chain rule and multivariable derivatives

Given $F\colon\mathbb{R}^2\to\mathbb{R}$ defined as $F(x,y) = x^2 y + y^3 + 2x-1$ and $g\colon\mathbb{R}\to\mathbb{R}$ such that $g(0) = 1$ and $F(x, g(x)) = x$, find $g'(0)$.
5
votes
1answer
144 views

Derivative of $f(x)^{g(x)}$ at points when $f(x)=0$

I am interested in understanding the general behavior of the derivative for $$f(x)^{g(x)}$$ at points where $f(x)=0$. For example, if $f^g=x^n$ we have $$\frac{d}{dx}f^g(0)=\begin{cases}0 & n\ge ...
0
votes
1answer
31 views

Twice Differentiable Function Proof

Let f:(a,b)→R be twice differentiable, and assume that |f '(x)-f '(y)|≤ |x-y| for all x, y ∈ (a,b). Show that |f(x) - f(y) - f '(x)(x-y)|≤ |x-y|^2 for all x, y ∈ (a,b). I am stuck and not quite ...
5
votes
2answers
121 views

Generalized power rule for derivatives

Background This background is not really necessary to answer my question, but I included it here to provide context. This question has some programming aspects to it as well, but since my question ...
1
vote
1answer
60 views

Is the function $f(x)=x$ on $\{\pm\frac1n:n\in\Bbb N\}$ differentiable at $0$?

This is really a question of definitions. If a function $f$ is not defined on an open set containing $x$, how do we define the derivative of $f$? Is it sufficient to be locally approximable by linear ...
0
votes
2answers
32 views

Prove that $2\arctan x + \arcsin \frac{2x}{1+x^2} = \pi$ for every $x\geq1$

Prove that for every $x\geq1$ $$f(x) = 2\arctan x + \arcsin \frac{2x}{1+x^2} = \pi$$ My idea is to firstly calculate $f(1)$ which is actually $\pi$. Then I need to show, that for every $x\geq1$, ...
4
votes
2answers
59 views

Continuity, derivatives of rational function

I'm studying the continuity of a function and its derivatives checking if the function is continuous, differentiable and calculating some derivatives. The function is \begin{cases} ...
0
votes
1answer
42 views

Differentiation of an infinite series of functions

I want to show: If $$f(x) :=\sum \limits_{n=0}^{\infty} g_n(x)$$ where the series is convergent for every $x \in [a,b]$ and $g_n(x)$ is a nondecreasing function on $[a,b]$ for each $n$ ; then ...
1
vote
0answers
14 views

Differentiation of vector-function

Let $f(x) = e^{-x^Tx},$ where $x \in \mathbb{R}^n$. What will be the second derivative? The first is $~f'(x) = 2x^T e^{-x^Tx}$, and when I try to find the second, I confuse. It will be $$f''(x) = ...
1
vote
0answers
39 views

Prove that if the derivative $f'(x)$ of a function exists on the measurable set $E$, then $f'(x)$ is measurable on $E$.

Prove that if the derivative $f'(x)$ of a function exists on the measurable set $E$, then $f'(x)$ is measurable on $E$. We are told to only consider 1 dimensional spaces,that f is a measurable ...
1
vote
1answer
20 views

Gradient of draws from random variables

In the context of neural networks there has recently been some interest in differentiation incarnations of random variables. Example. Given a random variable $y \sim \mathcal{N}(\mu, \sigma^2)$. Now ...
0
votes
1answer
42 views

Lagrange and Leibniz notation.

Suppose $g=g(x,y)$ is a certain function and we need to find the new function $g_x(x^2y,y)$, say. How would one write this in Leibniz notation. Is it $\cfrac{\partial g(x^2y,y)}{\partial x}$ or ...
0
votes
2answers
70 views

How to find $\frac{dy}{dx}$ for $\sqrt{xy} = 1$? [closed]

What approach would be ideal in finding $\frac{dy}{dx}$ for $\sqrt{xy} = 1$?
0
votes
1answer
25 views

Finding units of measurement of coefficients in ODE's

If we have a question where we have to find the coefficient's units such as K in this case. The actual formula contains more parts but it is simply the derivatives that I am unsure about. ...
1
vote
0answers
41 views

Find k such that the function $f(x)=|x|^3$ is $C^{k}$ but not $C^{k+1}$

I've been working on this for a while, it is on a practice sheet so there are no answers. A proof would be appreciated! Find k such that the function $f(x)=|x|^3$ is $C^{k}$ but not $C^{k+1}$ ...
0
votes
2answers
32 views

Series differentiation

$\displaystyle e^x= \sum_{j=0}^{\infty} \frac{x^j}{j!}$ The textbook says that when we differentiate this, we obtain the same series, so that $(e^x)'=e^x$. But why is this? Isn't the derivative ...
0
votes
1answer
47 views

Why is My answer Wrong?

I have a question here $\frac{d}{dx}\left(\frac{8}{e^{1-4x}}\right)$ I simplify this $8\left(\frac{1}{\left(e^{1-4x}\right)^2}\right)\left(-4e^{1-4x}\right)$ to $\left(\frac{32}{e^{1-4x}}\right)$ ...
0
votes
0answers
40 views

Order of differentiation on a power series

I encountered something strange to me just now. Say we have $$f(x)=\ln(1+x^3)$$ Now, I want to find the power series expansion for $f'(t^2)$. I get two different answers for when I take the ...
5
votes
1answer
106 views

Is the standard part function another devil's staircase?

The devil's staircase or Cantor function is an awesome function that increases value but has derivative zero everywhere (or "almost", whatever that means). I was incredibly amazed when I found out ...
2
votes
2answers
93 views

Properties of the function defined by $g(x) = \sum\limits_{n=0}^{\infty} \frac{1}{1+n^2x^2}$

I am looking at the function $g:\mathbb{R} \rightarrow \mathbb{R}$ defined as $$g(x) = \sum\limits_{n=0}^{\infty} \frac{1}{1+n^2x^2}$$ I would like to know if this function is convergent, continuous ...
4
votes
4answers
196 views

Rigorously prove that the tangent line is indeed tangent?

Let $f$ be a function of $x$ and $f'$ be the derivative, at a point $(x_0, f(x_0))$ the slope is $f'(x_0)$, we know from any calculus book that the line $g(x) = f(x_0) + f'(x_0)(x - x_0)$ is tangent ...
1
vote
2answers
25 views

Find the derivative with respect to $x$ of $y=\log_4 (x^3)$

I got $\frac{3x^2}{x^3(\ln 4)}$. Then $x$ cancels and left with $\frac{3}{x \ln4}$ ? That felt too easy so I'm sure its wrong. Or am I actually correct on this one?
0
votes
1answer
19 views

Proving a function is bounded above.

Hi all, while doing this question ,I feel that I understand the concept of the question, but can't seem to formulate it into a viable answer. If the limit as $x \rightarrow \infty$ is the same as $x ...
0
votes
2answers
20 views

Parametric Eqn / Differentiation

Parametric eqns of a curve are $x = t + \frac{1}{t}$ , $y = t - \frac{1}{t}$, where $t$ cannot be $0$. At point $P$ on curve, $t = 3$ and the tangent to curve at $P$ meets the $x$-axis at $Q$. The ...
3
votes
1answer
37 views

What is the largest class of measurable functions $f$ s.t. $f'$ a.e.?

We know by Lebesgue Theorem that monotone functions on interval [a,b] has finite derivate almost everywhere and different of two monotone functions have finite derivative a.e. $\textbf{My Question}$ ...
1
vote
1answer
124 views

Real Analysis: Derivative of the function

Find k such that the function $f(x)=|x|^3$ is $C^{k}$ but not $C^{k+1}$ I am SO lost...any help would be appreciated
3
votes
4answers
65 views

How do you work with infinitesimal exponents in synthetic differential geometry?

I just read this paper by Andrej Bauer, which discusses the basic tenets of synthetic differential geometry. Namely, that for any function $f$, any real number $x$, and any infinitesimal $\epsilon$ (a ...
0
votes
1answer
62 views

Approximating area under/above the curve

I got this problem and not sure how to relate the derivative and the integral: Let $f : [0, 1] \rightarrow \Bbb R$ be twice differentiable, and assume that $f(0) = f(1) = 0$, $f''$ is continuous, ...
2
votes
3answers
48 views

Help with a derivitive

Im studying for my final in calculus and there is one derivative that I am having an issue with: $$ 1\over x(\ln x)^p $$ The solution according to the book: $$ -{p+\ln x\over x^2\ln x^{p+1}} $$ ...
0
votes
0answers
74 views

k-times differentiable functions on [0,1]

Is $C^k[0,1]$ (the set of all k-times differentiable function, not necessarily continuously) complete with respect to the norm $\|f\|_\infty + \|f'\|_\infty +\cdots+\|f^{(k)}\|_\infty$? I know the ...
2
votes
2answers
62 views

Is this a valid proof for eulers formula?

I am wondering whether this proof is a valid proof of Eulers formula: $e^{ix}=i\sin(x)+\cos(x)$ $$\frac{d}{dx}e^{ix} = i(e^{ix})$$ $$\frac{d}{dx}(i\sin(x)+\cos(x)) = i\cos(x)-\sin(x) = ...
2
votes
1answer
29 views

Calculate the derivative of $Γ(z,v)$ with respect to $z$

Let $Γ(z,v)=∫_{v}^{+∞}t^{z-1}e^{-t}dt$ be the incomplete $Γ$-function. My question is: Calculate the derivative of $Γ(z,v)$ with respect to $z$.
2
votes
1answer
26 views

Is finding the tangent plane to a surface made any more complicated if the surface $\neq 0$?

So I have $x^2 + y^2 - z^2 = 4$ as my surface and the point I'm looking at is $(2,1,1)$. So if it was $0$, I'd do my partial derivatives and get the equation: $4(x - 2 ) + 2(y-1) - 2(z-1) = 0 $ ...