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

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0
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2answers
15 views

Proving a function has second derivative / Uniform convergence of a series

I am studying sequences and series of functions and in the course notes there is this excercise: Prove the function $$f(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$$ has second derivative. ...
-1
votes
2answers
31 views

How to prove polynomial has repeated roots? [on hold]

if we have a increasing function, say $f(x)$, so we can say $f'(x) \geq 0,\space \forall x\in \mathbb{R}$. We take a special case: if $f'(x)=0$ has a root $\alpha$ and $f(\alpha)=0$ this ...
0
votes
1answer
27 views

Show that a differentiable function $f:\mathbb{R} \to \mathbb{R}$ has a global max in $a$ if $a$ is its local max

My task is this: Let $f:\mathbb{R} \to \mathbb{R}$ be a differentiable function and assume that the only stationary point $f$ has is a local max in the point $A = (a,f(a))$. Show that $A$ must be a ...
1
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1answer
16 views

Formula for the gradient of $f(A) = u^TA^kv$

Given a function of the form $$f(A) = u^TA^kv,$$ where $A$ is an $n\times n$ real-valued matrix, $u$ and $v$ are real vectors, and $k$ is some positive integer power. Does there exist a general ...
1
vote
1answer
7 views

Multivariate Non-Differentiability

This example says that "continuous partial derivatives imply differentiability but not vice-versa". Based on transposition logic, I would then assume that if a multivariate function has discontinuous ...
0
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1answer
16 views

Lipschitz continuity of continuously differentiable function

Is it true that a continuously differentiable function in a Banach space $X$ is locally lipschitz in $X$?
0
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2answers
27 views

Differential Equation - Where does the solution end?

I was asked to solve the differential equation $y'+\frac{y}{x+1}=\frac{2y-1}{x}$, given the starting point y(0.5)=5/6. The equation meets the criteria for Existence and Uniqueness for every x>0 (as y' ...
0
votes
1answer
29 views

Prove that the derivative of the function f(x) = e^(-1/x^2) at x is 0

Let $f$ be defined such that $f(x) = e^{-1/x^2}$ when $x \neq 0$, and let $f(x) = 0$ when $x = 0$. How can I go about proving that $f'(0) = 0$? I know that: $f'(0) = \lim\limits_{x \to 0} ...
1
vote
0answers
10 views

Deriving the Normalization formula for Associated Legendre functions: Stage $3$ of $4$

The question that follows is a continuation of this Stage $1$ question and this previous Stage $2$ question which are needed as part of a derivation of the Associated Legendre Functions Normalization ...
0
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0answers
36 views

Is this partial derivative correct?

I think that the $-x_1$ in the underlined segment should be a $-x_3$, unless I am understanding the notation wrong.
0
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0answers
30 views

Derivative of L2 norm

I am reading a paper about image processing and I have a question. In the paper we have equations like below. $X_{C1} = 0.596X_R - 0.274X_G - 0.322X_B$. $X_{C2} = 0.211X_R - 0.523X_G + 0.312X_B$ ...
0
votes
0answers
22 views

differentiation of a norm of matrix function

I need to differentiate the following function W.r.to $x$ $y=\|x (\mathbf{I-W}-x \mathbf{Diag(v_2)W})^{-1}\mathbf{v_1} - b\|_2$ where $0<x<\frac{2}{max_i{|{v_2}_i|}}$,$\mathbf{v_1}\in ...
2
votes
0answers
13 views

Prove $f(x_1,\dots,x_n) = g(x_1-x_na_1/a_n,\dots,x_{n-1}-x_na_{n-1}/a_n)$ iff $a\bullet \nabla f(x)$

Let $f:\mathbb{R}^n \to \mathbb{R}$ be differentiable and $a=(a_1,\dots,a_n),\; a_n\neq 0$. Prove $a\bullet \nabla f(x) = 0 \;\forall x\in\mathbb{R}^n$ if and only if there's a differentiable ...
0
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2answers
34 views

Use the rule for differentiating a product to prove that the derivative of $x^n$ is $nx^{n-1}$ for all $n∈N$.

I know the rule of differentiation, but to proving why the derivative is that is my problem. Should I be proving this question by induction because that's what I've been learning.
0
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1answer
33 views

Find $dy/dx$ if $xy + y^2 = 2$. [on hold]

I just can't understand when I try to differentiate it comes $$x\frac{dy}{dx}+y\cdot1 + 2y\cdot \frac{dy}{dx} = 0$$ for some reason in the self-help book. Please explain thoroughly.
1
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1answer
13 views

How to compute the following sum of the differentiable map?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable map such that $f(x) = x$ for $x \notin [-T, T]$ for some $T>0$ and such that $0$ is a regular value. Compute the $$\sum\limits_{x\in ...
1
vote
1answer
52 views

Find 2 continuous functions $F$ and $G$ defined on $[a;b]$, such that $F'(x) = G'(x)$, but $F(x) - G(x) \neq \text{const}$

The problem: Find 2 continuous functions $F$ and $G$ defined on $[a;b]$, such that for every $[\alpha;\beta] \subset [a;b]$ there exists an interval $[\alpha';\beta'] \subset [\alpha;\beta]$, where ...
0
votes
1answer
37 views

Continuously differentiable functions

Let $f, g,$ be $ C^2$ functions $\mathbb{R} \rightarrow \mathbb{R}$, $ F: \mathbb{R}^2 \rightarrow \mathbb{R}, F(x,y) = f(x+g(y))$ Check that $(D_1F)(D_{12}F)=(D_2F)(D_{11}F)$ I know how to ...
0
votes
3answers
34 views

An increasing smooth map $f:(0,1)\rightarrow(0,1)$ which does not extend to any smooth function on a larger domain

Although I'm not sure it's related, I have found a smooth map $f:(0,1)\rightarrow(0,1)$ which does not extend to any continuous function on a larger domain, namely ...
3
votes
1answer
48 views

Deriving the Normalization formula for Associated Legendre functions: Stage $2$ of $4$

The question that follows is a continuation of this previous Stage $1$ question needed as part of a derivation of the Associated Legendre Functions Normalization Formula: ...
0
votes
2answers
34 views

AP Calculus BC - Derivative of inverse problem

Let $g(x)$ be the inverse of the function $f(x)$. Given the following values on the table below, at which value $x=a$ will $g'(a)=1/6$? (No calculator allowed) ...
0
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1answer
33 views

AP Calculus BC - Polar curve question

A particle moving along the polar curve given by $r = 2 + 2\sin(\theta)$ has position $(x(t),y(t))$ at time $t$, with $\theta = 0$ when $t = 0$. This particle moves along the curve so that ...
0
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1answer
24 views

Chain rule for $f(X(t), Y(t))$ where $X, Y : R \to R^2$

I'm having some trouble on understanding how to calculate the derivative of $g(t)$ with regards to $t$, where $g(t) := f(X(t), Y(t))$ and $X(t)$ and $Y(t)$ are $2d$ vectors. That is $X,Y: R \to R^2$. ...
1
vote
1answer
29 views

Equation of a line tangent to $g(x)$ and parallel to line connecting endpoints of $g(x)$

Let $g(x)$ be a differentiable function defined on the interval $0 \le x \le 16$. Some values of $g(x)$ and its derivative $g'(x)$ are given below. Which of the following is the $x-intercept$ of the ...
0
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3answers
20 views

Alternating sign Nth derivative

Say I have a function $$ f(x) = \dfrac 1x$$ and I'm looking at its $n^{th}$ derivative and trying to come up with a formula. I can easily get it because if forms a very consistent pattern and it ...
0
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0answers
28 views

velocity and acceleration of a disk rotating at a constant speed

A disk of radius 1 is rotating in the counterclockwise direction at a constant angular speed ω. A bug starts at the center of the disk and moves directly toward edge. The position of the bug at time ...
1
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4answers
75 views

$\int \frac{1}{\sqrt{x^2+1}} dx$

So I've seen some options on the internet that are fairly good, but I have this substitution: $x^2+1=t-x$, you square both sides and get $x = (t^2-1)/t$ and $x + 1 = (t^2-1)/2t + 1$. If we call that ...
2
votes
1answer
33 views

Find the derivative of each of the the following functions

Find the derivative of each of the the following functions. $f(x)=\sqrt{7+\sqrt{x^3}}$ $\frac{d}{du}\left(\sqrt{u}\right)\frac{d}{dx}\left(7+\sqrt{x^3}\right)$ A: ...
2
votes
2answers
47 views

The Cantor staircase function and related things

The Cantor staircase function https://en.wikipedia.org/wiki/Cantor_function has an interesting property: $\{x\colon f'(x)\neq 0\}$ is a nowheredense nullset. But it it differentiable almost ...
2
votes
1answer
40 views

The set where a derivative vanishes is G-delta

If $f:I\to R$ ($I$ - interval) is differentiable, then $\{x\colon f'(x)=0\}$ is a $G_{\delta}$ set. The lecturer didn't prove this fact and I found no proof in my books. How it can be proven?
0
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2answers
43 views

Derivative of n x n Invertible Matrix

For an invertible $n$ x $n$ matrix $A$, define $f(A):=A^{-2}$. Calculate the derivative $D\space f(A)$. (i.e. give $D\space f(A)B$ for arbitrary $B$.) I'm not super sure how to go about this?
0
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0answers
5 views

Accuracy Rebonato Swaption Approximation Formula among Different Strikes

Can somebody explain me if the Rebonato swaption volatility approximation formula is accurate for only ATM strikes, and if yes why? Can it also be used for ITM and OTM strikes? My foundings: Let $0 ...
0
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0answers
15 views

Equality of mixed partial using double limit

As a student, my teacher told me to give a proof for the equality of mixed partial. The Theorem stated that, Supposed $f$ is a real value function of 2 variable $x$ and $y$ and $f(x, y)$ is defined ...
1
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0answers
23 views

Logistic model - solution verification

I'm looking at the Logistic model: $$\begin{cases} \dot{X} = X(1-X)\\ X(0) = X_0 \end{cases}$$ where the phase space is $M = \mathbb{R}$. The solution appears to be $X(t) = \dfrac{1}{1 + ...
4
votes
3answers
137 views

Prove that $f(ab) = f(a) + f(b)$

Question : Assume only that $f: (0,\infty)\to{\mathbb{R}}$ is differentiable and that $f'(x) = 1/x$, and $f(1)=0$. Prove that for all $a,b \in(0,\infty)$, $f(ab)=f(a)+f(b)$. [Hint: Let $g(x)=f(ax)$] ...
1
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0answers
23 views

How to prove second order differentiation matrix is of the form..

Given that the matrix: $$D2 = \left[\begin{matrix}a11&a12&a13\\a21&a22&a23\\a31&a32&a33\end{matrix}\right]$$ is a second-order differentiation matrix in the sense, for a ...
2
votes
3answers
33 views

I want to find whether the expression $D = \sqrt{5t^2 - 40t+125}$ is increasing or decreasing when $t=5$.

I want to find whether the expression $D = \sqrt{5t^2 - 40t+125}$ is increasing or decreasing when $t=5$. My logic is I want to find whether is $f'(5)>0$ or $f'(5) < 0$. I need to use the ...
1
vote
0answers
32 views

Why should $\phi'$ and $\phi''$ be $\mathcal O(1)$?

As Strogatz writes in his book Nonlinear Dynamics And Chaos (p. 64) There are often several ways to nondimensionalize an equation, and the best choice might not be clear at first. Therefore we ...
1
vote
1answer
8 views

What does it mean for the difference $V(\overline{x}_L -dx) -V(\overline{x}_L)$ to be of the second order in $dx$

What does it mean for the difference $V(\overline{x}_L -dx) -V(\overline{x}_L)$ to be of the second order in $dx$, where $dx$ is some tiny increment of $x$? What we know about $V$: $V(z) = U(z) - ...
0
votes
0answers
45 views

Find the points on the curve $y=x+e^x$ at which the tangent line is horizontal.

Find the points on the curve $y=x+e^x$ at which the tangent line is horizontal. The answer was $(0,1)$, but I don't get it. I tried to take the derivative of the function and equal it to $0$ ...
0
votes
1answer
18 views

Fourier Integral Theorem?

I have this function $x(t)=\left|\frac{t}{T}\right|rect(\frac{t}{2T})$ The book states that given the function x(t) is piecewise linear, we can use the Fourier theorem to calculate X(f). They get the ...
0
votes
0answers
23 views

Compute $\lim_{t\to0}\frac{f(2+t,3+t)-f(2,3)}{t}$ using the partial derivatives of $f$

How can I solve the question? I know that I need to work with the definition of the partial derivatives with respect to $x$ and $y$. $$f'_y(2,3)=-3,\quad f'_x(2,3)=2\\ ...
2
votes
2answers
71 views

How to properly find supremum of a function $f(x,y,z)$ on a cube $[0,1]^3$?

Solving an applied problem I was faced with the need to find supremum of the following function $$f(x,y,z)=\frac{(x-xyz)(y-xyz)(z-xyz)}{(1-xyz)^3}$$ where $f\colon\ [0,1]^3\backslash\{(1,1,1)\} ...
2
votes
0answers
74 views

Functions $g(x)/h(x),h(x)/f(x)$ are constant [duplicate]

Suppose $f$, $g$, $h$ are functions from the set of positive real numbers into itself satisfying $f(x)g(y)=h(\sqrt{x^2+y^2})$ for all $x$, $y\in (0,\infty)$. Show that the functions $g(x)/h(x)$, ...
1
vote
1answer
18 views

Find the maximum value of the function

So I was just messing around with finding the maximum and minimum values of functions, and I came across this: $$ \text{Find the maximum value of} \,\, f(x)=\frac1{x^{2x^2}}.$$ Any ideas?
4
votes
3answers
44 views

Differentiability of function for $\Bbb{Q}$ and $\Bbb{R}\setminus \Bbb{Q}$

A function $f:\Bbb{R}\to\Bbb{R}$ is defined by $f(x)=x$, if $x$ is rational; $\sin(x)$ if $x$ is irrational. Show that $f$ is differentiable at $0$ and $f'(0)=1$. Here I'm thinking to apply ...
1
vote
2answers
22 views

$f(x, y) = \prod_{i = 1}^n (1 + xy_i)$, what is ${{{\partial f}\over{\partial x}}\over f}$, geometric series?

Let$$f(x, y) = \prod_{i = 1}^n (1 + xy_i).$$What is$${{{\partial f}\over{\partial x}}\over f}?$$What happens when we use the geometric series?
6
votes
2answers
106 views

Chain Rule and Vector valued functions?

Let $f: R^n \to R$ be given by $f(x) = \frac{||x||^4} {1 + ||x||^2}$ . Use the chain rule to show that $f$ is differentiable at each $x \in R^n$ and compute $Df(x)$. This vector valued stuff just ...
0
votes
0answers
24 views

Convergence of a function of two variables

The following question has been posed to me by a student in an analysis class. For which real numbers $\alpha \gt 0$ is the function $f : \Bbb R^2 \to \Bbb R$ given by $f(x, y) = (x^2 + y^2)^\alpha$ ...
0
votes
2answers
28 views

Prove that there exists some real number θ satisfying 0 < θ < 1 for which f '''(θ) = 0

Let f: D → R be a 3-times differentiable function defined over an open interval D, where 0 ∈ D and 1 ∈ D. Suppose that f(0) = f '(0) = 0 and f(1) = f '(1) = 0. Prove that there exists some real ...