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

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3
votes
1answer
32 views

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

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: ...
1
vote
1answer
51 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 ...
-2
votes
1answer
119 views

Double obstructing wall problem, what is the optimal walk path and length?

Every day, you walk from point A to point B which are exactly $2$ miles apart straight line distance, however, each day, there is a $50$% chance of there being an obstructing wall perpendicular to the ...
9
votes
5answers
294 views

Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
0
votes
0answers
27 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
votes
0answers
18 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
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
0answers
14 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 ...
0
votes
2answers
740 views

Inverse functions and tangent line

Let $f(x) = \frac14x^3 + 12x + 6$ and let $y = f^{-1}(x)$ be the inverse function of $f$. Determine the $x$-coordinates of the two points on the graph of the inverse function where the tangent line is ...
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
votes
2answers
31 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
votes
1answer
177 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 ...
0
votes
1answer
30 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
vote
2answers
480 views

Show the derivative of an activation function

I am learning about neural networks and am using the sigmoid activation function $$q(z)=\frac{1}{1+e^{-z}}.$$ The problem is that I need to use its derivative $q^{\prime}(z)$. Would anyone have any ...
3
votes
0answers
312 views

Derivatives on hidden layers in backpropagation (ANNs)

I'm working on understanding all the math used in artificial neural networks. I have gotten stuck at calculating the error function derivatives for hidden layers when performing backpropagation. On ...
1
vote
1answer
12 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
2answers
895 views

Hessian after coordinate changing

Let $f\colon \Bbb R^n\to\Bbb R$. Let $z=Px$ coordinate changing. $P$ is $n\times n$ constant matrix, $x$ and $z$ are the variables in $\Bbb R^n$. Does anyone know a formula which express how the ...
1
vote
1answer
43 views

Integration and differentiation of Fourier series

I am interested in the properties of Fourier series under integration and differentiation, and I've noticed a "strange" phenomenon. Suppose I have a Fourier series which I Integrate, and suppose that ...
0
votes
3answers
33 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 ...
6
votes
3answers
358 views

Deriving the Normalization formula for Associated Legendre functions: Stage $1$

The question that follows is needed as part of a derivation of the Associated Legendre Functions Normalization Formula: ...
0
votes
2answers
27 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
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
1answer
31 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
votes
1answer
23 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
2answers
28 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 ...
3
votes
3answers
71 views

Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$

I was given a task to prove that inequality is true for x>0: $(e^x-1)\ln(1+x) > x^2$. I've tried to use derivatives to show that the $f(x) = (e^x-1)\ln(1+x)-x^2$ is greater than zero, but has never ...
1
vote
1answer
80 views

Holomorphicity of $f(x + iy) = x^2 + iy^2$

By definition: $f: E \rightarrow \mathbb{C}$, where $E$ is an open subset of $\mathbb{C}$ is holomorphic on $E$ if $f$ is $\mathbb{C}$-differentiable at all points of $E$. The key point being ...
0
votes
3answers
17 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
votes
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
vote
4answers
74 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 ...
0
votes
0answers
44 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 $0$ ...
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
32 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
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
votes
2answers
40 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
votes
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
votes
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 ...
4
votes
3answers
133 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
vote
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 + ...
2
votes
2answers
70 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)\} ...
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
0answers
22 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
32 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
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) - ...
2
votes
2answers
1k views

Differentiating with respect to the limit of integration

I'm confused about problems involving differentiation with respect to the limit of an integral, I just want to check that my understanding is correct. For example, are the following statements ...
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\\ ...
4
votes
3answers
43 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 ...
11
votes
5answers
1k views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
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
2answers
66 views

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

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 $f(x)/g(x), ...