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

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1answer
17 views

gradient norm of a simple function

In this answer Derivation of soft thresholding operator how can I derive that $\nabla(||x-b||_2^2)=b-x$?
3
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2answers
133 views

Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
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1answer
32 views

Derivatives of conditionally defined functions

I was asked in an exercise to show on what intervals of $\mathbb{R}$ a function $f(x)$ is solution to certain differential equations. The function is defined as: $$f(x) = \left\{ \begin{array}{rl} ...
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0answers
25 views

Partial Derivatives of Vector Function

Let $z(t)=[z_1(t),z_2(t)]^T$ Also, $V(z)=az_1^2+2bz_1z_2+cz_2^2$ I am trying to find $V'=\frac{d}{dz} V$. I know this means I have to take partials with respect to $z_1$ and $z_2$, but the middle ...
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2answers
42 views

Find the function $h(x) = g(2g^{-1}(x))$

Show that the function $g(x) = x^4 + x^3 + 1$ is one-to-one on [0, 2]. In addition, for the function $h(x) = g(2g^{-1}(x))$, find h′(3). For the first part, I manage to prove that g(x) is increasing ...
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1answer
28 views

Convex function almost surely differentiable.

If f: $\mathbb{R}^n \rightarrow \mathbb{R}$ is a convex function, i heard that f is almost everywhere differentiable. Is it true? I can't find a proof (n-dimentional). Thank you for any help
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3answers
64 views

Differentiation of functions w.r.t. a composed argument

I need help with the following derivative involving inner products: $$\frac{d\, \log(x)^T\,y}{d\,x^T\,y}$$ Here $x$ and $y$ are $n$-dimensional vectors, $T$ indicates transpose, and the logarithm of ...
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1answer
22 views

A Question about Hessian of log function (general form)

Sincerely hope to ask how to obtain the RHS? Should I consider ln(10) among the process of d(log(x))/dx? Thanks!
2
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1answer
86 views

Where is piecewise dirichlet function with $|x|^2$ continuous or differentiable?

If $|x|^2$ is continuous and differentiable on all of $\mathbb{R}^n$ (already shown differentiability by showing all $n$ of its partial derivatives are continuous), then... Question: For the function ...
2
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0answers
31 views

Implicit differentiation and rules

I'm supposed to write the rules used for some differentiable functions. I got all of them correct except for the last one which is $d(x^c)$. I put in $cx^{c-1}$ because I thought it was the power ...
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2answers
69 views

How do I find $(f^{-1})'(a)$? [closed]

if $$f(x) = 3x^3 + 3x^2 + 6x + 9 $$ $$a = 9$$ and also $$f(x) = 2x^3 + 3\sin x + 3\cos x$$ $$a = 3$$ I know I have to find the inverse but I think I’m getting overly complicated answers and my ...
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3answers
48 views

Solve $\lim_{x\to0}{\frac{x^2\cdot\sin\frac{1}{x}}{\sin x}}$

Find the limit: $$\lim_{x\to0}{\frac{x^2\cdot\sin\frac{1}{x}}{\sin x}}$$ After treating it with l'Hopital rule, we get: $$\lim_{x\to0}{\frac{2x\cdot\sin \frac{1}{x}-\cos\frac{1}{x}}{\cos x}}$$ Now, ...
1
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1answer
375 views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
0
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1answer
68 views

How do I properly set up this optimization equation?

So I've been the given the task to fully optimize any packaging. I chose a DS game box. So first I took the measurements of the cartridge itself ($3.5 \text{ cm} \times 3.3 \text{ cm} \times 0.38 ...
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2answers
31 views

Function composition and differentiability

This problem asks for an example of functions $f$ and $g$ such that $g$ takes on all values, $f \circ g$ and $g$ are differentiable, but $f$ is not differentiable. I'm having trouble jumping straight ...
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1answer
22 views

Calculate the Derivative of a univariable integral at a point $4$

Considering the function below: the objective is to calculate $F'(4)$ (the derivative of $F(x)$ in the point $4$)? we know that: and that: so if I try to replace $x$ by $4$ in $F'(x)$ I get ...
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1answer
33 views

$-2(\sin x+2\cos 2x)=0$

I am finding the 2nd derivative critical values for graphing a trig function. So far I have it simplified to $$-2(\sin x+2\cos 2x)=0$$ What values for x make this equal zero? And is there a ...
1
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1answer
64 views

A function that is differentiable at a single point

Consider the function $$f(x)=\begin{cases} x^2 & x \in \mathbf{Q}, \\ 0 & x \notin \mathbf{Q} \end{cases} $$ $f$ is continuous only at $0$ and now I need to show that at this point it is ...
4
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4answers
80 views

L'Hospital's rule problem

$$\lim_{x\to 0^+}(x^{x}-1)\ln(x)$$ I need to solve this by L´Hopital´s rule: this is an indetermination of the type $0 \cdot \infty$: $$\lim_{x\to 0^+}(x^{x}-1)\ln(x)=\lim_{x\to 0^+}{(x^{x}-1)\over ...
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0answers
20 views

Example of class 1 function, $f$ bijective but does no exist $(f^{-1})´$

give an example of a class 1 function, $f$ bijective but does no exist $(f^{-1})´$ for some $y\in f[D]$ I can´t find such a function I would really appreciate your help
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5answers
645 views

Intuition behind chain rule [duplicate]

What is the intuition behind chain rule in mathematics in particular why there is a multiplication in between?
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1answer
65 views

Difficult example of functions

give an example of two functions: $g$ discontinuous at $t_0$, and f continuous but not derivable at $g(t_0)$ so that $f\circ g$ is derivable at $t_0$ Do this two functions exist? I would appreciate ...
0
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1answer
17 views

Product or chain rule

$f(x)=\frac{(y')^2}{x^3}$ Find $\frac{d}{dx} \frac{\partial f}{\partial y'}$ I don't understand how to take this derivative properly. Can someone describe step by step?
0
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1answer
26 views

Is this differentiation correct?

$J(x,y')=\int_1^2 xy'(x)+(y'(x))^2dx = \int_1^2{f(y,y^\prime,x)}$ Need to find $\frac{d}{dx}(\frac{\partial f}{\partial y^\prime})$ $\frac{\partial f}{\partial y^\prime}=x+2y'(x)$ ...
1
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1answer
35 views

Help with multivar. chain rule

I am having trouble with the following problem. I feel that I do understand the multivariable chain rule in general, but applying it here is more difficult. I am lost on where to start. Any help would ...
0
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1answer
18 views

The equation$(x^2+y^2)^3-3(x^2+y^2)-2=0$defines the var$y$as a function of$x$,$y=f(x)$,in the vecinity of the point$(x,y)=(1,1)$Find$f'(1)$and$f''(1)$

I have this solved problem and I don't quite understand something, either it's a mistake or I'm missing something. So the problem is: The equation $(x^2+y^2)^3-3(x^2+y^2)-2=0$ defines the variable $y$ ...
1
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1answer
133 views

Calculus Area Problem: Shortest length of a fence…

Hello all this is my first question on this website! A rancher wants to fence in an area of 1,000,000 square feet in a rectangular field and then divide it in half with a fence down the middle ...
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0answers
56 views

variational derivative

Let $\Omega \subset \mathbb{R}^n,\ n=1,2 \mbox{ or } 3$. Define the following energy $$E=\int_{\Omega} \frac{1}{\varepsilon}\left[f(u)+\frac{\varepsilon^2}{2}|\gamma(n)\nabla u|^2\right]\,dx$$ ...
1
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1answer
320 views

Leibniz Notation Second Derivative Chain Rule?

I believe I understand the chain rule better from a few tutorials as the following: $$\frac{d}{dx}(f(g(x)) ) = \frac{\partial f}{\partial g}\frac{\partial g}{\partial x}$$ But how would you ...
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0answers
34 views

Finding the number of derivatives for series problems

I have the following problem: How smooth are the following functions? That is, how many derivatives can you guarantee them to have? $$a)\;\;\;\;\; ...
10
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1answer
218 views

Define second derivative ($f''$) without using first derivative ($f'$)

The question I'd like to ask is this: If $f''(0)$ exists, does $f'$ exist in a neighborhood of $0$? Of course, under the standard definition of $f''(0)$, we have already assumed that $f'$ exists ...
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0answers
49 views

Simpson's rule error rate for N-dimension

I'm doing a project that involves numerical method, but I'm not too familiar on calculus. I'm using Simpson's rule to integrate n-dimension gaussian, I was able to get the integration result for ...
2
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2answers
75 views

Prove that $\lim_{n\to \infty} {f(x_n)-f(x_0)\over x_n - x_0}= f´(x_0)$

Problem: Prove that if $f$ is continuous at $x_0$ and$$\lim_{n\to \infty} {f(x_n)-f(x_0)\over x_n - x_0}$$ exist for any sequence ${x_n} \to x_0$ and $x_n\neq x_0$ $\forall n\in \mathbb N$, then ...
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2answers
38 views

Linear Approximations

Can't figure out where I'm going wrong here. Isn't it just f(x)+f`(x) dx?
1
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1answer
39 views

What does the adjoint operator do? Is this Frechet derivative correct?

Problem statement Let $x \in l^2$ and $J(x) = \sum_{n = 1}^{+\infty} x_{2n - 1}^2$ Find first and second Frechet derivatives. Attempted solution Let's note that $J(x) = \sum_{n = ...
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2answers
23 views

Writing out chain rule for the following function

$\frac{dh}{dx}$, where $h(x) = f(x, u(x), v(x))$. First of all, this function doesn't even make sense to me. It's a function of one variable, with domain $\mathbb{R}$ and range $\mathbb{R}$. How can ...
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2answers
51 views

Prove that if $f$ is derivable on $[a,b]$ and $f$ is lipschitz continuos then $f$ has bounded derivative

Prove that if $f$ is derivable on $[a,b]$ and $f$ is lipschitz continuous (LC) then $f$ has bounded derivative My proof: $f$ is LC $\Rightarrow$ f has bounded derivative: there exist $M\gt 0$ ...
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2answers
184 views

Simplifying Second Derivatives

I can't seem to figure out how my professor simplified this second derivative. Any help is much appreciated. I'm having trouble simplifying the second derivatives of most problems so step by step ...
0
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1answer
42 views

Frechet derivative of double integral.

Problem statement Let $u(t) \in L^{2}(0, 1)$ and $J(u) = \int_0^1 tu(t) \int_0^t u(s)dsdt$ Compute first and second Frechet derivatives. Attempted solution $$ \begin{split} J(u + h) - J(u) &= ...
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2answers
37 views

Is there an easier way to prove a multivariate function is differentiable?

$f\colon U \rightarrow \mathbb{R}, (x,y) \mapsto \sqrt{1 - x^2 - y^2}$ where $U = \{(x,y) \mid x^2 + y^2 < 1\}$. So the definition of differentiability I have is: $$\lim \limits_{(x,y) ...
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0answers
32 views

Second derivative of vector composition

I found the following result in Boyd. For the function $f(x) = h(g(x)) = h(g_1(x),\ldots, g_k(x))$, where $h:\mathbb{R}^k\to\mathbb{R}$, $g_i:\mathbb{R}^n\to\mathbb{R}$, $x\in\mathbb{R}^n$. ...
4
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0answers
64 views

Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
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2answers
183 views

What is the difference between “differentiable” and “continuous”

I have always treated them as the same thing. But recently, some people have told me that the two terms are different. So now I am wondering, What is the difference between "differentiable" and ...
0
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1answer
68 views

What trick to calculate this Frechet derivative?

Let $u(t) \in L^{2}(0, 1)$. I need to calculate the first and second Frechet derivatives of $$J(u) = \int_0^1 \left(\int_0^{t^3}u(s)ds\right)^2dt$$ I am completely at a loss here: I know several ...
1
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1answer
132 views

Show a function is not continuous at a point

$$ f(x,y) = \begin{cases} \dfrac{x^2 y^4}{x^4 + 6y^8}, & \text{if }(x,y)\neq(0,0) \\ 0, & \text{if }(x,y)=(0,0) \end{cases} $$ For the definition of differentiability, I have: $$\lim_{h ...
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1answer
78 views

Inverse Function Theorem for Manifolds with Boundary

In Lee SM it is written that the inverse function theorem can fail for manifolds with boundary.As hint it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^n\to\mathbb{R}^n$ ...
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0answers
46 views

Partial Derivative

$$f(x,y,z)=x^2+\ln(1+y)+e^{yz}$$ Why $$f^\prime_y=\frac{1}{1+y}$$ and not $$f^\prime_y=\frac{1}{1+y}+e^{yz}z$$ having in mind that the third addend in $f$ also contains $y$ (which is a real ...
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1answer
74 views

Frechet derivative of shift operator in $l_2$?

Let $x \in l_2$ and $J(x) = \sum_{k = 1}^{+\infty} x_k x_{k + 1}$. Find $DJ(u)$ and $D(DJ(u))$. Attempted solution Since $x \in l_2$, then $\sum_{k = 1}^{+\infty}x_k < \infty$. Another fact: ...
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0answers
61 views

Totally differentiable function - definition

I know for a function of several variables, if all partial derivatives exist and they are continuous at and around a point $a$ then the function is totally differentiable at that point. I ...
1
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1answer
28 views

Represent derivation as a standard matrix (Linear mapping)?

Given a matrix $a$ of coefficients $\left( \begin{array}{cc} a_0 \\ a_1 \\ .. \\a_n\end{array} \right)$representing $a_0 + a_1 x + a_2 x^2 + ... a_n x^n$, how can I find a standard matrix D such that ...