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

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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
3
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5answers
595 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
63 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 ...
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1answer
16 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?
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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 $\frac{d}{dx}(\frac{\partial f}{\partial y^\prime})$ $\frac{\partial f}{\partial y^\prime}=x+2y'(x)$ ...
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1answer
31 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 ...
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1answer
17 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 $y$ ...
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1answer
105 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
53 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
182 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
52 views

find the the variable that maximizes a function

I have a function that I am trying to find for what input it maximizes. $$ f(n) = {\binom{S}{2}}^{n/S}$$ I need to find the $S$ for which this function maximizes (for infinite $n$). more generally, ...
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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
205 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 ...
1
vote
0answers
43 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
70 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
35 views

Linear Approximations

Can't figure out where I'm going wrong here. Isn't it just f(x)+f`(x) dx?
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1answer
35 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 = ...
-1
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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 ...
1
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2answers
49 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
153 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 ...
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1answer
36 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
34 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
31 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
60 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
55 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
61 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
101 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 ...
1
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1answer
71 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
41 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 ...
1
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1answer
68 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
57 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 ...
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1answer
26 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 ...
1
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1answer
25 views

prove that a function is not bounded

Prove that $$2x\sin{1\over x^2}-{2\over x}\cos{1\over x^2}$$ is not bounded when $x\to 0$ I tried to find two sequences that converges to $\infty$ and $-\infty$ but I can´t; I also derived the ...
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1answer
32 views

Another differentiation question

$f=\frac{y'(x)^2}{x^3} dx$ I don't understand how to take the derivative if y is some unknown function. Could someone solve step by step?
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2answers
38 views

Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)

Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$) i) Suppose $\sum a_i$ converges. Must $f'(0)$ exist? ii) Suppose $f(0) = f'(0) = 0$. ...
1
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2answers
31 views

Could someone please help me find the derivative of the inverse of $f$ at $0$?

The problem is: for $\displaystyle f(x)= \int_0^{\ln x} \frac{1}{\sqrt{4+\mathrm{e}^{t}}} \, \mathrm{d}t$, $x > 0$, find $(f^{-1})'(0)$. I know that I should use the fundamental theorem of ...
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1answer
67 views

Very interesting multivariable calculus question.

If $\displaystyle z = \frac{f(x-y)}{y}$, show that $\displaystyle z + y \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 0$.
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1answer
91 views

Inverse function theorem question - multivariable calculus

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) ...
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2answers
85 views

Why is this proof of 1D the Chain Rule wrong?

This proof of the one dimensional chain rule was pointed out as inaccurate (if not utterly wrong) by our calculus teacher a couple years ago: denoting the composition as $f(g(x))=(f\circ g)(x)=f(y)$ ...
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0answers
38 views

Convert function to table values

I'm trying to remember the name of the method for finding a discrete function from its derivative. I want to create a table of values by difference equations. (not by step on the x or y axis) ...
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1answer
30 views

Critical Numbers Problems

Okay so I found the critical number no problem, it being cos x=-1/2, but on my answer sheet it says that the critical numbers are ...
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1answer
49 views

Could someone please help me solve this calculus problem?

For f(x)= integral 1/sqrt(4+e^t) dt from 0 to lnx, with x>0, find (f^-1)'(0) (that is, the derivative of the inverse of f at 0)
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5answers
150 views

How to solve $\frac{d}{dx}\int_{\pi}^{x^2}\sin(t) \ \text{dt}$ using the Fundamental Theorem of Calculus?

I want to find the value of: $$\frac{d}{dx}\int_{\pi}^{x^2} \sin(t) \ \text{dt}$$ I have recently been taught Fundamental Theorem of Calculus Part $1$ (I learnt Part $2$ first I don't know why). I ...
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2answers
83 views

partial differentiation of a variable w.r.t. its time derivative

What will be the partial derivative : $$\frac{\partial\theta}{\partial\dot\theta}$$ and $$\frac{\partial\dot\theta}{\partial\theta}$$ where, $\theta = \theta(t)$ and $\dot\theta = ...
15
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8answers
3k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
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3answers
78 views

What is the derivative of $\cos^4(x)$?

I'm not sure if we use the power rule, or if the chain rule is needed for this particular problem.
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2answers
189 views

The derivative of $x^2 \cdot \cos(x)$

I want to know how to derive this function. Can someone explain the steps? I know most derivative rules but I'm clearly not seeing how this works: $$\frac{d}{dx}(\ x^2cos(x)) = x(2\cos(x) - ...
7
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1answer
238 views

Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
0
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3answers
136 views

Infinite derivative

I have just discovered the second derivative of $\frac{d^2}{dx^2}$. However now I have a curiosity for the infinite derivative. I am asking for a proof on if the infinite derivative is possible. I got ...
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4answers
104 views

The Derivative of a Derivative?

For homework, one of the critical questions asks, "Is it possible to find the derivative of a derivative, why or why not? Provide a proof in your explanation." My first thought was that you could do ...