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

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76 views

derivative of the integral

I am working on a few problems, just need some help to see if I'm working them correctly, $$(1)\;\;\;\;\;g(x)=\int_0^x(x-u)e^{u^2}du$$ find $g'(x),g''(x)$ $$(2)\;\;\;\;\;\psi(x,y)=\int_1^xe^{ty}dt$$ ...
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2answers
48 views

Differentiation of a Vector with respect to a vector

I am studying a paper and I am going crazy about one differentiation which it is written on it but not explained. I think I am missing something and probably something easy. I would love if someone ...
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0answers
15 views

Minimising the surface area of a rectangular prism [Solution Verification]

A packaging company is going to make open topped boxes, with square bases that hold $100$ centimetres$^3$. What are the dimensions of the box that can be built with the least material?
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30 views

Derivatives of semi-elasticities

I want to find the sign of $\partial_z (\partial_z \log F(T1) - \partial_z \log F(T2) )$ where $T1<T2$ and $F(T) = \int_0^T e^{g(t) + z\, h(t)} dt$ All we know is that $h(t)>0\forall t$ ...
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0answers
60 views

Why can't we differentiate constants like variables?

I understand why we the derivative of $x$ is $1$. Similarly, the derivative of a constant function $y=a$ is $0$, because it's slope is flat, since for every $x$, $y=a$. But can you please tell me why ...
2
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0answers
62 views

Div, grad, curl in curvilinear coordinates

I've a lot of different formulas for div, grad, curl, and laplacian in different coordinate systems. How are these formulas derived? What's the general procedure for finding the formula of say the ...
2
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1answer
31 views

Numerical approximation of differentiation

I have the following task to solve: Let $b>x$ be defined, determine $w_0,w_1$ and $w_2$ in dependency of $b$ such that the approximation $f''(x) \approx w_0 f(x-h) + w_1 f(x) ...
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0answers
24 views

derivative of a tensor function

If e is a second order tensor and it's symmetrical, assuming abs() is a tensor function such that returns all the components of e be positive, I am interested the derivatives of the function abs(e) ...
2
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1answer
40 views

Why this particular definition for the derivative on a group ring?

I am reading Introduction to Knot Theory by Crowell and Fox, and I am a bit confused at the way they define a derivative on a group ring (and on a group). I understand a derivative (or derivation ...
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1answer
17 views

Maximize yield given two products and price

Just started learning to use derivatives to apply to real world, and this homework problem is really tripping me up, and getting a formula to start is even giving me a bunch of trouble. If fertilizer ...
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1answer
29 views

Difference Between Differentiation on Time vs Position

I'm hoping somebody could help me understand the difference between the following: $$∂_tc(x,t)$$ $$∂_xc(x,t)$$ My understanding is the the top derivative would be something like velocity but what ...
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2answers
28 views

A question involving Frechet differentiability

Let $X, Y$ be real normed spaces and $U \subset X$ open subset. In "Nonlinear functional analysis and applications" edited by Louis B. Rall, we have the followint definition (page 115) A map $F : U ...
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2answers
62 views

Differentiating the function $\arcsin(3x-4x^3)$

When I have to differentiate the function $\arcsin(3x-4x^3)$ which of the following methods is more appropriate ? Putting $x=\sin θ$,simplifying and then differentiating for certain ranges of $x$. ...
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1answer
37 views

Derivative of an expodential function

Let the function: \begin{equation} \phi(x)=\begin{cases} e^{-\frac{1}{1-x^2}}, & x\in (-1,1)\\ 0, &x \not \in (-1,1)\end{cases} \end{equation} How can I show that $\phi(x)$ is differentiable ...
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0answers
15 views

Gateaux derivative vs Frechet derivative - what is the essential difference?

To a non-mathematician with a background in calculus, what is the primary difference between the two derivatives? What is happening in the infinitesimal?
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1answer
85 views

The importance of being real

Let $\Sigma$ be a collection of holomorphic, one-to-one function from some simply connected region $\Omega$, which map $\Omega$ into the open unit disc $U$. Fix $z_0 \in \Omega$ and put $$\eta = ...
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1answer
53 views

Derivative of a trace function

Let $K$ be a Hermitian matrix, and $X$ be a positive one. What is the derivative of the trace function $$ \mbox{ Tr } X|e^{itK} - X|^3$$ with respect to $t$ at $t = 0$ ? There is a nice formula for ...
4
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2answers
63 views

Binomial Theorem of Differentiation? [duplicate]

I noticed that $$\frac{d^{n}}{dx^{n}} f(x)g(x)=\sum_{i=0}^n {n \choose i} f^{(i)}(x)g^{(i)}(x)$$ and it's had me scratching for a little bit. It's easy to see how the cross terms add up but can anyone ...
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1answer
38 views

How to turn a system of first order into a second order

So I have two equations $X' = aX + bY$ $Y' = cX + dY$ I want to convert it back to a second order equation with the form $X'' + \alpha X' + \beta X$ with $\alpha,\beta$ in terms of a,b,c,d. I ...
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1answer
29 views

Let $I \subseteq \Bbb R$ be a non-degenerate open interval

Let $I \subseteq \Bbb R$ be a non-degenerate open interval, and let $n\in \Bbb N$. Let $f:I\rightarrow \Bbb R$ be defined by $f(x)=x^n$ for all $x\in I$. Prove that $f$ is differentiable and ...
2
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1answer
21 views

Solving equation with functions inside the function?

I've been given the problem: For h(x) defined below, find h′(2), given that: f(2)=−3, g(2)=3 , f′(2)=−1 and g′(2)=7. h(x) = f(x)g(x) I was thinking h'(x) = (-1)(7) = -7 Is this right? If ...
2
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1answer
38 views

How to prove the limit of “the exponential of a sequence”

So given a convergent sequence $\{a_n\}_{n=1}^\infty$ with limit $a$, I'd like to prove that $$\lim_{n\to\infty} \left(1+\frac{a_n}{n}\right)^n=e^a.\quad(1)$$ Knowing that $e$ is defined by ...
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1answer
28 views

Derivative of hyperbolic tangent equation?

Suppose I have the following hyperbolic tangent function: $$\ f(x)=\frac{(1-e^{-2x})}{(1+e^{-2x})}$$ What will be the first derivative of this function?
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1answer
34 views

Is the derivative of a exponential function a^x always greater than the derivative of a polynomial x^n as x approaches infinity

with n and a being any constants > than 1. I have tried taking the $\lim\limits_{x \to \infty} a^x / x^n$, and l'hopitals is telling me than $x^n$ can always be reduced to 1 with multiple iterations, ...
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3answers
81 views

Analyzing if function is “onto”

I have some function $g$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $g: [a,b] \to [a,b]$ is onto. How can I find out if this is true or not? P.S. I am not saying all $g$ ...
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3answers
47 views

How do I find the critical points of this function involving e?

I have the function: $$g(x)={{1 \over \sqrt{2 \pi}} \cdot e^{{-(x-2)^2}/2}}$$ Through very tedious differntion, I got to: $$g'(x) = {{{-(x+2)} \cdot {e^{{-(x-2)^2}/2}}} \over {2 \pi}}$$ Setting ...
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2answers
34 views

Analyzing derivative of function.

I have some function $g: [a,b] \to [a,b]$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $\forall x \in [a,b]: |g'(x)| \lt 1$. How can I find out if this is true or not? P.S. I ...
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4answers
73 views

$n^\text{th}$ derivative of $\tan^{-1} x$

Find $$\frac{d^n(\tan^{-1}x)}{dx^n}$$ Or find the $n^\text{th}$ derivative of $\tan^{-1}x$ w.r.t. $x$. Differentiation 4-5 times did not patternize so as to find out the $n^\text{th}$ derivative. ...
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1answer
69 views

Why are the differentiation/integration rules what they are?

So I understand what rules you use where, and the general forms of the rules like: $$\left(\frac{d}{dx}\right)^nx^k=\frac{k!}{(k-n)!}x^{k-n}$$ My question is why are these the formulas that give us ...
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2answers
212 views

Differentiation Tricks

Since most derivatives are trivial to take, it's understandable why integrals get most of the mathematical tricksters' attention. However, not all derivatives are trivial to take and I think it's good ...
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2answers
32 views

At what argument $x$ is the tangent to the graph $y=\frac{1}{2}x^2-\ln x$ horizontal?

At what argument $x$ is the tangent to the graph $y=\frac{1}{2}x^2-\ln x$ horizontal? Well this is a question which I found in a website. I found the Derivative to be $(x^2+1)/x$. As far as I ...
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1answer
9 views

How to derive $r, \theta, \phi$ for the sperical coordinate gradient?

I'm trying to figure out how to get the gradient in spherical coordinates. I'm as far as the author writes in this answer: http://physics.stackexchange.com/a/78514 and I understand how and why to get ...
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2answers
67 views

Derivative $(1-x)^{-2}$ [duplicate]

I'm getting answer of $\frac{2}{(1-x)^3}$ , but online calculators suggest it's $\frac{-2}{(1-x)^3}$. I've tried it as $(1-x)^{-2}$, which results $-2*(1-x)*-1 = \frac{2}{(1-x)^3}$. Same result with ...
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1answer
35 views

The first assumption leads to the third one that looks inconsistent at a glance. Can you explain it better?

Background I am trying to solve the following problem: > Given 2 distinct curves $C_1: y=f(x)=e^{6x}$ and $C_2: y=g(x)=ax^2$ where $a>0$. The objective is to find the range of $a$ such that ...
0
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1answer
63 views

Is an everywhere differentiable function locally Lipschitz?

If we have a differentiable function $ f:\mathbb{R}^n \to \mathbb{R}^n $, does it have to be locally Lipschitz? It's obviously true for continuously differentiable functions, but what happens without ...
8
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2answers
131 views

Derivative of $\int_0^1 e^{\sqrt{x^2+t^2}}\,\mathrm{d}x$ at $t = 0$

Let the real-valued function $\phi:\mathbb{R}\to\mathbb{R}$ be defined by $$\phi(t)=\int_0^1e^{\sqrt{x^2+t^2}}\,\mathrm{d}x,$$ it can then be shown that $\phi$ is continuous and differentiable. I ...
1
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1answer
74 views

Dimension of garden to minimize cost

Math question: A homeowner wants to build, along her driveway, a garden surrounded by a fence. If the garden is to be $5000$ square ft, and the fence along the driveway cost $6$ dollars per foot while ...
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0answers
56 views

How to differentiate $y$ with logarithmic differentiation

I am asked to find the differentiate $y$ using logarithmic differentiation $$y=\frac{ x(x^5+1)^{1/2}}{(x-1)^{1/3}}?$$ I tried it 3 times and I got three different answer each time Any help
2
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1answer
39 views

Solve an initial value problem using the directional derivative

In my notes there is the following example of solving an initial value problem using the directional derivative. The problem is the following: $$u_t(x,t)=u_x(x,t), x \in \mathbb{R}, t>0 \\ ...
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1answer
35 views

Using log to take derivative of a function

Is it safe to say that if $\frac{d}{dx}ln(f)= g $ for some functions f and g, then $\frac{d}{dx}f = e^{g}$? Why or why not? (novice high schooler here)
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21 views

limit of recurrence serie by derive

how to define limit this series $$U_{n+1} = {df(U_n)\over dU_n}$$ $f(x)$ function derivable in Class $C^n$, $f^{(n)}(x)$ function not null and $U_0 \gt 0$
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3answers
49 views

Inequalities and Differentiation

Having become so accustomed to differentiation and integration being applied just like normal algebraic operators, I was somewhat suprised yesterday when I realized that $f(x) \geq g(x)$ does not ...
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1answer
22 views

Finding a solution for $(2\sin y-2x)y'-y=0$ that goes through the point ($\frac{2}{\pi},\pi$).

How do I find a solution for the differential equation: $(2\sin y-2x)y'-y=0$ that goes through the point ($\frac{2}{\pi},\pi$) by using the property of inverse function: $\frac{dx}{dy} = ...
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1answer
54 views

Understanding and teaching the concept of derivative

I need to prepare an introductory lecture about derivatives and the concept of differentiation to a class of people with a general mathematical background (who have also studied calculus a few years ...
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0answers
34 views

How to prove a function is negative over specified interval

I have a function $f$ as follows: $f(x)=\frac{((b-1)^2-x^4)}{x^2\sqrt{(x^2-b-1)^2-4b)}}+1$ where $b\gt0$ is a positive constant. I know that $f(x)\lt0 \text{ for } x\ge(\sqrt{b}+1)$ , but I don't ...
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0answers
19 views

If $f$ is derivable in $(a,b)$, proof that if $f=0$ in $k$ points, then $f'=0$ in at least $k-1$ points.

Let $f:[a,b]\longrightarrow \mathbb{R}$. Prove that if $f$ is derivable in $(a,b)$, proof that if $f=0$ in $k$ points, then $f'=0$ in at least $k-1$ points. I have to use Rolle's theorem for this, ...
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0answers
33 views

Derivative of a linear basis function over a moving mesh

Given a moving mesh $0=x_0(t)<x_1(t)<\cdots<x_N(t)<x_{N+1}(t)=1,$ where $t$ denotes the current time so that the mesh is moving with time. The linear basis function is then defined as ...
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2answers
27 views

In differentiability

Let $f(x,y)$ and $g(x,y)$ are differentiable functions in $x$ and $y$. Suppose $f(x,y) = F(g(x,y))$.My question, Is $F$ differentiable function?!.
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1answer
39 views

Find the directional derivative using $f(x,y,z)=xy+z^2$. [closed]

Find the directional derivative using $f(x,y,z)=xy+z^2$, at the point $(2,3,4)$ in the direction of a vector making an angle of $\frac{3\pi}{4}$ with grad $f(2,3,4)$. PS - I am having trouble ...
10
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2answers
121 views

Can you prove My conjecture about Invertiblity of the Derivative Matrix ?! (to use Inverse function Theorem)

In the Analysis2 midterm exam, we had the following problem: Let the equation $a_nx^n+\cdots+a_1x+a_0=0$ has $n$ simple real roots (distinct) $\{\alpha_1,\cdots,\alpha_n\}$. Prove that the above ...