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

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176 views

What is $dx/dF(x)$ where $F(.)$ is a continuous, increasing function.

I was wondering if it is possible to find $dx/dF(x)$, that is, the derivative of $x$ with respect to $F(x)$, which is an increasing, continuous function. Does it involve finding the derivative of the ...
2
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5answers
68 views

for each $x>1 , \frac{x-1}{x}\ < \ln x < x-1$

I tried to prove this with differentiation: when $x >$ 1, all 3 functions are positive and when $x = 1$, all 3 reaches zero. And the derivatives are varying like ...
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1answer
36 views

n-th derivative with respect to $\frac{1}{x}$

Is it any easy way to calculate : $\frac{d^n x}{d\left(\frac{1}{x}\right)^n}$ for arbitrary $n\in\mathbb{N}$ ? (for $n=1$ it is obvious, but for $n>1$ the formula for $n$-th derivative of ...
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3answers
66 views

$n \text{th}$ derivative of $f(x)$?

Let $ \ f(x) \ = \ x^4 e^x \ $ . Determine the nth derivative of $ \ f \ $ at $ \ x \ = \ 0 \ $. I know by working it out that the first, second, and third derivative will be 1. The fourth, fifth, ...
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1answer
53 views

Derivation of the dirac delta

I have the following function: $$\int^{a+\epsilon}_{a-\epsilon}f(x)\dfrac{1}{\epsilon} dx$$ When I take the limit $\epsilon \to 0$, I want to show that this becomes equivalent to the behavior of the ...
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2answers
155 views

The third derivative of the first principles definition of of a derivative

So the $\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}$ this is what I learned to find the first derivative and by taking this concept and trying to find the second derivative using this method I came up ...
3
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4answers
143 views

l'Hospital's rule with trigonometric functions

$$\lim_{x\to0^+}\frac{1-\cos(x)}{x^2\sin(x)}$$ I keep running in circles using the L'Hospital rule. After the third time applying it I got 0 but this isnt true from the graph. I can see it goes to ...
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0answers
50 views

Chebyshev spectral derivation with 16 nodes for $\,f(x)=e^{\,\text{sin}^{2}\,(x)+\cos(x)}\,$ defined in $\,[0,2\,\pi].\,$

I'm making the following exercise in Matlab, and I'm having trouble expresing my result in $x\in[0,2\pi]$ not in $x\in[-1,1]$. I first done this (as shown below) in Gauss-Lobatto points, but I don't ...
0
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1answer
29 views

Gradient of a function defined on a curve

Given: $f=f(\vec{r})$ and $\vec{r}=\vec{r}(s)$, therefore $f = f(\vec{r}(s)) = f(s)$, $f$ is defined only on the curve $\vec{r}(s)$. How then does one express the gradient of $f$ $\nabla ...
1
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1answer
45 views

How do we get the last relation?

I am looking at the conservation of momentum. The force at $W$ from the tensions at the boundary $\partial{W}$ is $$\overrightarrow{S}_{\partial{W}}=-\int_{\partial{W}}p \cdot ...
0
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1answer
33 views

Interchange of partial derivatives in financial mathematics

We have an equation (Black-Scholes): $\dfrac{\partial V}{\partial t} + \dfrac{\sigma^2 S^2}{2} \dfrac{\partial^2 V}{\partial S^2} + r S \dfrac{\partial V}{\partial S} - rV = 0$ Let's say we want to ...
0
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1answer
28 views

How to check if a function is partially differentiable

Sorry for this basic request. $$$$Could you please tell me how to check if a function is partially differentiable (I don't know if this is the right term), both over an interval, as well as at a ...
1
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1answer
72 views

Derivative of $(\ln x)^e$ [duplicate]

In Randall Munroe's What If, he says that "if you want to be mean to first-year calculus students, you can ask them to take the derivative of $(lnx)^e$" He says, as I would expect, that the result ...
2
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1answer
47 views

Show that $\|Du_{\lambda}\|_{L^2(\mathbb{R}^n)} = \|Du\|_{L^2(\mathbb{R}^n)}$

Let $x \in \mathbb{R}^n$. Given $$u(x) := \left(\frac 1{1+|x|^2} \right)^{\frac{n-2}2}, \quad u_\lambda(x):=\left(\frac \lambda{\lambda^2+|x|^2} \right)^{\frac{n-2}2},$$ I need to show that ...
0
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0answers
50 views

Convex functions-Comparison of derivative and second derivative

Let $\phi:(0,\infty) \to \mathbb{R}$ be a function with second derivative, strictly incresing and concave. Suppose that $f(t)=\phi(e^t)$ is convex. Then one can prove that $$ \lim_{x \to \infty} ...
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0answers
26 views

provable?$\frac{\partial f}{\partial\vec{m}}=\frac{\partial f}{\partial\vec{m_1}}+\frac{\partial f}{\partial\vec{m_2}},\vec{m} = \vec{m_1}+\vec{m_2}$

I have 3 vectors and 1 scalar function f. I need to have such equality to be working, but I am not sure it does. $$\frac{\partial f}{\partial \vec{m}} =?= \frac{\partial f}{\partial \vec{m_1}} + ...
2
votes
0answers
117 views

Find the derivative of f at point P in the direction of vector u.

Find the derivative of $f$ at point $(18,9)$ in direction of $\left<7,2\right>$. $$f(x,y) = \arctan \left(\frac{2y}{x} \right) + 3\arcsin\left( \frac{xy}{324} \right)$$ For this I got ...
1
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1answer
84 views

Showing that the gradient is orthogonal to level surface

It is well known that the gradient of a function (which is sufficiently well behaving) $g(x)$ is orthogonal to its level surface, for example $g(x)=0$. I have seen the following derivation of this ...
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1answer
21 views

Help with calculation with derivatives.

Ok, so this is probably a brain fart on my part, but anyway I have that $x=e^s$, and the second step of the following is unclear to me: \begin{equation} ...
2
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1answer
70 views

Circular definition of tangent line and derivative

I'm trying to understand the deep relations between the tangent line to the graph of a function $f$ at a given point $P$, and the derivative of $f$ at the same point. Indeed, in many books the ...
0
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1answer
77 views

Derivatives of operations on eigenvectors with repsect to matrix

My question is: Given a matrix $A$ and its eigenvector $v$ which corresponds to $A$'s maximum eigenvalue, is there a closed form formula to calculate the derivative $$\frac{\partial(u^Tv)}{\partial ...
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0answers
31 views

prove: $\triangledown(\triangledown\cdot u)-\triangledown \times (\triangledown \times u) =\vartriangle u$ [duplicate]

The claim is $\triangledown(\triangledown\cdot u)-\triangledown \times (\triangledown \times u) =\vartriangle u$ where $u:\mathbb{R}^3\to \mathbb{R}^3$ is a vector field, $ \vartriangle$ is the ...
0
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1answer
56 views

Directional derivative $f(x,y)=\frac{x^3}{1+x^2+y^2}$

I'm stuck on calculating the directional derivative of $f(x,y)=\frac{x^3}{1+x^2+y^2}$ in $(3,-1)$ along $(a,b)\in\mathbb{R}^2$. My try: $\lim\limits_{t\to ...
2
votes
3answers
94 views

The derivative of $x!$ and its continuity

is the factorial of fractions and negative numbers defined? If yes, then what is its graph? Also please find its domain. Our teacher said the factorial of a fraction is the fraction itself. He also ...
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1answer
43 views

Derivative of $|x|$

If $f(x)=|x|$ for $x = (x_1,x_2,x_3,\ldots,x_n) \in \mathbb{R}^n$, what is the derivative of $f(x)$ with respect to $x_i$ if $i\in\{1,2,3,\ldots,n\}$? I am confused, please show me a hint to get ...
1
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1answer
72 views

Proving a corollary of a corollary of the Mean Value Theorem (corollary-ception)

This is will a wordy question but here it goes: My analysis book states the mean-value theorem and then a corollary which we will label as (1): Let $f$ be a differentiable function on $(a,b)$ such ...
4
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1answer
156 views

A function with midpoint-linear derivative is a quadratic polynomial

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $$f'\left(\frac{a+b}{2}\right) = \frac{f'(a)+f'(b)}2,\quad \forall a,b\in\mathbb{R}$$ Prove that $f$ is a polynomial of ...
4
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1answer
76 views

What are higher derivatives?

From Wikipedia: Higher derivatives can also be defined for functions of several variables, studied in multivariable calculus. In this case, instead of repeatedly applying the derivative, one ...
2
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1answer
30 views

Existence of a smooth function with given derivative roots

Is there a smooth function $f$ that for all $n\in\mathbb{Z}_+$, $f^{(n)}(n)=0$ i.e. $n$th derivative at the point $n$ is zero and $f^{(n)}(x)\ne 0$ for all $x\in\mathbb R\setminus \{n\}$? If there is ...
3
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1answer
111 views

Extreme values of a two-variable polynomial

Is it possible to find a two-variable polynomial which has only two extreme values on the whole plane, one is a local maximum, another is a local minimum, and the local maximum is less than the local ...
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1answer
51 views

Derivative of the composition of two functions

Is the calculation below valid? \begin{align} f(x)=ax+b+g(f(x))\\ \frac{df(x)}{dx}=a+\frac{dg(f(x))}{df(x)}\frac{df(x)}{dx}\\ \frac{df(x)}{dx}-\frac{dg(f(x))}{df(x)}\frac{df(x)}{dx}=a\\ ...
2
votes
3answers
150 views

One point following another moving in a straight line?

There is a plane with two points on it, let's say A and B. A starts at an arbitrary constant point, let's say $(0, 0)$, and $B$ at a point that needs to be tested, which we'll call $(c, d)$. A moves ...
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2answers
34 views

How to evaluate limits

Let $f$ be a continuously differentiable function on $\mathbb R$. Suppose that $L=\lim\limits_{x\to \infty}(f(x)+f^{'}(x))$ exists. If $0<L<\infty$, and if $\lim\limits_{x\to \infty} f^{'}(x)$ ...
2
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3answers
69 views

Differentiating both sides of an inequality with monotonic functions

If $f(x)\le g(x)$ for all real $x$ for monotonic functions $f$ and $g$ (say, both increasing), does it follow that $f'(x)\le g'(x)$? (Note: I've seen several questions asking the same thing without ...
1
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1answer
51 views

Differential calculation in multiple variables function (cannot understand 2nd order differential form)

This question is somehow related to this question. Consider a multiple variables function $G(u, v) = \left(\matrix{x(u,v) &=& G_x(u,v)\\y(u,v) &=& G_y(u,v)\\z(u,v) &=& ...
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1answer
58 views

Antiderivative of $\frac {dy}{dx}$

This is probably a very simple question, but I think its interesting. What I would think, based on my intuition (which I think is correct in this case) is that $$\int \frac {dy}{dx}=y$$ However, ...
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1answer
23 views

Is the following property suffictient for second order differentialbility?

Let $U\subset R^n$ be an open set, and $f:U\to\mathbb R$ a $C^1$ function. Suppose that for any $x_0\in U$, there exists a $n$-variable-polynomial $T_{x_0}$ of degree at most $2$ satisfying that ...
0
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0answers
85 views

upper bound of an $L^\infty$ function's derivative

Consider a function $u:\mathbb{R} \longrightarrow \mathbb{R}^n$ that is essentially bounded, i.e., $u \in L^\infty$. There is an upper bound of its derivative? I think there is not allways ( i.g. ...
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2answers
83 views

A question about the vector space spanned by shifts of a given function

Let $E$ be the space of continuous real functions and $f\in E$ Let $T_t$ denote the shift operator: $T_t(f)(x)=f(t+x)$ Let $T(f)$ be the linear span of the set $\{T_t(f) \;|\; t\in ...
1
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1answer
39 views

Apply chain rule to $u = y^{1 - n}$ in order to find $\frac{du}{dx}$

Let $u = y^{1 - n}$. I know that, by using the chain rule: $$\frac{du}{dx} = \frac{du}{dy} \cdot \frac{dy}{dx}$$ Also, I know that $\frac{du}{dy} = (1 - n)y^{-n} = \frac{1 - n}{y^{n}}$ Now, for ...
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0answers
100 views

How fast is the distance between two points changing.

I am having a difficulty with the following question from my calculus unit. Bus station A is located 100km west of bus station B. At 12pm a bus leaves station A driving south at 70km/h and a bus ...
1
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1answer
64 views

$f '(x) = -f(x)$ and $f(1) = 1$, Solve for $f(2)$

I am honestly not even sure how to start this problem... My first thought was that $f(2) = 2$ ... But now I don't even know where to go from there.
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0answers
37 views

Mean Value Inequalities for vector-valued functions

Let $X$ and $Y$ be Banach spaces, and let $U\subset X$ be open. If $f\colon U\to X$ is continuously differentiable and $x,v\in X$ are such that the line segment $\ell=\{x+tv\mid t\in[0,1]\}$ lies ...
1
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1answer
63 views

Why is the chain rule applied to derivatives of trigonometric functions?

I'm having trouble to understand why is the Chain rule applied to trigonometric functions, like: $$\frac{d}{dx}\cos 2x=[2x]'*[\cos 2x]'=-2 \sin 2x$$ Why isn't it like in other variable derivatives? ...
3
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1answer
43 views

Stochastic calculus - Ito confusion

We have $W(t) = f(t)X(t)$. My textbook says that $dW = fdX + X\dfrac{df}{dt} dt$. I don't get how they arrived at this conclusion. I get the first part, because $\dfrac{dW}{dX}dX = fdX$. But for the ...
2
votes
1answer
85 views

Understanding the Definition of a derivative as slope of a tangent line

I'm trying to understand the derivative and am wondering why the derivative is described as the slope of the tangent line and not the slope of a function itself. Say $f(x) = 2x+5$ where ...
0
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1answer
91 views

What is the mean value theorem for the Fréchet (total) derivative?

What is the mean value theorem for the Fréchet (total) derivative? Off the top of my head, it's something like $$ \|F(x+h)-F(x)\|\leq \sup_{c\in[0,1]} \|F^\prime(x+ch)\|\|h\| $$ but the double ...
-1
votes
3answers
104 views

Maximum & minimum area of rectangle outside another.

Find the maximum & minimum area of an outer rotated rectangle when the inner rectangle has the side lengths $a$ and $b$. Here's an image: I have already tried to relate the side of ...
0
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0answers
31 views

Grand canonical derivative.

I've been trying to work out how to find the density in the thermodynamic limit of a nearest neighbour magnetic lattice gas in the grand canonical ensemble. I'll with hold the Hamiltonian for the ...
2
votes
0answers
16 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...