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

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1answer
25 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
71 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 ...
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0answers
47 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
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0answers
95 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
78 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
votes
1answer
52 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
76 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
30 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
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3answers
90 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 ...
1
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1answer
42 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
58 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
152 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
72 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
29 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
86 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 ...
1
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1answer
44 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
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3answers
145 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
32 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
62 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 ...
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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, ...
0
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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
75 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
83 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
33 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
61 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
42 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
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1answer
75 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
59 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 ...
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3answers
90 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 ...
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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
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0answers
14 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 $$ ...
6
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2answers
106 views

Can I interpret the exponential of the derivative operator, $e^D$, as infinite shift operators each shifting “infinitesimally”?

To better explain what I mean, an example can be very useful. Consider $e^{i\theta}$. We could express this using the series definition or the limit definition of $e^x$ instead: $$e^{i\theta} = ...
0
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1answer
53 views

Real Analysis Question- Differentiability on an interval

So this is the question I am trying to answer... At $(-2,0)$ and $(0,2)$ we just differentiate using normal rules of calculus, yes? Here is my attempt for at $x=0$. Is this correct? For d)I think ...
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3answers
154 views

Easy question : $\int (xdy+ydx)$

I am ashamed to ask such an easy question but, well: Lets say I got a function $$ f(x,y)=xy $$ Now let's compute the total differential of the function $$ d(f(x,y))=xdy+ydx $$ Now if I do $$ \int ...
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1answer
33 views

Find derivative using the definition of the derivative as a limit

$$f(x) = \frac{1} {\sqrt{x}}$$ find $f'(x)$ using the definition of the derivative as a limit. I know that $$ f'(x) = \frac{(x + \delta)^{-1/2} - (x)^{-1/2}}{\delta} $$ as $\delta$ goes to $0$. ...
2
votes
3answers
69 views

Real Analysis - differentiable

$f:[0,\infty]\rightarrow \mathbb{R}$ is twice differentiable. If $f''$ is bounded and exists the limit of $f(x)$ at infinity, then $\lim_{x\rightarrow \infty}f'(x)=0$. I tried to use the Taylor's ...
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7answers
439 views

Understanding derivatives

I don't know if this is written somewhere else. I've looked all over the internet so apologies if this has already been covered. I'm doing Year 12 Maths in Australia for what it's worth. In our ...
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0answers
18 views

selecting points on a domain which represent the derivative of a function

I'm working on some algorithm part of which entails me to subdivide a domain based on the derivative of a function. Let's just consider the 1D case with a closed and bounded domain $[a,b]$ for a ...
1
vote
2answers
129 views

derivative of $\ln(4)$

what is the derivative of $\ln(4)$? I am trying to find the derivative of this equation: $h(x)=\ln(\frac{x^3\cdot e^x}{4})$ by rules of logs I simplified the $h(x)$ to the following: ...
0
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0answers
49 views

Generalized “softmax”

I'm looking for a function $f$ from $\mathbb{R}^n$ to $[0,1]^n$, similar to softmax in the sense that is satisfies these properties: $\sum_i f(x)_i = c$, where $c$ is a chosen constant (i.e., c=1 ...
7
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3answers
115 views

Limit of $\frac{f\left(x+g\left(x\right)\right)-f\left(g\left(x\right)\right)}{x}$ as $x\to 0$

I'm trying to answer the following question: Let $f$ be continuously differentiable in all of $\mathbb{R}$ and let $g:\mathbb{R}\to\mathbb{R}$ be a function satisfying ...
1
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2answers
89 views

Using the same limit for a second derivative

I've been trying to answer the same question answered here: Second derivative "formula derivation" And I'm stuck in a step that is not addressed both in the answer and in the comments of ...