New answers tagged derivatives
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Partial derivatives of nested function $y = f(x_n, f(x_{n-1},f(\dots, f(x_1, x_0))))$
Recall that $df= {\partial f\over dx} dx+ {\partial f\over dy} dy$
Let $f_n(x_n,...x_1)=f(x_n,f_{n-1}(x_{n-1},...x_1))$
Reporting $f_n$ in $(*)$, as
$df_n=df(x_n, f(x_{n-1},f(..,x_1))$ we get
$df_n= {\...
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3
votes
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Prove that $\sqrt{x} > \ln(x)$ holds true for all $x > 0$.
Geometric proof for $e^x>2x$:
the tangent line of $e^x$ at the point $(\ln2,2)$ is
$$y=2x+2(1-\ln2).$$
Because $e^x$ is strictly convex on $\mathbb R$, so
$$e^x\geq2x+2(1-\ln2)>2x,\quad \...
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Prob. 26, Chap. 5, in Baby Rudin: If $\left| f^\prime(x) \right| \leq A \left| f(x) \right|$ on $[a, b]$, then $f = 0$
Let $g(x)=e^{-Ax}f(x),x\in[a,b]$,
then $g'(x)=e^{-Ax}(f'(x)-Af(x))$.
So when $x\in[a,b]$, we have
\begin{align*}
g(x)g'(x)
&=e^{-2Ax}\big[f(x)f'(x)-Af^2(x)\big]\\
&\leq e^{-2Ax}\big[|f(x)f'(x)|...
- 10.3k
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Prove that $\sqrt{x} > \ln(x)$ holds true for all $x > 0$.
Substituting $x=e^{2t}$, the given inequality is equivalent to
$$
e^t \ge 2t,
$$
for all real $t$. Consider the function $f(t) = e^t -2t$. It’s a sum of two convex functions, so it is convex. In fact, ...
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Prove that $\sqrt{x} > \ln(x)$ holds true for all $x > 0$.
In what follows, I will assume you are familiar with elementary calculus.
Consider the function $f(x) = \sqrt{x} - \ln(x)$. Differentiating it, we get $f'(x) = \frac{1}{2\sqrt{x}} - \frac{1}{x}$. Thus,...
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Rate of change of water level as it's being poured into an upright right rectangular pyramid
Your approach is a reasonable one but more complicated than necessary. You should recognize that the rate of rise is the rate of volume increase divided by the top area. At the time of interest the ...
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4
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Why is a function not differentiable at the end-points of its domain?
Is there a source telling you that you can't say $x^2$ on $[0,8]$ is differentiable at the endpoints?
The way you pose the question makes me suspect you are studying the Mean Value Theorem, which is ...
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Is $\lim_{x\to 1^{-}}f'(x)=2$ same as saying LHD at $x=1$ is $2$?
There's a theorem that states:
Let $f:[a,b)\to\mathbb{R}$, continuous in $[a,b)$, derivable in $(a,b)$ and $\displaystyle\exists\lim_{x\to a^+}f'(x)=m\in\mathbb{R}$, then $f'_+(a)=m$
I think you are ...
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Deriving the equation of osculating circle starting from the definition using order of contact
I do not have the book by Zorich but here you have to consider the real variable function $t\to f(x(t),y(t))$ where $f(x,y)=(x-a)^2+(y-b)^2-R^2$ and the constant $0$ function and ask which is the ...
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A formula for higher order derivatives of inverse function
The Wikipedia article
inverse function rule gives some examples and is easy to digest.
Another article can be found on vixra: Higher order derivatives of the inverse function
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2
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Jacobian of elementwise multiplication $\mathbf{X}$ $=$ $\mathbf{A}$ $\odot$ $\mathbf{B}$
$
\newcommand\dd{\mathrm d}
\newcommand\m\mathbf
\newcommand\vecc{\mathrm{vec}}
\newcommand\diag{\mathrm{diag}}
$There is a straightforward way to turn
$$
\dd(\m A\odot\m B) = \dd\m A\odot\m B + \m ...
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An identity about derivatives of $C^{2}$ functions.
I'll use Einstein summation for this,
$$\delta^j_k = \frac{\partial x^j}{\partial x^k}(x) = \frac{\partial}{\partial x^k} \left( (\varphi^{-1})^j(\varphi(x)) \right) = \frac{\partial (\varphi^{-1})^j}{...
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Form the differential equation satisfying the relation $\sqrt{1+x^2}+\sqrt{1+y^2}=\lambda(x\sqrt{1+y^2}-y\sqrt{1+x^2})$.
You are on the right track; the question requires the use of the sum-to-product trig identities to simplify the expression for $\lambda$ in terms of $\theta$ and $\phi$. You forgot a small detail, i.e....
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Is the Wirtinger derivative just $\frac{\partial f}{\partial z} = \frac{\partial f}{\partial x}$?
$ \newcommand{\z}{\overline{z}}
\newcommand{\Rt}{\mathbb{R}^2}
\newcommand{\dx}[1][]{ \frac{\partial #1}{\partial x} }
\newcommand{\dy}[1][]{ \frac{\partial #1}{\partial y} }
\newcommand{\dz}[1][]{ ...
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Understanding the Transition from the Traditional Derivative Definition to Spivak's Reformulation OR Preparation for differentiation R(n)
Suppose $\lim_{x \to a} f(x) = L$. We want to show that $\lim_{x \to a} f(x) - L = 0$. This is equivalent to showing that, given any $\epsilon > 0$, there exists $\delta > 0$, such that $|x - a|&...
1
vote
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Chain rule for $\frac{df(\mathbf xt)}{dt}$
If $f_k$ denotes the partial derivative of $f$ with respect to its $k$th argument, then the chain rule says that
$$
\frac{d}{dt} f(t x_1, \dots, t x_n)
= \sum_{k=1}^n f_k(t x_1, \dots, t x_n) \, x_k
.
...
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1
vote
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Is there any solution $u\left(t,x\right)$ that satisfies the following system of equations?
Write it as $\nabla u = \omega$ with $\omega(t,x) := (a-\frac{b}{\sigma} \vartheta (\mu - x), b\sigma)$ with $(t,x ) \in \mathbb{R}^2$. It has a solution if and only if
$$\frac{\partial \omega_1}{\...
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Accepted
Differentiability of 2 variable functions
There's a good reason the "increment theorem" does not appear as a definition in (good?) mathematics textbooks. The result is false. Although a function satisfying this condition will be ...
- 121k
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Derivatives of Linear Functions vs Derivatives of Non-Linear Functions?
As a matter of fact, yes, this principle does have a name. The limit definition of the derivative, $$\lim_{h\rightarrow0}{\frac{f(x+h)-f(x)}{h}}$$ is just the idea of taking your difference over an ...
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Derivatives of Linear Functions vs Derivatives of Non-Linear Functions?
This is single variable calculus. Your example is too complicated to see what's going on.
For any differentiable function $f$, the derivative is the limit of the difference quotient, so for small $h$
$...
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Does zero derivative on a non-open connected set imply function is constant
This is not an answer, but an extended comment.
In your question you defined a function $f : E \to \mathbb R$ to be differentiable at a non-isolated point $x_0 \in E$ if there exists a linear $A: \...
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2
votes
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Derive a matrix derivative involved Hadamard product
$
\def\k{\otimes}
\def\h{\odot}
\def\o{{\tt1}}
\def\lR#1{\Big(#1\Big)}
\def\bR#1{\Big[#1\Big]}
\def\LR#1{\left(#1\right)}
\def\BR#1{\left[#1\right]}
\def\CR#1{\left\lbrace #1 \right\rbrace}
\def\op#1{\...
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Derive a matrix derivative involved Hadamard product
I use the differential to calculate derivatives involving multi-dimensional quantities, when I don't want to explicitly use coordinates.
In this case, for a variation $\delta M$ in $M$,
\begin{align*}
...
- 50.1k
2
votes
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Why is the inverse of a differentiable function $f: \mathbb{R}^n \to \mathbb{R}^n$ is also differentiable?
The essential point is the interpretation of "invertible". The minimal requirement is that $f$ is bijective so that the inverse function $f^{-1} : \mathbb R^m \to \mathbb R^n$ exists.
...
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1
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Alternative definition of derivative in real analysis
The idea comes from the fact that your definition is equivalent to the limit
$$f'(x)=\lim_{y\to x}\frac{f(y)-f(x)}{y-x}$$
existing; indeed rearranging your equation we have
$$\frac{f(y)-f(x)}{y-x}-\...
- 11.1k
3
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Derivation of infinite continued fraction
It's obvious that if $x>0$, $y>0$, and so the negative root can be rejected. Thus, for $x>0$,
$$y'=\frac{-1+\frac{x}{\sqrt{x^{2}+4}}}{2}$$
and likewise for $x< 0$,
$$y'=\frac{-1-\frac{x}{\...
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Equivalence of two definitions of differentiability on non-open sets
This is only a partial answer.
Your second definition has to be modified as follows:
$f$ is differentiable at $x_0\in E$ if there exists an open subset $U$ of $\mathbb{R}^n$ containing $x_0$, and a ...
- 81.9k
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vote
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Product rule applying on operators
You can apply the product rule indeed, but the unknown term vanishes actually, because the total derivative and the functional derivative commute. Let's consider a functional $F[x(t)] = F(t,x,\dot{x},\...
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Problem understanding proof about total derivatives
I started writing this before Asigan's excellent answer, but I did spend a bit of time on this so I'll leave mine still.
EDIT: I've added an explanation for the second-to-last equality in the ...
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Accepted
Problem understanding proof about total derivatives
Going back to the definition of $o()$, $$(\partial_1 f)(c_{x,y},y)(x-a) = (\partial_1 f)(a,b)(x-a) + o(|x-a|)$$
actually means
$$\lim_{(x,y)\to (a,b)}\frac{(\partial_1 f)(c_{x,y},y)(x-a)-(\partial_1 f)...
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votes
Water leaking from box and the relationship of volume and height.
$$
P_{atm} + \rho gh + {1 \over 2}\rho v_{up}^{2} =
P_{atm} + {1 \over 2}\rho v_{down}^{2}\quad
(\ Bernoulli\ )
$$
$$
\left({b^{2} \over a^{2}} - 1\right)v_{up}^{2} = 2gh
$$
$$
\color{#44f}{v_{up} = \,...
- 92.3k
2
votes
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$\frac{\Gamma'(\frac{s}{2}+1)}{\Gamma(\frac{s}{2}+1)}$ or $\frac{1}{2}\frac{\Gamma'(\frac{s}{2}+1)}{\Gamma(\frac{s}{2}+1)}$?
First of all, while writing something like "$\log(\xi(s))$" is okay for intuition, I think it is a terrible idea to work with such things in a loose way because there is no such thing as a ...
- 50.8k
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Confusions about Theorem 16.3 in Munkres' "Analysis on Manifolds"
The proof of Theorem 16.3 is in fact a bit sloppy.
The functions $\psi_i$ are defined on $\mathbb R^n$ and the series $\lambda(x) = \sum_{i=1}^\infty \psi_i(x)$ converges for all $x \in \mathbb R^n$. ...
- 81.9k
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Confusions about Theorem 16.3 in Munkres' "Analysis on Manifolds"
The supports of the $\psi_i$s, and thus of the $\varphi_i$s, are contained in $A$, so that, almost by the definition of a compactly supported smooth function, on $\mathbb{R}^n \setminus A$ you can ...
- 6,606
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Advanced calculus book recommendations.
An "anti-recommendation": The accepted answer recommends Loomis & Sternberg's Advanced Calculus. I strongly disagree with this and would advise people to avoid this book. The table of ...
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Can a function have a inflection point at somewhere non-differentiable?
The main point is that one can define convexity/concavity without reference to the derivative at all, although your text does not do so. A function $f$ is convex [resp., concave] on an interval if for ...
- 121k
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votes
Accepted
What is the point of taking out the constant "a" in the derivative problem d/dx(af(x))?
When the power rule is usually taught, technically all that has been proven is that $$\frac{d}{dx}(x^n) = nx^{n-1}$$ for a monic polynomial $x^n$ that is, a polynomial with leading coefficient of one. ...
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What is the simplest way to show that $f(z)$ is differentiable?
Most of the rules you know from calculus work in complex analysis as well, so that for instance knowing that $f$ and $g$ are differentiable tells you that $fg$ is as well, and that $(fg)' = f'g + f g'$...
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votes
Accepted
Does zero derivative on a non-open connected set imply function is constant
Further to my comment,
If there is a continuous injective $f:S^1\to\mathbb{R}^2$ such that $\frac{|f^{-1}(x)-f^{-1}(y)|}{d(x,y)}\to0$ uniformly as $d(x,y)\to0$, then $f^{-1}:\text{Image}(f)\to S^{1}$ ...
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Can a function have a inflection point at somewhere non-differentiable?
There's a flaw in the definition.
Let $f$ be a differentiable function defined on open interval $I$.
We say that $f$ concave-up on $I$ when ...
This gives us definitions of concavity on an open ...
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2
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Can a function have a inflection point at somewhere non-differentiable?
The french version of Wikipedia https://fr.wikipedia.org/wiki/Point_d%27inflexion is a bit less restrictive concerning the regularity assumptions on $f.$
In the sequel, "locally at $x_0$" ...
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Proof that $f(x) = \frac{xa^{x-1}(1-a)}{1-a^{x}}<1$ for $x>1$.
You can use Bernoulli inequality for real exponents as follows:
First, rearrange the given inequality:
$$f(x) = \frac{xa^{x-1}(1-a)}{1-a^{x}}=x\frac{1-a}{a}\frac{a^x}{1-a^x}< 1 \Leftrightarrow x\...
- 33.1k
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Proof that $f(x) = \frac{xa^{x-1}(1-a)}{1-a^{x}}<1$ for $x>1$.
Standard way: In order to prove that $f(x)=\frac{xa^{x-1}(1-a)}{1-a^x}<1, \forall x\in (1,\infty)$, it is equivalent to proving that $xa^{x-1}(1-a)<1-a^x, \forall x\in(1,\infty)$. Define $g(x)=-...
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Understanding and deriving the functional derivative
The integral should not be there. By looking at the proof of Euler-Lagrange equation
$$
\int \frac{\delta F}{\delta y} \eta(x) dx =\frac{d}{d\epsilon} F[y(x)+\epsilon\eta(x)]_{\epsilon=0}
$$
So I ...
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Prove that a multivariable function is differentiable $\iff$ each of its components is differentiable
Converting geetha290krm's comment into an answer to get this question off the unanswered list:
On (1), you are right that this is the composition of two functions without the notation that you are ...
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Why does differentiation with respect to time commute with the Fourier Transform?
As commented by Ted Shifrin, dominated convergence implies differentiation under the integral, see this question for example. The integrals there are over all of $\mathbb{R}$.
For example, for your $t$...
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Why does differentiation with respect to time commute with the Fourier Transform?
There is nothing that special about the Fourier transform, if you derive in the variable that you are not transforming. Differentiation commutes with bounded linear operators in general. Say, if you ...
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Why does differentiation with respect to time commute with the Fourier Transform?
This comes from a general property about integrals, namely that for a "nice/smooth" function $g(x,t)$,
$$
\frac{\mathrm d}{\mathrm dt}\int_Sg(x,t)\mathrm dx=
\int_S\frac{\mathrm d}{\mathrm ...
- 667
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Forming $PDE$ of $z=e^yf\left(\frac{x}{y}\right)+e^xg(xy)$
(Partial answer)
You are on the right track, however you will need to find a second-order PDE, because two characteristic lines are hidden behind this function actually, namely $xy = const$ and $x/y = ...
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votes
General solution to this coupled system of differential equations
Given the ordinary differential equations system:
$$
\begin{cases}
a\,x'(t)+b\,x(t)+c\,y'(t)+d\,y(t)=u(t)\\
e\,x'(t)+f\,x(t)+g\,y'(t)+h\,y(t)=v(t)\\
\end{cases}
$$
let's start by rewriting it this way:...
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