Questions tagged [derivatives]
Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
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Rolle's Theorem for functions defined on Banach space
In J.Dieudonne's "Treatise on analysis, vol. 1" Chapter 8.2, there's a problem which asks to prove Rolle's theorem for a function defined on a Banach Space. The problem is as follows:
Let f ...
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Nice $\epsilon$-$\delta$ proof that the derivative of $\sin(x)$ is $\cos(x)$?
Looked around a bit and all I see are proofs using the limit definition of a derivative. This is not for an assignment, I could just use the limit definition if I wanted to, but I was wondering how ...
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Check whether there is a global existence of solution of function $y'(x) = \frac{-y(x)}{1+e^{y(x)}}$?
I know that the derivative needs to be bounded in order to have a global solution but I got stuck here and don't know how to proceed further.
$$\left|\frac{df}{dy}\right|= \left|\frac{y.e^y-(e^y+1)}{(...
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Use of a substitution to prove that $e^{2xt-t^2}$ is the exponential generating function of the Hermite polynomials
The generating function encodes all the Hermite polynomials in one formula. It is a function of $x$ and a dummy variable $t$ of the the form:
$e^{2xt-t^2}=\sum^\infty_{n=0}\frac{H_n(x)}{n!}t^n. $
We ...
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Proving the Power Rule for differentiation when the power is a real number
Let, $f(x)=x^{n}$. Then,
$$\lim_{h\to 0}\frac{(x+h)^{n}-x^{n}}{h}=f'(x)$$
If $n\in\mathbb{N}$ we can use the Newton 's binomial theorem to prove the power rule but how to prove it when $n\in\mathbb{R}$...
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Differentiation of logLikelihood MLE
Given a log likelihood function,
$$l=\sum_{i=1}^m\log[\frac{n!}{X_{i,1}!X_{i,2}!(n-X_{i,1}-X_{i,2})!}p_1^{X_{i,1}}p_2^{X_{i,2}}(1-p_1-p_2)^{n-X_{i,1}-X_{i,2}}]$$
To derive: with respect to $p_1$, $\...
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Are there any scientific papers on discriminant of extrema points of derivatives of 3rd order
When we want to calculate the extrema points in one dimension we just need f'(x) =0 , f"(x)>0 to have local minimum for example
For 2 dimensions we have determinant
$D = f"_{xx}(x,y)f"_{yy}(x,...
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Find m so that f has one local extreme point.
Let $f\colon (-1,1) \to \mathbb R $ and $ f(x)=e^x(x^2+x+m) $.
The function f has only one extreme local point if and only if m belongs to the set:
a) (-5, 1); b) {-5, 1}; c) [-5, 1); d) ${\...
CommunityBot
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Derivative(s) of Cantor Measure (Donald L. Cohn ch. 6.2, exercise 6.2.4, related to lemma 6.2.5)
First, context: I'm doing a course on measure/integration theory following the book by Donald L. Cohn. In section 6.2, he defines the (upper and lower) derivates of a finite Borel measure $\mu$ on $\...
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Is it possible to differentiate a modified bessel function of the third kind?
Suppose that we have two variables, $\alpha\in \mathbb{R}^+$ and $\beta\in \mathbb{R}^+$. Then we have the following modified Bessel function of the third kind,
$$\delta = K_{1} \left(\sqrt{\alpha\...
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How to find the derivative of a function in the direction from point to point?
How can I find the derivate of a function? This is the exercise:
$z = x^3 - 3y^3$ at $M(3;1)$ in the direction from point $M$ to point $K(6;5)$.
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Calculate $\frac{\partial}{\partial \mathbf{A}}\lVert \mathbf{A}^{\top}\mathbf{AX}-\mathbf{X} \rVert _{F}^{2}$
Calculate $\frac{\partial}{\partial \mathbf{A}}\lVert \mathbf{A}^{\top}\mathbf{AX}-\mathbf{X} \rVert _{F}^{2}$ with $\mathbf{A}\in\mathbb{R}^{M\times N}$ and $\mathbf{X}\in\mathbb{R}^{N\times D}$, and ...
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Assume that $\mathop {\lim }\limits_{x \to \infty } f(x) + f'(x) = 0$. Prove: $\mathop {\lim }\limits_{x \to \infty } f(x) = 0$
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable everywhere. Assume that $\mathop {\lim }\limits_{x \to \infty } f(x) + f'(x) = 0$.
Prove: $\mathop {\lim }\limits_{x \to \infty } f(x) = 0$
...
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Derivative of $f(x)$ wrt $g(x)$
Let,
$f(x)=\operatorname{arcsec}\left(\frac{1}{2x^{2}-1}\right)$ and $g(x)=\sqrt{1-x^{2}}$.
Upon substituting $x=\cos\theta$ we get,
$$\frac{df}{dg}=\frac{2}{x}$$
But my textbook says that answer is $\...
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$\sum_{i,j=1}^nx_ix_j\frac{\partial^2f}{\partial x_i\partial x_j}=0$ and $\nabla f(0)=0$ implies constancy of $f$ in $B_1(0)$
Let $B_1(0)$ be the unit ball in $\mathbb R^n$ centered at the origin. Assume that the function $f\in C^2(B_1(0))$. Prove that
$1)$If $f$ satisfies $$\sum_{i,j=1}^nx_ix_j\frac{\partial^2f}{\partial ...
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Confusion regarding: exact differentials, inexact differentials, directional derivative and partial derivatives. What is the proper approach?
When studying multivariable calculus we got introduced to partial derivatives and concepts like directional derivative. This all made sense. The notation was also quite clear to understand e.g.:
$$ f(...
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Derivative of matrix-diag matrix product
I would like to take a derivative of the following expression wrt vector $x\in\mathbb{R}^d$
$$
W\mathrm{diag}(f(Ax+b))
$$
where $f$ is some smooth element-wise function, $A\in \mathbb{R}^{K\times d}$, ...
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Definition of a differentiable curve on a manifold.
Let me just set up some very standard notions. (I am working from the lecture series International Winter School on Gravity and Light 2015 on youtube)
$\textbf{Setup:}$
Given a d-dimensional ...
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How to tell if $f(x)=x^n|x|$ is twice differentiable or not by looking at the 2nd derivative?
I came across the following question in my test:
Let $f:R\rightarrow R, f(x)=x^n|x|$. Then number of integer values of $n$ for which $f(x)$ is not a twice differentiable function in it's domain is?
$...
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Closed form for $\lim_{m\to 0}\frac{\partial^{2a-1}}{\partial m^{2a-1}}\pi \csc(m\pi) \left[\operatorname{\psi}^{(0)} (1-m) + \gamma\right]$
How to show that
$$\lim_{m\to 0}\frac{\partial^{2a-1}}{\partial m^{2a-1}}\pi \csc(m\pi) \left[\operatorname{\psi}^{(0)} (1-m) + \gamma\right] =2(2a-1)!\sum_{k=0}^{a-1}\frac{\eta(2k)\psi^{(2a-2k)}(1)}{(...
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Can anyone provide a "hierarchy of functions/function sets" in terms of differentiability?
I'm studying Advanced Analysis II, specifically differentiation in higher dimension vector spaces. I'd like to fully understand the logical implications and relations between differentiable, ...
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Is the covariant derivative a tensor?
While studying general relativity, the covariant derivative is constructed (in no rigorous manner) in order to make the derivative of a tensor transform like a tensor. Symbolically, $$ \nabla'_{\mu} V'...
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Assumptions about Discrete Space
I have always had this question about "practical assumptions about discrete space".
Take for instance a classic problem in Integer Programming:
Had we been dealing with a function in ...
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zero of $\frac{\sin(x t)}{x} - \frac{\sin(y t)}{y}$ for fixed $x$ and $y$?
Let $x$ and $y$ be fixed nonzero real numbers. I want to show that
$$
\frac{\sin(x t)}{x} - \frac{\sin(y t)}{y}
$$
has infinitely many zeros for any choice of $x$ and $y$.
I have tried looking into ...
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Derivative of $\sqrt[5]{\sin(x)}$ from first principles.
I'm a newbie to derivatives using first principle. I've just learnt how to differentiate basic functions using first principles.
My problem is that, how can we differentiate $\sqrt[4]{\sin x}$ or $\...
CommunityBot
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Strong convexity definition based on subgradient
Suppose that we have $f: \mathbb R^d \to \mathbb R$ is convex and satisfies
$$f(\textbf y) \geq f(\textbf x) + \nabla f(\textbf x)^\top(\textbf y - \textbf x) + \frac{\mu}{2} \| \textbf x - \textbf y \...
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Convex function inequalities with gradient being Lipschitz continuous, starting from $f(y)\le f(x)+(\nabla f(x))^T(y-x)+\frac{L}{2}\Vert y-x\Vert_2^2$
Let $f$ be a continuously differentiable convex function defined on $\mathbb R^n$, i.e., $f :\mathbb R^n \rightarrow\mathbb R$ is continuously differentiable and for any $x,y \in\mathbb R^n$ and any $\...
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Prove not differentiable at $(0,0)$ but directional derivatives every direction.
Verify that the function $f: \mathbb R^2 \to \mathbb R$ given by $f(x,y) = \frac{x^2y}{x^4 + y^2}$ for $ (x,y) \neq 0$ and $f(0,0) = 0$ is not differentiable at $(0,0)$ and yet has directional ...
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Differentiation of dependent variable as integrand by just one independent variable
$$-{\partial{H}_{y}\over\partial\mathrm{z}}=\epsilon{\partial{E}_{x}\over\partial\mathrm{t}}~~~,~~~{\partial{E}_{x}\over\partial\mathrm{z}}=-\mu{\partial{H}_{y}\over\partial\mathrm{t}}\tag{1}$$
$${\...
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The value of this series $\sum_{i=0}^{\infty}\frac{i+n}{i+n+m}\frac{x^{i+n+m}}{i!}$
I am trying to calculate the value of this series
$$\sum_{i=0}^{\infty}\frac{i+n}{i+n+m}\frac{x^{i+n+m}}{i!}$$
If we get the derivative with respect to $x$, we get
$$\sum_{i=0}^{\infty}\frac{i+n}{i!}x^...
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Convergence of derivatives.
We have a course on complex analysis in the current semester.Our professor introduced a theorem in class which is as follows:
Theorem
Let $f_n:\Omega\to \mathbb C$ be a sequence of holomorphic ...
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The derivative of $\max (0,x)$
there is a very fundamental question that makes me really confused... Does the derivative of function
$$
f(x)=[x]^+
$$
exist? Specifically, $[x]^+=\max[0,x]$.
Any hint is appreciated!
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Proof involving Rolle's theorem and the MVT
I'm stuck on a problem that asks me to show that the equation, $$6x^4-7x+1=0$$
does not have more than two distinct real roots.
I've tried to set up a proof by contradiction using Rolle's theorem and ...
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What's wrong with the following definition of $\lim$ and derivative for the surreal numbers?
I am reading about surreal numbers, and was asking myself, what is wrong with the following definition of limit and derivative for surreal functions:
Let $x,p$ are surreal numbers and $f$ is a ...
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Conflicting inflection point conditions in my book
My 12th grade calculus book mentions that an inflection point is a point that:
• Function $f$ in point $(c, f(c))$ has a tangent line.
• Function $f$'s concavity shall change at point $(c, f(c))$.
...
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Help me understand what is and isn't exponential growth.
If the rate at which something grows is proportional to itself, then you would call it exponential growth. Don't quote me on this, but I think it has something of the form $y = e^x$.
Now take for ...
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Infinite tangent line and rate of change.
We know that if the tangent line exists then the derivative exists except for vertical tangent lines. For example $f(x)=x^\frac{1}{3}$ has no derivative at $x=0$.
How can we substantiate it both ...
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Curve sketching: if $\dot{x} = x- 2y-x^2$ , which is the dominant term?
$\dot{x} = x- 2y-x^2$
I am trying to find where $x$ increases or decreases.
I thought $x^2$ would be the dominant term because it has a high power however in my notes it says $-2y$ is the dominant ...
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Mapping from dual numbers to real numbers
Background
I was naively playing around with some interpolation ideas once again and came across the dual numbers as a way to perform differentiation implicitly. Naturally, I thought, okay, perhaps ...
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Derivative of the product of a matrix scalar function and a matrix with respect to a matrix
How to take derivative of
\begin{align*}
\dfrac{\partial\left[\operatorname{tr}(\boldsymbol{A}^2)\cdot\boldsymbol{A}^3\right]}{\partial\boldsymbol{A}}=\,?
\end{align*}
with respect to a matrix $\...
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what does it mean to ''apply linearity'' in differentiation and integration?
In integration,
let's say I want to integrate $2(x+5)$
I could then put the $2$ in front of the integral sign and integrate $x+5$,
only to multiply it again by $2$ later. is this called ''applying ...
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Regarding the derivative of Euclidean L2 norm, Definition of differentiation in Rudin.
I am trying to understand the answer posted by hemanth in this post.
I understand how he derived the derivative $f:\Bbb R^n \rightarrow \Bbb R$ defined as $f(x)=\rVert x \rVert$,
$$
Df(x) = \nabla_x\...
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Directional derivative and normalization of the direction vector.
This issue has been discussed before but I still have some doubt about it. I understand while defining directional derivatives, some authors normalize the direction vector v while some don't. However, ...
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Little-O notation on vector norm
I was given the following definition of a total derivative (in multivariable calculus).
Let there be $A\subseteq\mathbb{R}^{k}$, $f:A\to\mathbb{R}^{m}$, $x\in A^{\circ}$.
We say $f$ is totally ...
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Differentiating a matrix with respect to a vector
In multivariate linear model, I have come across the following matrix-valued function of $\beta \in \Bbb R^p$.
$$\beta \mapsto(y-X\beta)(y-X\beta)^{T}$$
where matrix $X \in \Bbb R^{n \times p}$ and ...
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Shortest distance between two joggers
Question
Two joggers, $A$ and $B$, start at either end of a $100$ m track. $A$ runs horizontally towards the other end at $3$ m/s and $B$ runs diagonally at $5$ m/s. The diagonal forms an angle of $30$...
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Simple partial derivative of function with summation - how to solve it?
I need some help with the computation of a partial derivative.
The function is
$$f(x) = \sum_{a=1}^A x_a \bigg(\sum_{b=1}^{a-1} x_b^2 + \delta\bigg)^{-\gamma}$$
I need to compute the partial ...
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The average rate of change of a function
The following is a multiple-choice question in the 2014 AP Calculus BC exam paper.
The function $f$ is given by $$f(x) = \int_{1}^{x}{\sqrt{t^3+2}} \,{\rm d }t$$ What is the average rate of change of ...
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Is the operator norm of the derivative the same as the Riemannian norm?
On a Riemannian manifold, $T_p M$ is the set of all derivative operators at that point. The norm is given by $$\left\|\frac{d}{dx}\right\|=\sqrt{g\left(\frac{d}{dx}, \frac{d}{dx}\right)}=\sqrt{g_{i,j}\...
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Partial derivative of a matrix of euclidean distances
I'm trying to compute the gradient of a cost function wrt. to some matrices. During the computations, I reached an expression for which I wasn't sure about its derivative.
Here's my problem :
Let $X =...
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