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

learn more… | top users | synonyms (2)

2
votes
2answers
36 views

derivative of arctan(u)

Im trying to find the derivative of $\arctan(x-\sqrt{x^2+1})$ here are my steps if someone could point out where I went wrong. $$\begin{align} \frac{\mathrm d~\arctan(u)}{\mathrm d~x} \;& =\; ...
0
votes
1answer
41 views

Is $\displaystyle\lim_{h \to 0} H_n(f(h), g(h)) = H_n(\displaystyle\lim_{h \to 0} f(h), \displaystyle\lim_{h \to 0} g(h))$ true for all $n$?

Consider the limit $\displaystyle\lim_{h \to 0} H_n(f(h), g(h)), $ where $H_n(a, b)$ denotes the $n$th hyperoperation $H_n(a,b) = a \uparrow^{n-2}b$ with both $f(x)$ and $g(x)$ being continuous and ...
0
votes
1answer
30 views

Tangent meets curve again

If the tangent at the point $(16,64)$ on the curve $y^2=x^3$ meets the curve again at at $Q(u,v)$ then $uv$ is ? If found the tangent to the curve at $(16,64)$ but then I cannot find $uv$.Give your ...
1
vote
2answers
56 views

Is there any way to differentiate such function?

Let $S$ be a set. If I had a bijection $f$ mapping each element $n\in \mathbb{N}$ to an element $s \in S$ such that: $$s = f(n) = \sum^{n}_{k=1} {1\over k}$$ Is the function differentiable in ...
4
votes
1answer
288 views

The root of summation function

This is a calculation I need for my statistics project Big edit: simplify the function $f(x)$ a lot. Define for $f(x)$, $x\geq 0$, $$ f(x):=\sum_{k=1}^\infty ...
1
vote
1answer
32 views

Partial derivative of x - is quotient rule necessary?

Let $$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to determine if the given function is harmonic. I know that the 2nd partial derivative with respect to $x$ should, when added to the 2nd partial ...
4
votes
3answers
69 views

why $|x|$ in $\frac{d}{dx}\sec^{-1}x=\frac{1}{|x|\sqrt{x^2-1}}$

I derived $$\frac{d}{dx}\sec^{-1}x$$ as follows: Let $$z=\sec^{-1}x$$ Then $$x=\sec z$$ differentiating both sides w.r.t $x$ we get $$1=\sec z \tan z \frac{dz}{dx}$$ so we get ...
1
vote
1answer
24 views

Riemann Sum-esque limit involving Radicals

Me and my friend were trying to evaluate the following limit: $$ I\equiv\lim_{N\to\infty}\sum_{k=0}^N\sqrt{\frac{1}{N^2}+\left(f\left(\frac{k+1}{N}\right)-f\left(\frac{k}{N}\right)\right)^2} $$ it was ...
4
votes
1answer
43 views

$\frac {\partial}{\partial t}T$ vs $\frac d{dt} T$.

Suppose we have a function $T_1=F(x,y,t)$. Now suppose that $x=g(t),y=h(t)$, so we have a new $T_2=F(x(t),y(t),t)$, so then we have that $\frac \partial{\partial t} T_2=F_t$ and $\frac d{dt}T_2=F_x ...
0
votes
0answers
12 views

A $C^2$ $f$ such that for every $x \in \mathbb R^n$, $t \in \mathbb R$, $f(tx)=t^2f(x)$. [duplicate]

I am trying to do the following exercise: Suppose $f:\mathbb R^m \rightarrow \mathbb R^n$ is $C^2$ and for every $x \in \mathbb R^n$, $t \in \mathbb R$, $f(tx)=t^2f(x)$. Show that there exists a ...
0
votes
0answers
17 views

Given $f:[a,b] \rightarrow R$, if $x'$ is a local minimum of $f$ and $x'<b$ then there exists a sequence $x_n$ converging to $x'$ with $x'<x_n<b$

I'm trying to understand the demostration of the folowing lemma: Is a function $f:[a,b] \rightarrow R$ differentiable in a local extrema $x'$ then $f'(x')=0$ Demostration: $x'$ is a local ...
0
votes
1answer
23 views

Find the versors of the Frenet Triad

I have the following curve: $C(t)=(\frac{t^5}{5}, \frac{t^3}{3}, t ^ 2).$ The problem is to find $T$, $N$ and $B$ of the Frenet Frame. I know the fact that $\vec{B} = \vec{T} \times \vec{N}$. I've ...
3
votes
5answers
77 views

How to evaluate $\lim\limits_{x\to 0+}\frac 1x \left(\frac 1{\tan^{-1}x}-\frac 1x\right)$?

How to evaluate $\lim_{x\to 0+}\dfrac 1x \Big(\dfrac 1{\tan^{-1}x}-\dfrac 1x\Big)$ ? I used L'Hospital's rule but with no success . Please help . Thanks in advance
2
votes
1answer
39 views

Directional derivative understanding

[Beginning multivariable question.] I have just been introduced to a theorem that says $$D_uf(x)=\nabla f(x)\cdot u.$$ So in the two-dimensional case, $$\nabla f(x,y)= \langle f_x(x,y),f_y(x,y)\rangle ...
0
votes
2answers
54 views

Derivative of function involving absolute value

Could anyone help me with differentiating $|x|^5$ and $\frac{|x|^3}{(1+x^2)^8}$? I used the way we differentiate $|x|$ via substitution, i.e. enter link description here It fails on the two ...
0
votes
1answer
31 views

A cylindrical reservoir

A reservoir has the shape of a vertical cylinder with height $3m$ and radius $4m$ and it is filled with water. i) Let x be the height in meters measured from the bottom of the reservoir. The weight ...
2
votes
0answers
37 views

Which of the following functions is not everywhere differentiable?

Today, I took a FAMAT test and had this question: Which of the following is not everywhere differentiable? $A) \sin\left(\frac 1{x^2 + 1}\right)$ $B) \ln(x^2)$ $C) \arctan(x)$ $D) \sqrt{1 + ...
1
vote
1answer
48 views

Is $(x^2+y^2+x)dx+xydy$ the same as ${dy\over dx}(x^2+y^2+x)+{dx\over dy}(xy)$ ? Is this just a different notation?

Is $(x^2+y^2+x)dx+xydy$ the same as ${dy\over dx}(x^2+y^2+x)+{dx\over dy}(xy)$? Is this just a different notation?
1
vote
1answer
50 views

derivative of a projection matrix

The projection onto a parametrised vector $v(\lambda)$ is $P_v = \frac{vv^{T}}{v^{T}v}.$ Its complement is $$P = I-\frac{vv^T}{v^{T}v}.$$ I've got an expression containing this complementary ...
1
vote
1answer
23 views

Taylor Series in Fractional Calculus

I recently studied fractional calculus, namely the possibility to define fractional derivatives of some functions, like $$\frac{\text{d}^{1/2}}{\text{d}x^{1/2}}\ f(x) ~~~~~~~~~~~~~ ...
1
vote
1answer
25 views

4th order mixed leibniz derivative

How exactly is the order of mixed partials read in Leibniz notation? In Lagrange notation, we just read from left to right. $$f_{xyzz} = (\frac {\partial} {\partial z}(\frac {\partial} {\partial ...
1
vote
0answers
54 views

Why the differential equations have a wave behavior?

The differential equation for string: $$\frac{1}{c^2} \frac{\partial^2 f}{\partial t^2}=\frac{\partial^2 f}{\partial x^2} \tag{1}$$ I have inital condition: $$f(x)=\begin{cases}20x, & 0\le ...
1
vote
4answers
66 views

Is there a way to evaluate the derivative of $x$! without using Gamma function?

Taking the factorial function $x!$ I wonder if there is a method to find the first derivative of this function without making any use of the Gamma function (or related integral representations of the ...
0
votes
4answers
31 views

Feynman lectures, Volume I, chapter 13-4

While reading Feynman lectures on Physics, volume I, Chapter 13-4, I found following assumption, which I don't understand: Then, since $r^2 = \rho^2 + a^2$, $\rho\,d\rho = r\,dr$. Therefore ... ...
0
votes
1answer
21 views

Problem calculating $(g\prime)^{-1}(y)$.

I have some problem understanding how to calculate the inverse of a function. I have an example below: Calculate the following: $(g\prime)^{-1}(y)$. The $y$-value is: $y(s)=g\prime=2s-1 $. ...
1
vote
1answer
79 views

$f\in C^1(\mathbb R)$ , having finitely many zeroes and $f'$ changes sign at exactly two of these points , solutions of $f(x)=y$ for given $y$?

Let $f:(0,1) \to \mathbb R$ be a continuously differentiable function having finitely many zeroes and $f'$ changes sign at exactly two of these points , then is it true that for any $y \in \mathbb R$ ...
0
votes
1answer
44 views

higher order partial derivative notation (Leibniz)

Which one of the following two are correct? $$ f_{xy} = \frac {\partial} {\partial y} (\frac {\partial f} {\partial x}) = \frac {\partial ^2 f} {\partial xy}$$ or $$ f_{xy} = \frac {\partial} ...
1
vote
1answer
29 views

Proving P(x) > 0 given a condition.

$P(x)$ is a polynomial function such that, $P(1) = 0, P′(x) > P(x), ∀ x > 1. $ Prove that $P(x) > 0, ∀ x > 1.$ I was trying to do by taking the P(x) in the denominator and then ...
1
vote
2answers
35 views

$(r-1)^{th}$ derivative of $x^{k+r-1}$

EDIT: added $x^k$ in final answer I want to find: \begin{align} \frac{d^{r-1}}{dx^{r-1}}\left(x^{k+r-1}\right) \end{align} Writing out the first few terms and what I think is the last term we get: ...
1
vote
0answers
26 views

Meanvalue theorem for quadratic arguments

I have trouble proving the following result and I would be happy about any kind of suggestion how the precise argument looks like. Let $f : [0,\infty) \rightarrow \mathbb{R}$ denote a continuously ...
0
votes
1answer
80 views

What is the primitive function of $xe^{x^2+2x}$?

I need to know the primitive function (Antiderivative) of this function: f(x)= $xe^{x^2+2x}$ without using integral please. Also, please how could I find the primitive functions of those kind like ...
2
votes
0answers
35 views

Application of Leibniz rule for Lebesgue integral

Consider the real-valued random variables $X,Y$ defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the function $f\colon\mathbb{R}\rightarrow [0,\infty)$. Let ...
1
vote
2answers
29 views

Finding the error in the surface area of a cube. when length = 3, error= ${1\over 4} $

Find the approximate error in the surface area of a cube having an edge of length 3ft if an error of ${1 \over 4}$ in. is made in measuring an edge I have to do this by using differentials and ...
5
votes
3answers
536 views

Relationship between factorial and derivatives

I was wondering if there is any relationship between factorials and derivatives because I notice that if we had $x^n$ and we take the $n$-th derivative of this function it will be equal to the ...
0
votes
1answer
59 views

Suppose a matrix valued function $A$ with $A(0) = I$, find $\frac{d^2}{dt^2} det(A(t)) $ at $t=0$

The original question is : Let $A(t)$ be a differentiable square matrix valued function with $A(0) = I$, find $\frac{d^2}{dt^2} det(A(t)) $ at $t=0$ in term of $\dot{A}$ and $\ddot{A}$. I know in the ...
0
votes
2answers
39 views

When does $P(x,y)$ is a function of $x+2y$?

Suppose $P$ is a polynomial of two real variables $x$ and $y.$ How can I prove that $P(x,y)$ is a function of $x+2y$ if and only if $P_y=2P_x$ ? Here $P_x=\dfrac{\partial P}{\partial x}.$ Is ...
0
votes
1answer
36 views

Tough probability distribution question with integral over sample space not 1

$\frac{df}{dlogw} = c w^{-0.5} $ where f is the fraction of patients with a particular disease and w is the ratio of weight of patient's liver/patient's weight. If the probability that a patient of ...
0
votes
2answers
97 views

Matrix differential of $AA^T $

I need to find the first and second partial derivative of $\dfrac{\partial \|AA^T\|_{F}^2}{\partial A_i}$ where $A$ is a $n$ by $n$ matrix and $A_i$ denote the $\textit{i}^{th}$ row of matrix ...
0
votes
1answer
27 views

I have to solve for dV which is the volume of a sphere that would be used to “construct” the earth

The Earth does not have uniform density; it is most dense at its centre and least dense at its surface. The simplest density function is liner; in particular p(r) = A - Br ; Where A and B are ...
1
vote
0answers
36 views

Using Fundamental Theorem of Calculus to find derivative of $\int_k^\infty (s-k)\pi(s)ds$

I am trying to run through this example as a learning exercise, but I am not getting the result. Define $$ C_t = \int_k^\infty (s-k)\pi(s)ds $$ Where $\pi$ is some function (that satisfies whatever ...
0
votes
0answers
20 views

Cauchy-Riemann: am I applying the equations correctly?

$$\text{Re}(z) +i\text{ Im}(z)^2$$ The problem states to apply C-R and to describe what can be concluded. However, I don't understand what I can conclude without a point $z_0$. My conclusion: ...
2
votes
1answer
70 views

Can the function $f(x,y) = \frac{xy}{\sqrt{x^2+y^2}}$ be defined so that f is differentiable at the origin?

Can the function $f(x,y) = \frac{xy}{\sqrt{x^2+y^2}}$ be defined so that $f$ is continuous and differentiable at the origin? I redefined the function piecewise so that $f=0$ at the origin and $f = ...
0
votes
2answers
111 views

How to find the surface area of a spherical cap by integration?

I don't really understand how they derived the formula in the following picture. The aim is basically to find the formula for the surface area of a spherical cap. Why do you differentiate the ...
0
votes
1answer
32 views

Lipschitz Continuity of a Function of a Matrix

Define $f(A): \mathbb{R}^{p\times m} \to \mathbb{R}$ as follows: $$ f(A) = \frac{1}{2}\|Y-XAB\|_F^2 = \frac{1}{2}\text{tr}\{(Y-XAB)^T(Y-XAB)\}, $$ where matrices $Y\in\mathbb{R}^{n\times q}, ...
0
votes
0answers
6 views

Signing *change* of probability that one random variable is lower than another

Let $\tilde{z}_L \in [0,1]$ and $\tilde{z}_H \in [0,1]$ denote two random variables, with $F_L(z|\theta) := \Pr\{\tilde{z}_L \leq z|\theta\}$ and $F_H(z|\theta) := \Pr\{\tilde{z}_H \leq z|\theta\}$. ...
1
vote
2answers
30 views

Differentiability of a function and its square root

Consider a function $f:\Theta \subseteq \mathbb{R}^l \rightarrow [0,\infty) $. Let (1) $\sqrt{f(\theta)}$ is differentiable at $\theta_0 \in \Theta$ (2) $f(\theta)$ is differentiable at $\theta_0\in ...
0
votes
1answer
17 views

Derivative of a summation.

If a function $E={1\over2}\sum_{n=1}^N(y_k-t_k)^2$ And if $a_k = y_k$ then how ${\partial E \over {\partial a_k}} =y_k - t_k$ Can anyone please tell me how final answer was obtained using partial ...
0
votes
1answer
52 views

Is there better way to solve this derivative: $((5\tan 5x - 3\cot 5x)\arcsin(\frac{x+3}{x-1}))'$?

I've done $$(5\tan 5x- 3\cot 5x)'\arcsin\frac{x+3}{x-1} - (5\tan 5x- 3\cot5x)(\arcsin\frac{x+3}{x-1})'$$ And I've gotten $$5\left(\frac{5}{\cos^25x}+\frac{3}{\sin^25x}\right)\arcsin\frac{x+3}{x-1} ...
1
vote
0answers
49 views

Understanding derivative notation in those equations

I am given the following set of equations from a physics course, which is about longitudinal waves in rods. My questions are: On the second line you have $ (\frac{\partial \Delta}{\partial x})dx ...
0
votes
1answer
30 views

Question on one-sided derivatives

Assume we have a function $f$, say on $\mathbb{R}$, such that $f$ is continuously differentiable in all $x$ smaller than some given $x_0 \in \mathbb{R}$. I am a bit confused about the connections of ...