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

learn more… | top users | synonyms (2)

0
votes
0answers
18 views

How to calculate the Bouligand derivative (B-derivative)

Let $H(x)=\min (f(x),h(x))$ where $f$ and $h$ are continuously differentiable functions from $\mathbf{R}^n$ to $\mathbf{R}^1$. The Bouligand derivative (B-derivative) $BH(z)$ at $z$ of $H$ is given ...
0
votes
1answer
20 views

Equality of first-order partial derivatives

Let $f(u,v)$ be a "sufficiently good" function of two variables. I need to find sufficient conditions on $f$ such that $$ \frac{\partial f(u,v)}{\partial u}=\frac{\partial f(u,v)}{\partial v},\...
2
votes
4answers
63 views

Are there any pairs of functions where $g(n,x)=f^{(n)}(x)$?

Are there any non-piecewise pairs of functions that satisfy this quality? $g(n,x)=f^{(n)}(x)$ Where $n\in \Bbb{Z}$ and is the $n^{th}$ derivitive of $f(x)$ This is a long shot but I'm just ...
1
vote
2answers
48 views

If $f'(x) = 0$ for every $x \in D$, then $f(x) = k$ for all $x \in D$, even when $D$ is not an interval.

Either give a proof or a counterexample to the following statement: If $f : D \to R$ is a differentiable function and $f'(x) = 0$ for every $x \in D$, then $f(x) = k$ for all $x \in D$, even when $D$ ...
2
votes
2answers
34 views

using L'Hospital solve $\lim_{x \to \infty} x - x^{2}\ln(1 + \frac{1}{x})$

I can't get this to $ = \frac{0}{0}$ form so I can use l'Hospital rule $$\lim_{x \to \infty} x - x^{2}\ln\left(1 + \frac{1}{x}\right)$$ tips? [EDIT] $$\lim_{x \to 0} \frac{1}{x} - \frac{\ln(1 + x)}{...
1
vote
2answers
76 views

Shouldn't l'Hopital's rule work for every limit, not just indeterminate forms?

Why does taking the ratio of $f'(x)$ to $g'(x)$ as $x \to a$ give you the correct limit when $f(a)$ and $g(a)$ $= 0, \infty, -\infty$ , but not for other values of $a$? If the rationale for using ...
1
vote
2answers
43 views

Computer Vision Models 4.3 - Derivative of Summation

I am reading through the Computer Vision: Models, Learning, and Inference book to get an understanding of computer vision. The author describes the high-level steps taken to arrive at one of the ...
1
vote
1answer
26 views

If $f$ is differentiable at point, then error term of linear approximation is continuous in neighbourhood around that point

In this post it is said that if $f : \mathbb R \to \mathbb R$ is differentiable at $a$ then there exists a continuous function $\varphi$ defined on an interval $[-\epsilon, \epsilon]$ such that $\...
3
votes
1answer
34 views

Applications of Derivatives problem

$$f(x) = x^3 + ax^2 + bx + 5\sin^2x $$ is an increasing function on the set $R$. Then $a$ and $b$ satisfy: $a^2 - 3b - 15 > 0$ $a^2 - 3b + 15 > 0$ $a^2 - 3b + 15 < 0$ $ a> 0$ and $b >...
1
vote
2answers
32 views

Differentiability class of Matern function (based on Modified Bessel Function of second kind)

I am working on some techniques using the Matérn covariance function: $h(r) = \frac{2^{1-\nu}}{\Gamma(\nu)}\Bigg(\sqrt{2\nu}\frac{r}{\rho}\Bigg)^\nu K_\nu\Bigg(\sqrt{2\nu}\frac{r}{\rho}\Bigg)$ with $...
3
votes
1answer
19 views

Finding function for capital interest

Haven't fully grasped derivatives and I believe this question really holds the gist of it Your bank account has a continuous capital interest rate of 7%. The formula for this is $$\frac{dB}{dt} = 0....
1
vote
0answers
18 views

Conditions for weak differentiability of composition of $C^1$ real function with weakly time-differentiable $H^1$-valued function

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain and $\mathbb{R} \ni T > 0$. I will abbreviate $X=H^1(\Omega)$ and write $X'$ for its topological dual. Given $$u\in L^2\left(0,T;X \right)$$ ...
0
votes
1answer
64 views

Vector function derivatives for discrete adjoint equation

I'm in the process of deriving a discrete adjoint equation. I'm trying to find the derivative of the vector $\textbf{X}_{1}$ with respect to the vector $\textbf{X}_0$ but I am not able to, $\textbf{...
4
votes
2answers
84 views

100th derivative of $(1-2x)^{2/3}$ at point $x=0$

$$\frac{\mathrm d^{100}}{\mathrm dx^{100}} (1-2x)^{2/3}$$ Without Taylor. I relay don't have any idea how to use General Leibniz rule in this case.
0
votes
2answers
44 views

100st derivative $(\sinh(x)*\cosh(x))^2$ at point $x=0$

$$\frac{\mathrm d^{100}}{\mathrm dx^{100}}(\sinh(x)*\cosh(x))^2$$ Without Taylor I try this :$\sinh(x)'=\cosh(x)'$ but that didn't help in using General Leibniz rule.
7
votes
1answer
62 views

$n^{th}$ derivative of $\cot x$

What is the $n^{th}$ derivative of $\cot(x)$? I tried to differentiate it may times: I can't see a pattern forming. Please help.
1
vote
1answer
27 views

Continuity of the directional derivatives implies continuity at the point ?

This might be a trivial question. Consider a function $f:\mathbb{R^2}\rightarrow \mathbb{R}$ and consider some point $(a,b)\in \mathbb{R^2}$. Suppose we know that all the directional derivatives $D_{...
0
votes
1answer
30 views

Show that $f$ is differentiable at $x=1$.

Let $f$ be a real valued continuous function defined on $[0,2]$ such that $f$ is differentiable at all point except possibly at $1$. Suppose that $\lim_{x\to 1}f^{'}(x)=5.$ Show that $f$ is ...
2
votes
2answers
80 views

Is the composition of the differentiating operator commutative?

First of all, can I check that $d\over dx$ can be considered an operator, or function (as it says in the title)? Is the composition of the differentiating operator commutative? In other words, if $u=...
2
votes
2answers
76 views

100th derivative of $\frac{1+x^2}{1+\tan^2(x)}$ at point 0

$$\frac{\mathrm d^{100}}{\mathrm dx^{100}}\frac{1+x^2}{1+\tan^2(x)}$$ Without Taylor Is there a way to solve this problem by using General Leibniz rule. I tried but numerator make problem.
0
votes
1answer
101 views

Symmetric matrix - Langrange Multiplier

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix. Let $S=\{x \in \mathbb{R}^n \mid \|x\|_2=1\}$ be the unit sphere in $\mathbb{R}^n$ with respect to the $2$-norm $\|x\|_2=\sqrt{\langle{x,x}\...
0
votes
0answers
19 views

Differentiability of multivariable functions.

I have the following two questions: (1) What are the most important techniques to show if a multivariable function is differentiable. (2) I know how to show that a multivariable function is not ...
1
vote
1answer
43 views

Does differentiability on a set imply continuous differentiability on the set? Counterexample?

Of course, differentiability implies continuity, but for a function to be differentiable on a set, say $[a,b]$, then, for the limit to exist, would we not need it to be defined on the set? I hear ...
0
votes
0answers
14 views

differential operator of composition functions

I got confused about the following derivative: $$d(df\circ g)(p)h(p)= d^2f(p)(h(p),g(p))+df(p)\,dg(p)h(p)$$ I can't see the relation between the left and right hand sides!
0
votes
0answers
45 views

Vector Calculus in Curvilinear Coordinates and Index Notation

I am trying to understand how would I use the index notation in curvilinear coordinates. Checking out this reference, I got until this point $$\vec{\nabla} = \sum_a \vec{e}_ah_a^{-1}\partial_a $$ ...
0
votes
2answers
34 views

The Critical Value of $f(x)= x^{6\over 5}-12x^{1 \over 5}$

I need to find the critical number of $$f(x)= x^{6\over 5}-12x^{1 \over 5}$$ Here's what I've tried. $$f'(x) = {6 \over 5}x^{1\over 5} - {12\over 5}x^{-{4 \over 5}}\\ = {6x^{1\over 5} \over 5} - {12 \...
1
vote
3answers
29 views

differentiation of $\operatorname{erfc}(\sqrt{ax})$

I need your help to figure out the derivative of $\operatorname{erfc}(\sqrt{ax})$ with respect to $x$. Based on my knowledge on Wolfram references, they cite that: $$\frac{d \operatorname{erfc}(z)}{dz}...
0
votes
1answer
34 views

Directional derivative of determinant at the identity is the trace of the matrix?

Let $f:A\mapsto \rm{det}(A)$, Prove that $\left(Df\right)_{{\rm id}}\left(H\right)={\rm tr}\left(H\right)$ for all $H\in\mathcal{L}\left(\mathbb{R}^{n}\to\mathbb{R}^{n}\right)$. The question ...
1
vote
1answer
50 views

Deduction of an inequality involving $\int_2^x\frac{dt}{\log t}$ and $\int_2^x\frac{dt}{\log^2 t}$

I try repeat the PROBLEMA 49, see page 149, here in spanish, to obtain an inequality, if it is possible, involving the logarithmic integrals $Li(x)=\int_2^x\frac{dt}{\log t}$ and $Li_2(x)=\int_2^x\...
0
votes
4answers
81 views

Is $x=0$ a local minimum of $f(x)=x^{2/3}(5-x)$?

It seems to be decreasing for values below zero and increasing for values between zero and 2. However $f'(0) = \frac{(-5 (x-2))}{3 x^{1/3}}|_{x=0}$ does not exist. So it's not a local minimum? Btw, ...
0
votes
1answer
9 views

Differentation on time scales

Find $g^\Delta$ from definition where $g(t)=\sqrt{t}$. Definition: $f^\Delta(t)$ is such number, that for every $\varepsilon>0$ neighbourhood $U_t$ of $t$ exists that $|f(\sigma(t))-f(s)-f^\Delta(...
1
vote
1answer
52 views

The rate of change of distance between two airplanes [closed]

An airplane passes over an airport at noon traveling $540$ mi/hr due north. At 1:00pm, another airplane passes over the same airport at the same elevation traveling due east at $580$ mi/hr. Assuming ...
0
votes
0answers
20 views

Why is there no Potential Function of this vector field.

$$F=2xy^2z^2i+2xy^2z^2j+2xy^2z^2k$$ Solve for the potential function. $$\frac{df}{dx}=2xy^2z^2;df=\int_{x}2xy^2z^2=x^2y^2z^2+Q$$ $$\frac{df}{dy}=2xy^2z^2;df=\int_{y}2xy^2z^2=\frac{2}{3}xy^3z^2+Q$$ $$\...
0
votes
2answers
35 views

Calculus: Problem on two continuously differentiable functions [closed]

$f$ and $g$ are two continuously differentiable functions on $[1,\infty]$ with $f(1)=1=g(1)$ and $\frac{{g'\left( x \right)}}{{f'\left( x \right)}} \geqslant f\left( x \right)$. Prove that $2g\left( x ...
1
vote
1answer
25 views

Where did i go wrong in trying to find the intervals where y is increasing and decreasing?

Question: Find the intervals in which the following function is strictly increasing or decreasing: $(x+1)^3(x-3)^3$ The following was my differentiation: $y = (x+1)^3(x-3)^3$ $\frac1y \frac{dy}{dx} ...
0
votes
2answers
21 views

How do you check which intervals a cubic function will increase and in which intervals it will decrease?

I was trying to find the intervals in which the cubic function $4x^3 -6x^2 -72x + 30$ would be strictly increasing and strictly decreasing. I managed to get the fact that at the values {-2,3} the ...
0
votes
3answers
46 views

How can I differentiate $f(x,z(x,y))$ w. r. to x

How can I differentiate $f(x,z(x,y))$ w. r. to x If $z(x,y)=c=\text{constant}$ and $\hat{y}=f(x,c)$ then what is $d\hat y/dx$ If I just differentiate $f$ w.r. to $x$ without knowing whether $z$ is ...
0
votes
0answers
135 views

Showing $x^*$ is a saddle point

Let $f \in C^2(U;\mathbb{R})$, where $U$ is an open subset of a normed space $\mathbb{X}$. Let $x^* \in U$ be a critical point of $f$. Suppose there exists $u^{-}$ and $u^{+} \in \mathbb{X}$ such that ...
4
votes
2answers
40 views

Prove that if $x = \sqrt{a^{\sin^{-1} t}}$ and $y = \sqrt{a^{\cos^{-1}t}}$ then $\frac{dy}{dx}$ = $-\frac{y}x$

Prove: If $x = \sqrt{a^{\sin^{-1} t}}$ and $y = \sqrt{a^{\cos^{-1}t}}$ where $\sin^{-1}$ and $\cos^{-1}$ are inverse trig function, show that $\frac{dy}{dx}$ = $-\frac{y}x$ Unfortunately I don'...
2
votes
1answer
35 views

Derivative with a “mixed” discontinuity

I read that the derivative of a function can never have a "jump" discontinuity, but only essential discontinuity. My question is, can the derivative have a "half essential and half jump" discontinuity,...
1
vote
1answer
49 views

Why can't I change an equation before I differentiate it?

So recently I was reviewing calculus, and I tried to differentiate the equation: $(x^2-y^2)/(x^2+y^2)=1/2$ The first thing I did was make the equation easier to differentiate by multiplying the whole ...
0
votes
1answer
30 views

How to “mix” differentiation under the integral sign with the fundamental theorem of calculus?

The following function appeared before me today, and I don't know how to differentiate it: $$f(t) = \int_0^t h(s,g(t,s))\,{\rm d}s.$$ Assume that all functions involved are $C^\infty$ (or whatever we ...
0
votes
0answers
43 views

Parital derivative of a **scalar** loss function w.r.t. a **row vector** of a matrix

Still struggling the partial derivative of a scalar function w.r.t a row vector of a matrix, what is the way to solve such question, though I learned from here about the partial derivative of a scalar ...
1
vote
2answers
42 views

Uniform limit of a sequence of bounded derivatives is a bounded derivative?

Let $\{f_n\}$ be a sequence of differentiable functions on $\mathbb R$ such that $f_n'$ is bounded for each $n$ ; if $\{f_n'\}$ converges uniformly to $f$ on $\mathbb R$ then is it true that $f$ is ...
1
vote
1answer
54 views

For the differentiation of $x^{\frac23} + y^{\frac23} = a^{\frac23}$, why is the substitution $x = a \cos^3\theta$ legal?

While looking at a solution for finding the derivative of $x^{\frac23} + y^{\frac23} = a^{\frac23}$, the book uses: Let $x = a \cos^3\theta$ and $y = a\sin^3\theta$ However, why would that ...
2
votes
2answers
59 views

A little advice on using the chain rule for differentiation

I must be a little rusty, but how would I evaluate the following: $$ \frac{d}{dr}\left(1-\frac{b(r)}{r}\right)^{-1} $$ My stickler is that $b$ is a function of $r$...
3
votes
1answer
35 views

If $f$ is a real valued function, complex differentiable at $z_0$, then $f'(z_0)=0$

Cannot understand this proof that a real-valued function which is complex differentiable must have derivative at that point equal to zero. I just don't understand how the last statement in bold is ...
1
vote
1answer
24 views

Partial derivative of $f(x,y) = x \arctan\left[\frac{x}{y}\right]$

Can someone help me calculating a partial derivative of the function: $$f(x,y) = \begin{cases} x \arctan\left[\frac{x}{y}\right] & \text{if } y \neq 0 \\ 0 & \text{if } y = 0 \end{cases}$$ ...
0
votes
2answers
58 views

Is the right-hand derivative equal to the right-hand limit of the derivative?

Let $f(x)$ be a function on the interval $[a,b]$ which is differentiable on $(a,b)$. Is it true that $$f'_+(a)=\displaystyle\lim_{x\to a^+}f'(x)$$ if both limits exist? Darboux's theorem seems to ...
1
vote
0answers
38 views

Complex analysis Derivatives

Are the derivative rules the same?I mean i have seen the definition etc it is the same but i cant stop thinking C as $R^2$ and functions as Vector Fields .So all i can think is Partial derivatives and ...