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

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2
votes
1answer
41 views

Explain the minus sign in the following formula.

I just read that: If $z=f(x,y)=c$, be the equation of a curve, then the slope of the tangent to the curve at any point (x,y), is given by $$m=\frac {dy}{dx}=-\frac{\frac{\partial z}{\partial ...
2
votes
4answers
129 views

When can I say that $f(x) \gt g(x) \implies f'(x) \gt g'(x)$?

Are there cases when this relation holds? $$f(x) \gt g(x) \implies f'(x) \gt g'(x)$$ I.e. what are the conditions on $f(x)$ and $g(x)$ for that to be true? Is it even possible to determine them? In ...
0
votes
1answer
38 views

If a differentiable function has bounded derivative, Must it be that its derivative continuous?

I got this question: Let $f$ be a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, If $f'$ is bounded on $(a,b)$, Must it be the case that $f'$ is ...
4
votes
1answer
60 views

The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
4
votes
1answer
38 views

Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
3
votes
4answers
64 views

Lines tangent to parabola at point.

I'm struggling to figure out what I'm exactly required to do. The problem states "Compute which lines through the point $(1, 0)$ that are tangent to the parabola defined by $y = x^2$." I believe ...
1
vote
2answers
52 views

What does d f(t,x) = 0 mean?

A differential equation that can be written in the form $d\phi(t, x) = 0$ for some continuous and differentiable function $\phi(t, x)$ is called exact. What does $d\phi(t, x) = 0$ mean?
2
votes
2answers
31 views

Second derivative of $\frac{\ln t}{\sqrt t}$ and derivative of $\arccos(1-2x^2)$

$f(t)=\dfrac{\ln t}{\sqrt t}$ I'm stuck on the algebra of finding the second derivative. For the first derivative, I got: $f'(t)=\dfrac{t^{\frac{-1}{2}}(1-\frac{1}{2}\ln t)}{t^2}$ For the second ...
0
votes
2answers
62 views

Proof that energy of a free body is constant, using the derivate

Ok, what I'm trying to prove is the law of conservation of energy for a free fall. Let the downward direction be positive. We want to prove that: $$mgh+\frac{mv^2}{2}=constant$$ For this, we try to ...
0
votes
1answer
36 views

prove that a function whose derivative is bounded also bounded

I got this problem: Let $f$ be a differentiable function on an open interval $(a,b)$ such that $f'$ (the derivative of $f$) is bounded on $(a,b)$ (meaning there exist $0<M$ such that $\forall ...
2
votes
0answers
23 views

Convexity in each argument and directional derivative

Let $f(x,y)$ be a continuous function, convex in each argument separately. Does this imply the existence of one-sided directional derivatives in any direction? For example, does there exist (finite or ...
1
vote
2answers
58 views

Finding $\frac{d}{dx} y^x$

$$\frac{d}{dx} y^x$$ How would you find the derivative with respect to $x$ of $y^x$ assuming that $y$ is a function of $x$? I know you will have to use the chain rule somehow, and I know that the ...
0
votes
1answer
14 views

Derivative of rigid motion like reflection?

Is it possible to define a derivative for rigid transformations eg. reflection and translation? I am especially interested on reflections shortly $\sigma$. Because I am trying to relate ...
5
votes
1answer
182 views

Pointwise boundedness is uniform for a sequence of derivatives

Let $f \in C^\infty ([a, b])$ be an infinitely differentiable function defined on a closed interval $[a, b]$ with the following property: for any $x \in [a, b]$ the sequence $|f^{(n)}(x)|$ is bounded, ...
0
votes
0answers
35 views

Maximizing sum with a constraint

Given the function $$ f(\alpha_{1},\ldots,\alpha_{k})=C\sum_{i=1}^k \alpha_i e^{-(b^2/d)\alpha_i}\text{ with } C>0,\ b>0,\ d>0,\ \forall i\in\{1,\ldots,k\}:\alpha_{i}\ge 0 $$ with the ...
6
votes
1answer
266 views

Can monsters of real analysis be tamed in this way?

Consider the Weierstrass Function (somewhat generalized for arbitrary wavelengths $\,\lambda > 0$ ): $$ W(x) = \sum_{n=1}^\infty \frac{\sin\left(n^2\,2\pi/\lambda\,x\right)}{n^2} $$ $W(x)$ is an ...
0
votes
1answer
42 views

How can I derive this summation?

I have the following equation, $$ K_r=\left ( \frac{P}{RT} \right )^{v}exp \left \{ \sum_{s}\left [ (\beta_{s,r}-\alpha_{s,r}) \left \langle \frac{h_s}{RT}-\frac{s_s}{R}\right \rangle \right ] ...
2
votes
1answer
82 views

How to derive $\frac{d}{dx}\left(x+1\right)^{\sin\left(x\right)}$

I need help to find derivative of: $\frac{d}{dx}(x+1)^{\sin x}$ i tried to do something like this.. $$(x+1)^{\sin x}\cdot \ln\left(x+1\right)=\sin x(x+1)^{\sin\left(x\right)-1}\cdot ...
1
vote
1answer
85 views

Why continuity at a point one of Dini derivatives implies the continuity at this point others Dini derivatives?

Let $f:[a,b] \rightarrow \mathbb R$ be a continuous function and $x_0\in (a,b)$. How to prove that if the Dini derivatives $D^+f(x_0)$ is finite and continuous at $x_0$ then also $D_+f(x_0)$ is ...
5
votes
1answer
127 views

How to show that $e^x$ is differentiable?

I tried to search for a few minutes but I didn't find this question so I hope it's not a duplicate. So I want to show that $(e^x)' = e^x$. To do that, I must proof that the limit: ...
10
votes
5answers
245 views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
1
vote
0answers
27 views

Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
9
votes
1answer
256 views

Find compressed form for cumbersome calculation

Given the three functions $u^{\mathrm{(I)}}(t)\;=t \left(t^2\right)^{k}\,e^{2\beta t^2},\\ u^{\mathrm{(II)}}(t)=\sqrt{\left(t^2\right)^{2k}-\left(t^2\right)^{2k+1}}\,e^{2\beta t^2},\\ ...
0
votes
0answers
14 views

Maximum Principle - Proof

We want to show the maximum principle for a function $f = f(x,t)$ on a n-dimensional hypersurface $M,$ that is, (Corollary) Let $f = f(X,t)$ be a function on M, let $\vec{a}$ be a vector field on ...
1
vote
1answer
39 views

Question about limits and Mean Value Theorem

Let $f:(a,b) \rightarrow \mathbb{R}$ and $g:(a,b) \rightarrow \mathbb{R}$ be differentiable on (a,b) with $g'(x) \neq 0$ for all $x$ in $(a,b)$. Suppose $\lim_{x \to b-}\dfrac{f'(x)}{g'(x)}$ ...
1
vote
2answers
50 views

Simplifying derivative with exponential involved

I have the equation: $$f(x) = (5x^2 - 17)e^{-0.5x}$$ I have differentiated it to this so far: $$f'(x)=10xe^{-0.5x} + (5x^2 -17)(-0.5e^{-0.5x})$$ This is using the chain rule and product rule of ...
4
votes
2answers
45 views

Evaluating a limit (involving derivative)

Evaluate the following limit: $$\mathop {\lim }\limits_{x \to 1} {\left( {{{f(x)} \over {f(1)}}} \right)^{{1 \over {\log x}}}}$$ My work: $$\eqalign{ & \mathop {\lim }\limits_{x \to 1} f(x) = ...
0
votes
1answer
68 views

How to simplify $\ln^2\left(x\right)+2 \ln x-3$

I dont know how to simplify $\ln^2\left(x\right)+2 \ln x-3$ I dont know how to get $(\ln(x)+1)(\ln(x)+3)$ But I am stuck and don't really know how to do that. I tried something like this: $2\ln ...
1
vote
1answer
41 views

Dini Derivative

Let $f$ be defined on $\mathbb{R}$ such that $$ f(x) = \begin{cases} |x|, & \text{if }x \in \mathbb{Q} \\ |2x|, & \text{if }x \notin \mathbb{Q} \end{cases} $$ Calculate ...
1
vote
2answers
41 views

Constructing a matrix that computes derivatives

Consider the subset of functions given by $S = \text{Span}(e^{2t}\, \sin\, 3t, e^{2t}\, \cos\, 3t)$: Show that the derivatives of $e^{2t}\, \sin\, 3t$ and $e^{2t}\, \cos\, 3t$ are also in $S$ and ...
2
votes
2answers
45 views

Calculating the derivative in terms of $y(x)$

It has been a few years since I have taken calculus, and I must have forgotten how to do this. I need to find the derivative of the Sigmoid function in terms of $y(x)$. I have found this page which ...
0
votes
1answer
19 views

Find the absolute minimum and absolute maximum values

$$f(x) = x - (1/x) ; [3,1]$$ What is the first and second derivative of this function? I think I found the first derivative $f'(x) = (x-1)(x+1)/x$
0
votes
0answers
26 views

Gradient; how to do this?

I want to do this gradient, but I just don't get the right result: $\phi: \mathbb{R}^3 \rightarrow \mathbb{R}$ and $F(Y) = - q \ \text{grad}\phi(Y) = \frac{1}{4 \pi \varepsilon_0} ...
-2
votes
1answer
61 views

Finding the second derivative of a first derivative.

so I have a $ \frac{ dy}{dx}=\frac{-2x+y}{2y-x} $ from the original equation $ \ x^2+y^2-xy=1 \ $. I have to find the second derivative given a coordinate point $\left(\sqrt{\frac{1}{3}},2\sqrt{ ...
1
vote
1answer
61 views

Frechet/Gateaux differentiability of an integral operator L^2 --> R

Let $f: R \rightarrow R$ be a continuously differentiable function on the real numbers (if needed also infinitely many often differentiable). Define the Operator $F : L^2([0,1]) \rightarrow R$ for $x ...
1
vote
1answer
47 views

derivative of $y=(x^2+x^3)^4$

I can't figure out where I am going wrong. $$y=(x^2+x^3)^4$$ chain rule it first $$4(x^2+x^3)^3* \frac d{dx}(x^2+x^3)$$ which should become: $$4(x^2+x^3)^3(2x+3x^2)$$ factoring out should give me: ...
1
vote
2answers
49 views

total least squares derivation with matrices

Taken from a computer vision book: "to minimize the sum of the perpendicular distances between points and lines, we need to minimize $$ \sum_i (ax_i + by_i +c)^2$$ subject to $a^2 +b^2 =1$. Now using ...
0
votes
2answers
45 views

Why $\dfrac{d}{dt} \dfrac{dy}{dx} = \dfrac{d}{dx} [ \dfrac{dy}{dx} ] \quad \dfrac{dx}{dt} $ ? [Stewart P206 3.4.95, BDP P165 3.3.34]

If $y=f(x)$, and $x = u(t)$ is a new independent variable, where $f$ and $u$ are twice differentiable functions, what's $\dfrac{d^{2}y}{dt^{2}} $? By the chain rule, $\dfrac{dy}{dt} = \dfrac{dy}{dx} ...
0
votes
1answer
33 views

Formula for area under the curve

I don't know that the equation that I am going to explain below is correct or not, and this is why I am asking this question. So, I have found out that area under the curve could be found out by ...
0
votes
1answer
32 views

continuous partial derivative implies total differentiable (check)

Let $f: \mathbb{R^n}\rightarrow \mathbb{R}$ to have continuous partial derivatives, it suffice to show that $f$ is total differentiable at $(0,..,0)$ with $f_{x_i}(0,..,0) = 0$. Since each partial ...
2
votes
1answer
37 views

right derivative of a continuous function

Let $f:(a,b)\longrightarrow \mathbb{R}$ be continuous. Suppose $D_+f(x)=\lim_{h\to 0+}\frac{f(x+h)-f(x)}{h}\geq 0$ for any $x\in (a,b)$. Prove that $f(x_1)\geq f(x_0)$ whenever $x_1\geq x_0$. How to ...
13
votes
2answers
213 views

How prove that there exists $\xi\in(a,b)$ with $f'(\xi)=\frac{f(\xi)-f(a)}{b-a}$

Let $f(x)$ be continuous on $[a,b]$, differentiable on $(a,b)$, and with some $c\in(a,b)$ such that $f'(c)=0$. Show: There exists $\xi\in(a,b)$ such that $$ f'(\xi)=\dfrac{f(\xi)-f(a)}{b-a} $$ ...
0
votes
1answer
65 views

Derivative of $\frac{x}{(1+x^2}$ using the limit definition.

I just started learning Calculus on my own and understand where $\lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$ comes from but I'm having trouble with this one, I think my Algebra skills are letting ...
1
vote
3answers
99 views

Why is $\cos(x)$ the derivative of $\sin(x)$?

The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$. Is there a simple proof of this, preferably using pictures?
1
vote
1answer
22 views

Is there any trick you can use to derive f( h(x),x)

I was just wondering, is there a way to derive $ \frac{d}{dx} f( h(x),x)$ without knowing how the function looks? For example by some trick of using multivariable diferentiation of $f(h(x),y)$? Thank ...
0
votes
1answer
16 views

Rate of change, square area by time

The question describes a square with sides defined by $s = 2+t²$, where $t = time$. It asks to define the variation rate of the area when $t = 2$ My result: $ t = time $, $ s = side $, $ A = area $ ...
0
votes
1answer
32 views

$Df(x_0)$ is one-to-one. show $f$ is one-to-one on a neighborhood of $x_0$

Suppose $f:\mathbb{R}^n \to \mathbb{R}^m$ is $C^1$ and $Df(x_0)$ is one-to-one. Show $f$ is one-to-one on a neighborhood of $x_0$. I think it's about inverse function theorem. but i cannot prove ...
0
votes
1answer
24 views

Prove that function f has a local minima and maxima

$f:R->R, f(x) = (x^2+mx)e^-x$ Show that, for every m in R, the function f has a local ...
1
vote
0answers
37 views

Proof of pointwise convergence of derivative of power series

I proved: If $\sum_{n=0}^\infty a_n x^n$ converges for all $x \in (-R,R)$ then the differentiated series $\sum_{n=1}^\infty na_n x^{n-1}$ converges for all $x \in (-R,R)$. Please could somebody tell ...
1
vote
0answers
38 views

Meaning of this differentiation operators

I have been just reading this paper here: paper and was wondering how they carry out the differentiation in (4.9). In principle, this should be just the differentiation of 4.8 with the help of 4.7a. ...