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

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0
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0answers
27 views

Using Frobenius' Theorem for 3 functions in 2 variables [closed]

i) 1) $v= \frac{\partial u}{\partial x}$ 2) $w= \frac{\partial u}{\partial t}$ 3) $\frac{\partial v}{\partial t}= \frac{\partial w}{\partial x}$ 4) $\frac{\partial v}{\partial x}= \frac{\partial ...
1
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1answer
23 views

Showing that $F^*(L_{\mathbb{Y}}\omega)=L_{\mathbb{X}}(F^*\omega)$

Let $F:U \rightarrow V$ be a diffeomorphism between open sets in $\mathbb{R}^n$. Let $\mathbb{Y}$ be a vector field on $V$ and $\omega$ a $k$-form on $V$. Show that ...
0
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0answers
16 views

Borel sets and absolutely continuous functions - second part

Borel sets and absolutely continuous functions This question is a part of the question of this link. So, in order to show that $F'=0$ on set of positive measure, what I did was mentioned here: Let ...
2
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2answers
33 views

What's the rule for knowns and unknowns when dealing with derivatives

So a rule of thumb when doing basic algebra is you must have as many equations as you have unknowns. For example: $0=4x+6y^2$ $3x=2\sqrt{y}$ You have two equations and two unknowns and thus can ...
1
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2answers
65 views

For a $C^1$ function, the difference $|{g'(c)} - {{g(d)-g(c)} \over {d-c}} |$ is small when $|d-c|$ is small

Suppose $g\in C^1 [a,b]$. Prove that for all $\epsilon > 0$, there is $\delta > 0$ such that $|{g'(c)} - {{g(d)-g(c)} \over {d-c}} |{< \epsilon }$ for all points $c,d \in [a,b]$ with $0 ...
2
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0answers
45 views

Directional derivative (Vector)

Given $f:\mathbb{R}^2 \to \mathbb{R}^2$ is a map $f(x,y)=(u(x,y),v(x,y))$ and $\alpha=(\alpha_1,\alpha_2)$ is a point, then how does one show that $f$ is differentiable (or not) in the direction ...
1
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4answers
51 views

Applying math knowledge [closed]

Currently I'm in the middle of my first year of college studying informatics engineering. I was never great at math, but if I put some effort, I understand it and constantly get good grades. However, ...
1
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1answer
27 views

Show that $i_Yi_Xd\omega=d\omega(X,Y)$ for $\omega$ a $1$-form

If $\omega$ is a $1$-form, how does $i_Yi_Xd\omega=d\omega(X,Y)$? I get that $d\omega$ is a 2-form. So $i_X(d\omega)=d\omega(X,v_{2})$. So how do we proceed? I dont see how the step ...
1
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2answers
46 views

Differentiation under the integral sign, please help

Find the partial derivatives of the function: $$\int_{x^2e^{5y}}^{\ln(x^3-2)}\cos(t^2)dt$$ Maple responds: $$-2\,\cos \left( {x}^{4} \left( {{\rm e}^{5\,y}} \right) ^{2} \right) x {{\rm ...
1
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3answers
54 views

Economically computing $d\beta$

$\displaystyle \beta = z\frac{x dy \wedge dz + y dz \wedge dx + z dx \wedge dy}{(x^2+y^2+z^2)^{2}}$ Show that $d\beta=0$. So, let $r=x^2+y^2+z^2$, $\begin{align} \displaystyle d\beta &= ...
0
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1answer
35 views

Two statements about one-sided derivative and monotony

The statement 1 is: $f\colon [a,b]\to\mathbb R$,continuous on $[a,b]$,$f'_-(x)$ exists and is $\le0$ for all $(a,b]$.Can we infer that f is non-increasing on $[a,b]$? My attempt is: Assume $f$ is not ...
2
votes
4answers
64 views

Taking the derivative of $(1+x^2)^{(\sqrt{x})}$

As stated above, I'm having trouble taking the derivative of $(1+x^2)^{(\sqrt{x})}$. I know that I should somehow be using the exponential derivative form of $\dfrac{d}{dx} ( a^x ) = a^x\ln(a)$, but I ...
0
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2answers
36 views

differentiation of $g(x) = \lvert f(x)\rvert$ where $f(x)$ and $D(f(x)) = 0$

I'm really stumped on this problem and don't know how to go about it. It says $g(x)$ = $|f(x)|$ and to show that if $f(c) = 0$ and g is differentiable at c, then one must have $D(f)(c) = 0$. ...
1
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2answers
40 views

A limit question involving power of positive numbers

I'm trying compute the following limit: $$\lim_{t\to0}\left(\frac{1}{t+1}\cdot\frac{b^{t+1}-a^{t+1}}{b-a}\right)^{1/t},\quad b>a>0.$$ I know $\displaystyle\lim_{t\to0}(1+t)^{1/t}=e$ and ...
0
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1answer
21 views

Derivative of a function containing indicator function?

Consider $\delta\in \mathbb{R}$ and $X \in \mathbb{R}$. Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a map defined as $$ f(X; \delta):=\delta*1\{X\geq 0\}+X $$ where $1\{X\geq 0\}$ is $1$ if $X \geq ...
1
vote
1answer
36 views

Tangent Space: Identifications

Given a manifold $M$. Denote a chart by $\kappa$. Introduce the directional derivative: $$\partial:\mathbb{R}^n_a\to T_a\mathbb{R}^n:v\mapsto\partial_v\rvert_a$$ That is an isomorphism with inverse ...
0
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2answers
28 views

Assume $f$ is differentiable on all of $\mathbb{R}$, $f(0) = 5$ and $\forall x, f’(x) \neq 0$. Prove $f(x) \neq 5, \forall x \in \mathbb{R}$

The problem with my solution is that it only works for small values of h. How would I account for large values of h?
0
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4answers
27 views

First Derivative vs Second Derivative in relation to minima and maxima

I am struggling with understanding the difference between them and I need to write about them intuitively. The way my teacher explained it, the sign of the first derivative is used to determine if ...
1
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2answers
23 views

Find the derivative of a piecewise function.

I would just like to know if my proof here is valid. I know I left out some computational details, but I'm more concerned about the structure of the proof than those details.
0
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3answers
48 views

Find the derivative of $F(x) = x^7 \ln(x^3 e^{3x^2 -8})$

Find $F'(x)$ for $$F(x) = x^7 \ln(x^3 e^{3x^2 -8})$$ Here is what I have so far: $$(7x^6)(3x^2)$$
2
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2answers
28 views

Partial differentiation of a integral

If I have: $$f(x,y)=\int_{x}^{y}e^{-t^2}dt$$ To calculate that, I change the t to y and x to get: $$e^{-y^2}(e^{-y^2})'-e^{-x^2}(e^{-x^2})$$ With the differentials being in respect to x and y for each ...
0
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2answers
50 views

Seamlessly connect a sine curve and a parabola

I want to seamlessly connect an unknown parabola to a known sine wave. The equations are: s(x) = a sin(bx + c) p(x) = Ax^2 + Bx + C I want to draw ...
1
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0answers
24 views

I need help with a partial derivative

I was given a function and I need to find a partial derivative of it. The result I got is different from the answer, and I don't know why. Here's the function: $$sin(\theta_{a}) = ...
0
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0answers
20 views

Applications of the Lebesgue differentiation theorem

The Lebesgue differentiation theorem states that for every locally integrable function real valued $f$ defined on $R$, the set of Lebesgue points of $f$ has full measure. I understand quite well what ...
3
votes
3answers
91 views

A concave positive function on $[1,\infty)$ is uniformly continuous

Let $f$ be a concave positive function on $[1,\infty)$, then $f$ is uniformly continuous on $[1,\infty)$. This was a true or false problem that I couldn't prove to be true, so I'm thinking that maybe ...
1
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0answers
39 views

Poincaré Lemma problems and computing contractions in an economical way

Let $x=(A,B,C,D)$ be coordinates on $\mathbb{R}^4$. $\displaystyle \beta = \frac{(AdB-BdA)\wedge(dC \wedge dD)+(dA \wedge dB)\wedge(CdD-DdC)}{(A^2+B^2+C^2+D^2)^2}$ I would like to compute ...
6
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4answers
137 views

Why is the derivative of $x^2$ not $2x+1$?

If the derivative is the change of the function at each step, it could be expressed as: $$f(x)+f'(x)=f(x+1)$$ Therefore if $f(x)=c$ $$c+f'(x)=c \implies f'(x)=0$$ This is also correct for ...
1
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2answers
50 views

Evaluating $\frac{d}{dx}\int_1^{x^2}\sqrt{y^2+3}dy $

How to Evaluate $$\frac{d}{dx}\int_1^{x^2}\sqrt{y^2+3}dy $$On doing differentiation of integration we obtain the same function but here limit of integration is also to be applied .How can we do it ...
1
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2answers
23 views

Evaluating $\frac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt$

$$\dfrac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt$$ The answer I come up with is: $5\sin(3x)(3)-7e^{4(3x)}(3)$, however this was not on the answer choice. What is the ...
3
votes
1answer
63 views

Real roots of $p(x)=x^n+ax+b$

What can we say about the real roots of $p(x)$? My Work: If $n$ is odd I found that $p$ has at most $3$ real roots if $a<0$ and $p$ has at most $1$ real root if $a\geq 0$. How can I classify the ...
2
votes
1answer
43 views

Evaluating derivative of $\int^{3x}_{2x} \sin(t^3 + 1) \,\mathrm dt$

Maybe I'm not very good at my trig rules but I'm having a tough time finding derivative of $$\int^{3x}_{2x} \sin(t^3 + 1) \,\mathrm dt$$ I believe that $u = t^3 + 1$ and $du = 3t^2$, but I'm not ...
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2answers
63 views

Differentiability proof of exponential function $\sum_{n=0}^ \infty \frac{x^n}{n!}$

$$f(x)=\sum_{n=0}^ \infty \frac{x^n}{n!}$$ I want to prove that $f$ differentiable on $x$ in $[0,1]$. I am not clear with using the definition of differentiability. I can prove it is ...
1
vote
1answer
49 views

Evaluating $\frac{d}{dx}\int_{7-2x}^3 \frac{u^3}{1+u^2}du$

Evaluate following expression by using The Fundamental Theorem of Calculus $$\frac{d}{dx}\int_{7-2x}^3 \frac{u^3}{1+u^2}du$$
0
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2answers
22 views

Simple Integration Question - Integral of the product of a function and its derivative.

Why does $\int y'(x) y(x) dx = y(x)^2/2 + C$? This seems to be true, at least according to Wolfram, but I do not understand how it is derived.
-1
votes
0answers
13 views

Tangent line parallel to a chord in a parabola

In a parabola with equation $f(x) = ax^2+bx+c$, a chord $AB$, $A=(xa,f(xa))$ and $B = (xb,f(xb))$, is parallel to the tangent line at $x = \frac{xa+xb}{2}$. It's easy to verify that using ...
3
votes
3answers
103 views

$dy$ by $dx$ or $dy$ divided by $dx$

I was always taught do not say "$dy$ divided by $dx$", instead "$dy$ by $dx$" because it's not really dividing. I then studied differentiation from first principles, where one takes two points on a ...
4
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0answers
107 views

Use of Poincare Lemma in solving $\nabla \times \textbf{A}(\textbf{r})=\frac{\textbf{r}}{r^3}$

Let $U = \mathbb{R}^3 \setminus \{(0,0,z) \}$ (ie $\mathbb{R}^3$ with the $z$-axis removed ) and consider $\beta$ on $U$ given by $\displaystyle \beta = \frac{x dy \wedge dz + y dz ...
0
votes
1answer
34 views

Finding arc length by approximating

Let's say that a have a smooth curve in 2D and I want to find it's lenght. I split the curve in sections like here: http://www.whitman.edu/mathematics/calculus_online/section10.03.html. The only ...
2
votes
1answer
19 views

Recursive functions.

If you have a recursive function $$g(x) = f(f(x))$$ and you know that $$f(0) = 0, f'(0) = 1, f''(0) = 2$$ Will then $$g(0) = 0, g'(0) = 1, g''(0) = 2$$ ?
0
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1answer
20 views

Use MVT to prove $(k+1) \cos \left ( {\frac{\pi}{k+1}} \right ) - k \cos \left( \frac{\pi}{k} \right) > 1 \qquad \forall k \geq 2 $

I want to prove $$(k+1) \cos \bigg ( {\frac{\pi}{k+1}} \bigg ) - k \cos \bigg ( \frac{\pi}{k} \bigg ) > 1 \qquad \forall k \geq 2 $$ using Mean Value Theorem. My thought was to apply MVT to $f(x) = ...
1
vote
0answers
38 views

When the function f is constant?

Let $ f: \mathbb R^n \to \mathbb R^n$ be a differential function. Let $Df(x)$ be the derivative of $f$ at $x$ in $\mathbb R^n$. Then which of the following is/ or correct? $Df(0)(u) = 0$ for all $u$ ...
0
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1answer
61 views

Identity concerning Lie derivative of $k$-form $\omega$

Let $X$ and $Y$ be vector fields on $\mathbb{R}^n$. Show that for $\omega$, a $k$-form on $\mathbb{R}^n$, $(L_XL_Y-L_YL_X)\omega=L_{[X,Y]}\omega $. I try using Cartan's magic formula and get that ...
1
vote
0answers
61 views

Identity of the pushforward of a vector field using a Jacobi bracket.

Let $Z(u,v)$ be the vector field $Z(u,v)=(u^2+u,v^2+v)$, let $\Gamma_t$ denote its flow. I have shown that $[X,Z]=Z-X$. Show that $(\Gamma_t)_*X=e^{-t}X-(e^{-t}-1)Z$. Could someone please show me ...
0
votes
0answers
23 views

Derivative with constraint

Consider function $F=F(q(t),p(t))$ with constraint $p(t)=q'(t)$, where $'$ denotes time derivative. Let $\displaystyle G=F'=\frac{\partial F}{\partial q}q'+\frac{\partial F}{\partial p}p'$, I want to ...
1
vote
1answer
34 views

is there a way to simplify $x^{2} ( v' (x^{n})' )$?

so I have what is probably an algebra question, if I have $x^{2} ( v' (x^{n})' )$ where the ' denotes a derivative, is there a way to simplify this expression?
2
votes
1answer
50 views

use fundamental theorem of calculus to find a function $f(x)$ and a number $a$

I thought I understood the fundamental theorem of calculus but I'm confused on the following problem.. Use the Fundamental Theorem of Calculus to find a function $f(x)$ and a number $a$ so that ...
4
votes
2answers
69 views

There is a function which is continuous but not differentiable

I have a function which is a convergent series: $$f(x) = \sin(x) + \frac{1}{10}\sin(10x) + \frac{1}{100}\sin(100x) + \cdots \frac{1}{10^n}\sin(10^nx)$$ This function is convergent because for any E ...
0
votes
1answer
29 views

How can we find another path

Can you help me about this question please, Thank you..
2
votes
4answers
36 views

How can I differentiate this equation?

I need to differentiate this: $$ y = b(e^{ax}-e^{-ax}) $$ I've got the solution from a book, but I don't found the process to differentiate it. The solution is: $$ y = ab(e^{ax}+e^{-ax}) $$ Here ...
1
vote
1answer
32 views

If $f(z)$ is an entire function, prove that it has a zero at $z_0$ of order $k\ge 1$ iff $z_0$ is a simple pole of $\frac{f'(z)}{f(z)}$

Let $f(z)$ be an entire function. Prove that $f(z)$ has a zero at $z_0$ of order $k\ge 1$ iff $z_0$ is a simple pole of $\frac{f'(z)}{f(z)}$ and the residue of $\frac{f'(z)}{f(z)}$ at $z_0$ is $k$. ...