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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1answer
21 views

How to calculate this derivative $D^{\alpha}f(x)$?

Let $v\in\mathbb{R}^n$ be a fixed vector, and $f$ a function given by $f(x)=\cos(x\bullet v)$, where $x\bullet y$ is the dot product. What is the derivative $D^{\alpha}f(x)$ for $x\in\mathbb{R}^n$ ...
0
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2answers
28 views

Calculus: simpler way of showing that derivative is negative?

I want to show that $\frac{1-(1-\beta)^N}{\beta}$ is strictly decreasing in $\beta$ for $\beta \in (0,1)$ and $N \geq 2$. My approach so far is as follows: I take the derivative with respect to ...
0
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2answers
18 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ I get an answer of $\frac ...
2
votes
0answers
14 views

Existence of differentiable functions on $\mathbb R$ whose derivative is constant on the complement of uncountable set but not everywhere

Let $ A $ be a countable subset of the set of real numbers and $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f'$ is constant on $\mathbb R \setminus A$ , then I know that $f'$ is ...
0
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0answers
8 views

How to drive the membership function in fuzzy clustering nean

I am learning about Fuzzy c-means (FCM) which is a method of clustering which allows one piece of data to belong to two or more clusters. This method (developed by Dunn in 1973 ) is frequently used in ...
1
vote
0answers
10 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
1
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0answers
24 views

How to take derivative of $F(u)=\sum_{i=1}^{N} \int f^2(x) u_i^q(x) dx $

I have to find the derivative of a function. Could you help me to find it $$F(u)=\sum_{i=1}^{N} \int_{\Omega} f^2(x) u_i^q(x) dx $$ where $q \ge 1$, $f(x): \Omega \to R$, $u_i$ is membership ...
1
vote
2answers
39 views

Find maximum of a function

I want to find the maximum of a function. $$ d = \frac{35}{3} + \frac{7}{3}\sin( \frac{2\pi}{365}t ) $$ I don't know if I applied the chain rule correctly. $$ w = \frac{2\pi}{365}t $$ $$ w' = ...
1
vote
3answers
49 views

Slopes of inverse functions

I have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ...
0
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2answers
43 views

Lebesgue differentiation

Some days back I was doing the lebsegue integration. I was amazed by the integration's ability to maximize the potential of Reinman integration. Are there any new differentiation (except the metric ...
1
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3answers
23 views

Constructing Polynomial Function from Set of Points and Slopes

I only have a basic knowledge of calculus but I would like to know if it's possible to, given a set of points each with their own slopes, construct the simplest (or any) polynomial function that ...
11
votes
3answers
938 views

Are polynomials infinitely many times differentiable?

Are polynomials infinitely many times differentiable? If so, does it only mean that at some point we reach 0 and then we keep on getting 0? Thank you!
-1
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2answers
119 views

If $f(x+y^3)=f(x)+[f(y)]^3$ and $f'(0)\ge0$, what can $f(10)$ be? [on hold]

A real valued function satisfies the condition: $f(x+y^3)=f(x)+[f(y)]^3$ for all real $x$, $y$. If $f'(0)\ge0$ how to find $f(10)$ ?
2
votes
4answers
64 views

Derivative of the function $y = 2^{\sqrt{\tan x}}$

How to find derivative of the following function: $y = 2^{\sqrt{\tan x}}$ , $y' = ?$ I did the following $$\frac{d}{dx}2^{\sqrt{\tan x}} = 2^{\sqrt{\tan x}}\ln{2}(\sqrt{\tan}x)'$$ and stopped here. ...
-2
votes
5answers
90 views

How to find $f(\frac{1}{\sqrt3})$ and $f'(1)$? [on hold]

It is given that $$f(x)+f(y)=f\left(\frac { x+y }{ 1-xy } \right)$$ for real values of $x$ and $y$ $(xy \neq 1)$ and $$\lim _{ x\to0 }{ \frac { f(x) }{ x } }=2$$ How do we find ...
3
votes
1answer
51 views

Derivative by Definition of $\frac{\sin^2(x)}{e^x-1}$

I have to prove the derivative by definition of $$\frac{\sin^2(x)}{e^x-1}$$ $$f^{\prime}(x)=\lim_{\Delta x \to 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}}$$ $$\large f^{\prime}(x)=\lim_{\Delta x \to ...
0
votes
1answer
40 views

Derivative of a trigonometric function

What is the derivative of $$\cos^2 a (\tan a - \tan b)$$ Please anyone explain in detail. The differentiation is with respect to $a$. I tried to obtain the answer using chain rule, but didn't get it. ...
0
votes
1answer
16 views

Estimating the size of the difference function from the derivative

Suppose that $f(x)$ is a function defined in the interval $[1,N]$ for some integer $N$. We know that the derivative $f'(x)\approx L$ for some constant $L$. I want to express the function ...
-4
votes
1answer
38 views

Chain rule differentiation [on hold]

Can any one show me the steps to differentiate $v^2$ according to chain rule? Why is derivative of $v^2$ found out by chain rule and not by exponent formula?
-2
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1answer
32 views

Positive derivative on [0,1] implies a continuous derivative on [0,1]

If a real-valued function F defined on [0,1] is differentiable with positive derivative f everywhere on [0,1], can we conclude that f is continuous?
-1
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0answers
23 views

Continuously differentiable operator

if i consider the operator $A$ defined on $H^1_0$ by $$Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$$ where $$ G(t,s)=\begin{cases} t(1-s),~~t\leq s\\s(1-t),~~s\leq t\end{cases}$$ What is the expretion of $A'u$ ...
2
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3answers
59 views

derivative integral $\int_0^{x^2} \sin(t^2)dt$

I want to know how I derivative this integral: $$\int_0^{x^2} \sin(t^2)dt$$ what are the steps to derivative it?
0
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3answers
66 views

n-th derivative test.

Let $f(x)$ be a function such that it is $n$ times differentiable and $f^{'}(a)=f^{''}=(a)f^{'''}=(a)....=f^{n-1}(a)=0$ and $f^{n}\ne0.$ The $n^{th}$ derivative test tells us about the concavity of ...
1
vote
2answers
31 views

What is the cosine of angle of intersection of following functions?

1st Function: $\displaystyle 3^{x-1}\log x$ 2nd Function: $\displaystyle x^x-1$ How to find the cosine of angle of intersection of these two curves? Their $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ are not ...
1
vote
3answers
40 views

Multiplicity of a root of a polynomial

:) It's true that, if a polynomial has a root (let's say, k, for example) with multiplicity n (n>1, for n integer), then it's true that the derivate polynomial have k as a root with multiplicity ...
4
votes
1answer
39 views

If a function is left- and right-differentiable everywhere, how much can the one-sided derivatives disagree?

Just for fun, I was proving some results about convex functions the other day. I was able to show that for a convex set $E\subseteq\Bbb R,$ if $f:E\to\Bbb R$ is convex, then $f$ is left- and ...
0
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0answers
37 views

Second Order Differentials: Using $y = A + Bxe^x$

I've went over some of my math work which I'm currently doing at Uni and came across a rather confusing example. The example I went over is based on Second Order Differentials. So basically what I ...
7
votes
5answers
928 views

Proof of the derivative of ln(x)

I'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + h) ...
4
votes
2answers
104 views

Function that is continuous and its differential is continuous

Let $ f: \mathbb{R} \rightarrow \mathbb{R}$ . Show that $f$ is continuously differentiable if and only if, for every $x \in \mathbb{R}$ there exists a $l \in \mathbb{R}$ with the property that ...
1
vote
2answers
54 views

Proving that $\lim_{x {\to} \infty}f(x)=\infty $

Given $\ f(x)$ that is differential in $\ (x_0,\infty), f'(x)\ge a,$ for every $\ x> x_0$ and $\ a>0$, trying to show that $\lim_{x\to\infty}f(x)=\infty$. So far I've tried using Mean Value ...
0
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2answers
28 views

How do I find the equations of line tangent to a unit circle that have a slope of 1?

I'm supposed to find the equations of the lines tangent to a circle of radius $1$ and centered at the origin that have a slope of $1$. I know these things: this is a unit circle, the equation for a ...
0
votes
1answer
68 views

How can I find $f'(0)$ of this function?

I need to find $f'(0)$ if $$f(x)={x^2\sin x-\cos\left(3x\right)\over e^{-3x}+1}$$ How do I do this? When I tried using the quotient rule it became messy very quickly so I thought that there must be ...
2
votes
1answer
35 views

What is the proper use of Leibniz notation for one-sided derivatives?

The only notation I've seen has been restricted to either Lagrange's prime notation or Euler's $D$. Here are some of the variants: $$f'(a^+):=\lim_{x\to a^+}\frac{f(x)-f(a)}{x-a}$$ ...
0
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0answers
13 views

Proving multi-variable differentiability using the limit definition

I'm doing advanced calculus and I find it challenging to solve multi-variable limits while proving differentiability, more specifically 2 variable limits. could you show me how do I solve this limit?: ...
1
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3answers
49 views

In proving the product rule, how do we know to add and subtract f(x+h)g(x) from the numerator in the derivative definition?

I watched two YouTube videos to try to get a proof that makes sense, but in both videos, the authors said something to the effect of "add and subtract f(x+h)g(x)" without a good explanation as to how ...
2
votes
0answers
26 views

Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
3
votes
1answer
151 views

What is the minimum degree of a polynomial for it to satisfy the following conditions?

This is the first part of a problem in the high-school exit exam of this year, in Italy. The differentiable function $y=f(x)$ has, for $x\in[-3,3]$, the graph $\Gamma$ below: $\Gamma$ exhibits ...
1
vote
1answer
36 views

application of inverse function theorem (first I thought implicit function theorem and then corrected it), how to continue?

Let $f=(f_1,f_2,f_3):\mathbb{R}^2\to\mathbb{R}^3$ continuously differentiable, $\det\begin{pmatrix} D_1f_1 & D_2f_1 \\ D_1f_2 & D_2f_2 \end{pmatrix}\not=0$. How to prove: In every point ...
3
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2answers
28 views

$\|f(x)-f(y)\|\ge c\|x-y\|$ for all $x,y\in U, c>0$-> continuously differentiable inverse function $g:f(U)\to U$

Let $U\subset\mathbb{R}^n$ open, $f:U\to \mathbb{R}^n$ continuously differentiable, $\|f(x)-f(y)\|\ge c\|x-y\|$ for all $x,y\in U, c>0$. Why is $\det(Df(x))\neq 0$ for all $x\in U$ and $f\colon ...
1
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2answers
68 views

On proving the total differential.

I am following an open-course on multi variable calculus provided by MIT taught by Denis Auroux. The question I am about to ask is from this lecture. In the lecture Denis Arnoux gives a sketch proof ...
0
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2answers
25 views

Domain of derivative on open interval is open

Let $f : (a, b) \to \mathbb{R}$. Suppose that the derivative $f'$ exists at every point of a set $E \subseteq (a,b)$. Is it true that the domain $E$ of $f'$ is open? And if it is not true, is it true ...
0
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1answer
46 views

Can ∂x and ∂y in a derivate be seen as ∂ times x or ∂ times y?

I'm watching some tutorials on machine learning and know just enough calculus to have an intuition on what a derivative is, but that's it. But this question is bugging me so much that now I'm pretty ...
0
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1answer
24 views

What is $\nabla\cdot A\nabla u$ for $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$?

Let $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$. How can we compute $\nabla\cdot A\nabla u$? I assume we need to apply some kind of product rule, but I ...
2
votes
1answer
28 views

What is $\nabla Au$ for $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$?

Let $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$. How can we compute $\nabla Au$? I assume we need to apply some kind of product rule, but I wasn't able to figure out ...
0
votes
1answer
85 views

If $f(0)=0$ and $f''\ge 0$, then $f(a+b)\ge f(a)+f(b)$

Given $\ f$ so $\ f''(x) \ge 0$ for every $\ x \ge 0$, also $\ f(0)=0$. Trying to show that if $\ a,b \ge 0 \Rightarrow f(a+b) \ge f(a) + f(b)$ Using Taylor I used $\ f(0)=0$ and got ...
0
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2answers
34 views

Uniqueness of $c$ in mean value theorem

The mean value theorem says that If $f(x)$ is continuous on the closed interval $x\in [a,b]$ and differentiable on open interval $(a,b)$ then there exists $c\in (a,b)$ such that ...
0
votes
2answers
41 views

Find gradient for a 2D slice of a 3D function

I've got a mathematical problem that should have a general solution, but trying to solve it with mathematical software tools like Wolfram/Mathematica/Matlab etc. gave either complex or no solutions, ...
2
votes
1answer
29 views

Given that $f(1) = f'(1) = 1$, use Taylor polynomials to show that $\lvert f(x) - x \rvert \leq A(x - 1)^2$

Given that $\ f$ has continuous second derivatives in$\ [0,2]$ and $\ f(1)=f'(1)=1$, I'm trying to prove that for every $\ x \in [0,2]$ exists an A so that: $$ |f(x)-x| \le A(x-1)^2 $$ The second ...
1
vote
2answers
89 views

Solving a given complex integral

I am trying to solve a problem that involves solving the integral $$\int\frac{1}{\sqrt{y^2 + a^2}} \left(\frac{\sqrt{y^2 + a^2}}{k} - 1\right)^pdy$$ Where $$p=1-\frac{1}{1+n}, n>1$$$, $n$ is an ...
30
votes
14answers
3k views

Why does the derivative of sine only work for radians?

I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm ...