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

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0
votes
3answers
25 views

Differentiation question (quotient rule)

Find $f'(x)$ if: $$f(x)=\displaystyle {6x \over \sqrt{ 1+x^2}}$$ Ans: $\displaystyle {6 \over (1+x^2) \sqrt {1+x^2}} $ My problem is that after applying the quotient rule, i can't simplify it to ...
3
votes
1answer
37 views

What is the formal name for the conformal laplacian?

\begin{align} L=R-4\dfrac{n-1}{n-2}\nabla^k\nabla_k \end{align} What is the formal name for $L$? I have seen it referred to as the conformal laplacian, however I thought I once read $L$ with a formal ...
3
votes
2answers
219 views

If the derivative is zero on [a, b] so the function is constant - using Heine-Borel?

I know the proof using MVT but I was wondering if it can be proofed using Heine-Borel Lemma, that "Every open cover of close interval has a finite subcover". (without compactness, simple as that). ...
5
votes
2answers
2k views

What is the $1469^\text{th}$ derivative of $x^{532}-5x^{37}-4$?

I'm doing some basic calculus exercises on higher derivatives. But I'm stuck at a problem. The question is to find the 1469th derivative of $f(x)=x^{532}-5x^{37}-4$. I've read something about using ...
0
votes
1answer
18 views

Total derivative product rule

Definition: Let $U\in \mathbb{R}^n$ be an open set. Let $a\in U$ and $f:U\to \mathbb{R}^m$. We say that $f$ is total differentiable at $a$ if there exists a matrix $T\in \mathbb{R}^{m\times n}$ and a ...
3
votes
0answers
102 views

Using a sequence of measures to create simple functions which approximate the Radon-Nikodym derivative of the limiting measure

I have a bunch of discrete probability measures with finite support: $\mu_1,\mu_2,\dots$, which strongly converge to an absolutely continuous probability measure $\mu$ in $\mathbf{R}^2$. That is, for ...
2
votes
1answer
42 views

Matrices derivative

I have a linear product of matrices, I did solve most of it, however, I stop at this component $(X^T W^T D W X)^{-1}$. Given that $X$ is $n \times p$ matrix and $D$ is $n\times n$ matrix. $W$ is a ...
2
votes
1answer
65 views

Gâteaux derivate of the Tikhonov functional

Let $X,Y$ be Hilbert spaces, and let $A\colon X\to Y$ be a compact operator. The Tikhonov functional is given by $$ F(x)=\lVert Ax-y\rVert_X^2+\alpha\lVert x\rVert_X^2. $$ Calculate the Gâteaux ...
11
votes
2answers
233 views
+100

Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, ...
0
votes
1answer
34 views

Continuously differentiable operator

I consider an operator $A:H^1_0\to H^1_0$ defined by $$Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$$ where $$ G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to know what ...
1
vote
2answers
189 views

Finding the scalar derivative of a matrix product

I'm trying to find $$\frac{\partial}{\partial \lambda}y^T \left(\sigma^2 I + \lambda^{-1}K_{\theta}^{-1}\right)^{-1}y$$ where $y \in \mathbb{R^n}$ is fixed, $\lambda \in \mathbb{R}$ and ...
1
vote
2answers
41 views

Is 'a' differentiable in f when f is a product of a differentiable and non-differentiable function?

Recently, I was studying differentiable and non-differentiable functions and I wondered whether this "conjecture" of mine is true: 1) "If $f(x)$ is a function that is the product of $g(x)$ and $h(x)$ ...
33
votes
16answers
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 ...
-5
votes
0answers
43 views

Need help for calculating two derivatives analytically

During building a Jacobian matrix for a numerical simulation I need to calculate following two derivatives where I hesitate about the correct answer: $$\partial( \partial X/ \partial\theta)/\partial ...
2
votes
1answer
37 views

Classifying peak and valley *regions* of a histogram

I've been playing with a few ways of classifying contiguous regions of a histogram as: 1) peak, 2) valley, or 3) in-between bit. Global thresholding has worked minimally well for me so far, but I'm ...
0
votes
0answers
17 views

Estimating the derivative of the difference function from that of a function

Suppose that $f$ is a twice differentiable function in an interval $(N,2N)$. We write $f_1(n,h)=f(n+h)-f(n)$, i.e., $f_1$ is the difference function. Then, a proof I'm reading estimates that if ...
1
vote
1answer
31 views

How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$

Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is: $$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$ where $u_i(x)$ is a value that indicate ...
1
vote
3answers
115 views

How to prove that $f(x)$ is constant if $|f(x)-f(y)|^2 \le (x-y)^3$?

Let $\mathbb R$ be the set of real numbers and $f:\mathbb R \to \mathbb R$ be such that for all $x$ and $y$ in $\mathbb R$ $|f(x)-f(y)|^2 \le (x-y)^3$. How can we prove that $f(x)$ is constant? I ...
38
votes
10answers
3k views

Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator?

$1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$. I got this question in an Application of Derivatives test. I think log might be used here to compare the values, but even then the values ...
2
votes
1answer
29 views

derivative of a recursive vector-valued function

I have a recursive vector-valued function $$\mathbf{y}(t)=\mathbf{W}\mathbf{y}(t-1).$$ To compute the derivative of $\mathbf{y}(t)$ with respect to $\mathbf{W}$, do I need to use the product rule? ...
2
votes
3answers
100 views

Derivative of $(-1)^x$

I'm taking a summer calc 2 class and we're getting into alternating patterns. I was interested in seeing the graph of $(-1)^x$ so I typed it into my TI-84 for $y = (-1)^x$. Surprisingly, the graph is ...
2
votes
2answers
42 views

Why is 1 not a critical point for this function?

For the function $f(x) = \frac{x^2}{x-1}$, why is $1$ not a critical point, along with $0$ and $2$? Don't critical points include discontinuities?
0
votes
2answers
30 views

Solving the logistic equation [on hold]

Please help to solve the following logistic equation: $$y'=y \cdot (1-y), \ ∀(t>0 \land y(0)>0)$$
3
votes
4answers
403 views

Derivative of $\sqrt{\sin (x^2)}$

I have problems calculating derivative of $f(x)=\sqrt{\sin (x^2)}$. I know that $f'(\sqrt{2k \pi + \pi})= - \infty$ and $f'(\sqrt{2k \pi})= + \infty$ because $f$ has derivative only if $ \sqrt{2k ...
0
votes
5answers
190 views

derivative of $x\cdot|\sin x|$

I have the function $f(x)=x|\sin x|$, and I need to see in which points the function has derivatives. I tried to solve it by using the definition of limit but it's complicated. It's pretty obvious ...
2
votes
4answers
54 views

Can we take out a constant while differentiating?

In the solved example above, rather than taking $a^2x^4$ together and differentiating $a^2 = 0$, we differentiated $x^4$ and took out $a^2$. Why? Couldn't we have differentiated $a^2$ and gotten the ...
1
vote
0answers
12 views

Derivative of sum of two functional derivatives with different ranges

I have a functional of the the following form, $(o<a<1)$ : $F(g(x)) = \int_0^a \! g(x) x \, \mathrm{d}x. + \int_a^1 \! (g(x)-k)x^2 \, \mathrm{d}x. $ I want to find $ \frac{\partial ...
-12
votes
1answer
50 views

Please, find derivative and type how to solve [on hold]

$F(x,y)=(3xy^3-x^2y^2)^3$ $F'_x(x,y)= ?$ can someone help me((
0
votes
0answers
19 views

Derivation of energy function

Given the following energy function $E(d)$ (also found here on page 3): $$ E_d = \sum_{x,y \in \Omega} \left(d_{x,y} - \hat{d}_{x,y}\right)^2 + \lambda \sum_{x,y} \left( ...
3
votes
0answers
61 views

Differentiating an endomorphism

Let $(M,\rho)$ be a symplectic $2$-dimensional manifold, and let $J$ be a compatible complex structure on $M$, i.e. the symmetric $(0,2)$-tensor $$g(*,*) = \rho(*,J*)$$ is a Riemannian metric. Denote ...
1
vote
1answer
40 views

Verify $\frac {\partial B} {\partial T} =$ $\frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT}$

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

Question about differentiability, definition and consequences.

Let $E,F$ be normed spaces, we say $f:E \to F$ is differentiable in $x_0\in E$ if there exist $Df(x_0) \in \mathcal{L}(E,F)$ such that $$\lim_{h\to 0}\frac{f(x+h)-f(x)-Df(x_0)(h)}{\|h\|}=0$$ or ...
0
votes
1answer
39 views

Prove or Disprove a Property of $f$

Let $f:[0,1]\mapsto\mathbb{R}$ be a differentiable function, prove or give a counter-example that there exists a $c\in[0,1]$ such that $\frac{4[f(1)-f(0)]}{\pi}=(1+c^2)f'(c)$ attempt: I tried to ...
0
votes
2answers
23 views

second derivative of the composition of two multivariable functions

Let $U \subset \mathbb{R}^n$ be open, and let $\gamma: \mathbb{R} \to U$ and $f: U \to \mathbb{R}$ be to functions that are differentiable at least twice. I want to show that $\frac{d^2}{dt^2}(f ...
-1
votes
0answers
27 views

5 point derivative filter [on hold]

n many of the paper it is said that 5 point derivative filter transfer function is given by $$H(z) = \left(\frac{1}{8T}\right)(-z^{-2} - 2z^{-1} + 2z + z^{2}).$$ But no one has given detailed ...
1
vote
2answers
134 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 ...
0
votes
1answer
47 views

How to find the value of $2g(1)+2f(1)-h(1)$?

If $$\lim_{ m\to\infty }{ \frac { x^{ m }f(1)+h(x)+1 }{ 2x^m+3x+3 } }$$ is continuous at $x=1$ and $g(1)=\lim_{ x\to0}(\ln x)^{ 2/\ln(x) }$ then how to find the value of $2g(1)+2f(1)-h(1)$? Assume ...
5
votes
5answers
189 views

A unusual inequality about function $\ln$

These day,I met a unusual inequality when I solve a difficult problem, and proving the inequality means I have done the work! Could you show me how to prove it or deny it? By the way, I believe that ...
0
votes
3answers
56 views

Differentiate the Function $g(x)= \ln\ \frac{a-x}{a+x}$

$$g(x)= \ln\ \frac{a-x}{a+x}$$ $$\frac{dy}{dx}\ =\frac{d}{dx}\ \ln \frac{a-x}{a+x}$$ $$g'(x) = \frac{1}{\frac{a-x}{a+x}}\cdot\frac{1}{1}\ \ln\ \frac{a+x}{a-x}$$ $$g'(x)= \frac{a+x}{a-x}$$ This ...
0
votes
0answers
48 views

First derivative of energy function [on hold]

Given the following energy function $E(d)$ (also found here on page 3): $$ E_d = \sum_{x,y \in \Omega} \left(d_{x,y} - \hat{d}_{x,y}\right)^2 + \lambda \sum_{x,y} \left( ...
0
votes
1answer
152 views

Differential Calculus Problem - Sphere volume increasing (differentiation of algebraic functions)

The Air is pumped into a spherical ball which expands at a rate of 8cm^3 per second. Find the exact rate of increase of the radius of the ball when the radius is 2 cm. I have tried this question, ...
3
votes
1answer
29 views

diffeomorphism inbetween two subsets of $\mathbb{R}^2$

Consider the function $$f: \mathbb{R}^2 \to \mathbb{R}^2, \space\space f(x, y) := \pmatrix{x(1-y) \cr x y}$$ Now first, why is $f$ continuously differentiable? Then, I want to prove that $f$ ...
0
votes
0answers
9 views

Continuity/Differentiability of parametric function

I'm having a problem in understanding the nature of this function:Continuity and Differentiatiability of the following function parametrically defined. $x=2t-|t-1|$ and $y=2t^2+t|t|$ Will it be ...
-1
votes
4answers
53 views

How to prove $f(x)$ is differentiable at $x=0$ [on hold]

A real valued function satisfies $$|f(x)| \leq x^{2}\quad \forall \quad x\in R $$ then how to prove f(x) is differentiable at x=0 ?
3
votes
5answers
85 views
4
votes
1answer
92 views

Trigonometric derivative?

Hello everyone how would I solve the following derivative. $f(x)=5x^3\tan(x)+\cot(2x)$ I know the derivative of $\tan(x)$ is $\sec^2(x)$ So would I do $15x^2\sec^2(x)-\csc(2x)$ As my ...
3
votes
4answers
342 views

Am I using the chain rule correctly?

I'm supposed to find $y'$ and $y''$ of this function: $$y=e^{\alpha x} \sin\beta x$$ This is what I have done so far: $$y'=e^{\alpha x}\sin\beta x\cdot \alpha x'\sin\beta x\cdot \sin'\beta x \cdot ...
0
votes
0answers
19 views

Are these two compositions of two functions differentiable?

Assuming $U=\{x\in\mathbb{R}^2:x_1^2+x_2^2<1\}$ is the open unit circle in the plane and $f,g:U\rightarrow\mathbb{R}^2$ two functions with $f(0)=g(0)=0$. $f$ is Fréchet-differentiable in $0$, and ...
0
votes
1answer
65 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 $x\in\mathbb{R}^n$ ...