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

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-2
votes
3answers
32 views

Differentiation with respect to a constant variable?

Let $y=f(x)$. If we are trying to find $f^{\prime}(x)$ and we know that in the domain we are trying to find $f^{\prime}(x)$ in, $x$ is constant , then what is $f^{\prime}(x)$? Is it zero?
2
votes
1answer
59 views

What did I do wrong?

So, I have found the following problem. This problem is a multiple-choice one, and I have to pick the correct answer. The problem, gives a function $f:D \to R$, $$f(x)=\frac{xe^x}{e^x-a}$$ with $a$ ...
4
votes
5answers
115 views

Differentiate expression involving reciprocal of square roots.

I need to differentiate $$5\over 2+\sqrt{1+3x}$$ I can get the answer from Wolfram Alpha but I'm trying to understand the working. Do I use the chain rule? My calculus is at the basic level.
2
votes
1answer
25 views

Find all $n \in \mathbb N$ such that $g(x) = 100|x+1| - \sum_{k=1}^{n}|x^k+1|$ is differentiable $\forall x$

Find all $n \in \mathbb N$ such that $$g(x) = 100|x+1| - \sum_{k=1}^{n}|x^k+1|$$ is differentiable $\forall x$. It's my high school calculus problem. Is it possible to solve this problem in the high ...
2
votes
1answer
28 views

Example where partial derivatives commute but are not continuous.

I am looking for an example of a function $f:\mathbb R^2\to\mathbb R$ such that there is a point $x\in\mathbb R^2$ with the following properties: 1) All partial derivatives of second order exist in a ...
2
votes
2answers
21 views

Necessary condition for local maximum

Let $\Omega\subset \mathbb{R}^n$ open, bounded and let $f:\Omega\to\mathbb{R}$ be a $C^2$-function. I want to prove: Necessary for a interior maximum $x_0\in\Omega$ is that $D^2f(x_0)$ is negative ...
1
vote
1answer
32 views

Calculate the derivative

I'm asked to find the derivative of the following: $$ \sqrt[4]{x} + \sqrt[3]{3x} $$ I attempted to solve the problem and got the following result, but my book says I am wrong. $$ \frac 14x^{-\frac ...
0
votes
0answers
26 views

Derivative of indicator function and summation

I would like to take the derivative of $G = \sum_{i=1}^{n}\Big(\mathbb{1}\{i \geq x + k\} v(x) + \mathbb{1}\{i < x + k\} v(i)\Big)$ with respect to $x$, where $\mathbb{1}\{\cdot\}$ is the indicator ...
4
votes
4answers
162 views

Is My Proof that $\pi^e < e^{\pi}$ Valid? [duplicate]

The other day, a math teacher at my college gave me a challenge problem: Prove that $$\pi^e < e^{\pi}$$ without using a calculator. The next day, I found a valid proof, but I used a log table ...
0
votes
2answers
41 views

Differentiate the Function: $g(u)=\ln\left(\frac{\ln\ u}{1+\ln\ (2u)}\right)$

$$g(u)=\ln\left(\frac{\ln\ u}{1+\ln\ (2u)}\right)$$ $$=\ln\ (\ln\ u)-\ln(1+\ln\ (2u))$$ This is the part where I get a little confused. Keep in mind I am using this formula $$\frac{d}{dx}[\ln ...
0
votes
2answers
37 views

Measure of curve smoothness

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness? I was thinking that since curvature measures how fast a curve changes, ...
1
vote
2answers
53 views

Differentiate the Function $ h(z)=\ln\sqrt{\frac{a^2-z^2}{a^2+z^2}}$

Differentiate the function $$h(z)=\ln\sqrt{\frac{a^2-z^2}{a^2+z^2}}$$ My try: $$h(z) = \frac{1}{2}\ln\left(a^2-z^2\right)-\frac{1}{2}\ln\left(a^2+z^2\right)$$ so $$h'(z) = ...
1
vote
2answers
20 views

Solving for x using two derivatives and algebra.

There are two things I don't understand about the following: " Set these derivatives equal to each other and solve the resulting equation. $2\sqrt3\cos(x) = 2\sin(x)$ $= \sqrt3 = \tan(x)$ (since ...
1
vote
3answers
40 views

Differentiating the exponent power series

We know that $$ e^x = \sum\limits_{n=0}^{\infty}\frac{x^n}{n!} $$ We know that the series is uniformly convergent everywhere, and therefore we can differentiate term by term, i.e $$ ...
1
vote
2answers
54 views

Practical use for negative $dt.$

I am writing a section of notes for Calculus 1 on related rates. In the section where I discuss differentials, I write that the quantity $dt$ must be nonnegative. I imagined the only reason it would ...
-1
votes
4answers
36 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 ...
0
votes
1answer
19 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
1answer
243 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). ...
1
vote
2answers
43 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 $h(x)$ ...
-5
votes
0answers
46 views

Need help for calculating two derivatives analytically [on hold]

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 ...
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 ...
2
votes
3answers
102 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 ...
0
votes
1answer
37 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 ...
2
votes
2answers
43 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: $$y'=y \cdot (1-y), \ ∀(t>0 \land y(0)>0)$$
2
votes
4answers
55 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
1answer
32 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 ...
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
0answers
28 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( ...
2
votes
1answer
30 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? ...
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) ...
1
vote
0answers
13 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 ...
0
votes
0answers
29 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 ...
-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 ...
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 ...
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
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
55 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
4answers
343 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 ...
2
votes
1answer
44 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 ...
0
votes
0answers
19 views

Differential of square map in Lie group away from identity

I've looked everywhere for this specific example but couldn't find it. Probably simple but I only need it for a small application and my Lie theory is very rusty. Let $G$ be an arbitrary Lie group ...
1
vote
4answers
43 views

When can I use the natural log to help solve an integral?

Why is it okay to do this: $\int \frac{1}{x-2}dx = \ln(x-2)$ but not this: $\int \frac{1}{1-x^2}dx = \ln(1-x^2)$
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
2answers
26 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 ...
2
votes
4answers
110 views

What is wrong in my $f'(x)$?

We have $f:\mathbb{R}\rightarrow\mathbb{R}, f(x)=\frac{x^2-x+1}{x^2+x+1}$ and we need to find $f'(x)$. Here is all my steps: ...
1
vote
0answers
14 views

Hesse matrix is negatively semidefinite if a function has a local maximum

Let $U \subset \mathbb{R}^n$ be open, and let $f: U \to \mathbb{R}$ be at least twice continuously differentiable. Also, we assume $f$ has a local maximum $a \in U$. I now want to show that the Hesse ...
0
votes
0answers
52 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( ...
3
votes
5answers
88 views

Differentiate the Function $f(x)= \sqrt{x} \ln x$

Differentiate the Function $f(x)= \sqrt{x} \ln x$