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

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1answer
28 views

Problem with a derivative inside an integral

I am having some issues solving for the derivative of $$\int_0^1 q^{\alpha} e^{tq} dq$$ with respect to $t$ when $t > 0$. I tried to perform a direct method of computing: $$\lim_{t \rightarrow t_0} ...
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0answers
23 views

System of equations given a function as an input

I’m having a set of equation $f_1(y,t), f_2(y,t)… , f_n(y,t)$, I also know that $y(t)=e^{At}v$ where $A_{nxn}$ is a matrix and $v$ is a $nx1$ Vector. Can I bound $f(y(t))$ somehow? Or say anything ...
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0answers
20 views

Differentiation minimization

This question taken from web based engineering mathematics online test. My answer to this question as below. This system says it is incorrect. Is there any mistake? Plz help.
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2answers
83 views

Find the equation of the tangent to $y = 2^{x^2}$ at $(1,2)$?

I First Found the First Derivative and i got.. $\ln(2)x⋅2^{x^2+1}$ Then I don;'t know what to do.. the real answer is $y = (4\ln2)x+2-4\ln2$
2
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1answer
68 views

Continuous strictly increasing function with derivative infinity at a measure 0 set

Let $E\subset [0,1]$ with $\mu(E)=0$. Does there exist a continuous, strictly increasing function $f$ on $[0,1]$ so that $f'(x)=\infty$ for all $x\in E$ (in Lebesgue sense)? I think there exist such ...
6
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2answers
113 views

Is there any function continuous in $R$ and differentiable in rational numbers with zero derivative?

I'm looking for a function continuous in $R$ and differentiable in all rational numbers and it's derivative should be $0$.But not the constant function. And there is a same question about irrational ...
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1answer
28 views

Tangent parallel to the initial line for polar equation =, can r^2 be used instead?

Given a formula for a polar equation: $$\ r^2 = a^2 \cos^22 \theta $$ It could be said that to find the points parallel to the initial line, $$\frac{dy}{dx} = \frac{d (r\sin\theta)}{d\theta} = 0$$ ...
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1answer
43 views

If double derivative test fails can we go for Higher derivatives

if $f(x)=x^4$ we have critical points given by $$4x^3=0$$ which is $x=0$ Now $$f''(x)=12x^2$$ so $$f''(0)=0$$ and also $x=0$ is not Point of Inflexion since $f''(0^{+})$ and $f''(0^{-})$ have same ...
2
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2answers
59 views

find the rate of change $f(x) = 4\sin^3 x$ when $x = \frac{5\pi}{6}$

find the rate of change $f(x) = 4\sin^3 x$ when $x = \frac{5\pi}{6}$ To find the rate of change I need to find $\frac{dy}{dx}$ using the chain rule $h'(x) = g'(f(x)).f'(x)$ $g'(f(x)) = 12\sin^2 x$ $...
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0answers
28 views

What would be the meaning of the following expression?

If I have a function r(s). Then I could write down( in principle): $\dfrac{d}{dr} (r'(s))$ But I have troubles understanding what this would mean and how to work with it. Using leibniz notation I ...
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0answers
12 views

Criteria for Lipschitz continuity in case the function is not assumed to be (everywhere) differentiable.

There is a well known theorem that characterizes Lipschitz functions among the everywhere differentiable ones (we assume, for simplicity, to only discuss the single variable case): A function is ...
1
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1answer
20 views

Continuous $(x,y) \to k(x,y)$ with “discontinous slope-behavior” at one $x\to k(x,y)$-slice: Is this possible?

My question is if it is possible to construct a continuous function $k\colon[a,b]\times [c,d] \to \mathbb{R}$ such that for each $v\in (c,d]$ the function $k(\cdot,v)$ its slope (i.e. derivative) is ...
2
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1answer
27 views

Discontinuous derivative *not* by oscillation

All the differentiable functions that I have ever seen whose derivative is discontinuous, achieve this discontinuity by oscillating: See, e.g., this question. Is it possible to construct ...
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1answer
33 views

Difference between Increasing and Monotone increasing function

I have some confusion in difference between monotone increasing function and Increasing function. For example $$f(x)=x^3$$ is Monotone increasing i.e, if $$x_2 \gt x_1$$ then $$f(x_2) \gt f(x_1)$$ ...
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4answers
144 views

Dubious “proof” of $e^x$ derivative?

The proof to which I am referring is amply discussed here: Derivative of exponential function proof, but I remain unconvinced by the answers that pertain to the specific proof discovered by ...
2
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1answer
55 views

Distributional derivatives

I need to compute derivatives as distributions of following functions: $f(x) =$ $|x|$ $|x^2 - 1|$ $\mathrm{sgn}(x)$ $4$ Where $f : \mathbb{R} \to \mathbb{R}$. ad 1) $|x|$ is continuous, so it ...
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0answers
17 views

Dirichlet problem with separation of variables

$\triangle u=0$, with $0<r<2$ and $0\leq \theta <2 \pi$ $u(r=2,\theta)=\sin \theta $ There are three conditions that we need to check with separation of variables, which are: a) The ...
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0answers
16 views

derivative of an definite integral using taylor expansion

Let $f(x,y,t): \mathbb{R}^2 \times [0,1] \rightarrow \mathbb{R}: (x,y,t) \mapsto f(x,y,t)$, you can see $t$ as the time . Define for a specific $t \in [0,1]$, $f_t(x,y) := f(x,y,t)$ and assume we have ...
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1answer
28 views

Partial differentiation, solving an equation

I have this question about partial differentiation which I try to solve but I seem to get wrong all the time. The question is: if $x=t\sin(s)$ and $y=t\cos(s)$, find $d^2f(x,y)/dsdt$. I'm assuming ...
0
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1answer
23 views

find the equation of the tangent to $y = \sin x$ at the point where $x = \frac{\pi}3$

Check workings for find the equation of the tangent to $y = \sin x$ at the point where $x = \frac{\pi}3$. When $x = \frac{\pi}3$, $y = f(\frac{\pi}{3}) = \sin \frac{\pi}{3} = \frac {\sqrt{3}}2$ $\...
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0answers
22 views

Substitution when calculating limits

I tried to calculate: $$\lim_{k\to 0}\frac{e^{(1+k)t}-e^t}{k}$$ Apparently I can define: $h=k*t$ which gives: $$t*\lim_{h\to 0}\frac{e^{t+h}-e^t}{h}=t*\frac{d}{dt}e^t=te^t$$ QUESTION: Why can I ...
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2answers
31 views

show that $y = \cos x$, has a maximum turning point at $(0, 1)$ and a minimum turning point at $(\pi, -1)$

show that $y = \cos x$, has a maximum turning point at $(0, 1)$ and a minimum turning point at $(\pi, -1)$ Turning points occur when the gradient is 0 or $\frac{dy}{dx} = 0$ $f(x) = \cos x$ $\...
3
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4answers
39 views

Problem involving binomial coefficients where p+q=1

If $p+q=1$, then show that $$\sum_{r=0}^n r^2 \binom {n}{r} p^r q^{n-r}=npq+n^2p^2$$ I was able to solve this by differentiating the expression twice and then relating the given variables. But the ...
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2answers
40 views

simplification of square root of $\pi$

Using the power rule, my textbook differentiates this: $\frac{d}{dx}(\sqrt{x^{2+\pi}})$ like this, using the power rule: $$\begin{align} =& \frac{d}{dx}(x^{1+(\pi/2)}) \tag{1}\\ =& (1+\frac{...
0
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1answer
45 views

Determine if the function is continuous and differentiable on the closed interval $[0,\frac{1}{\pi}]$

Define $G(x)=\int_0^xg(x),$ where g is given by the following: $$g(x) = \begin{cases} \sin\frac{2}{x} & \textrm{ if $x\ne 0$} \\ 0 & \textrm{ if $x =0$} \\ \end{cases} $$ Is ...
0
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2answers
44 views

Finding the max/min turning points of $y=\sin x$

Use differentiation to show that $$y = \sin x$$ has a maximum turning point at $\left(\frac{\pi}2, 1\right)$ and a minimum turning point of $\left(\frac{3\pi}2, -1\right)$. I know that the ...
1
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1answer
42 views

Application of Schawarz lemma??

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. We have $|f(z)|\leq 1-|z|$ for all $z$ ...
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2answers
52 views

twice differentiable functions

Let $f:\mathbb R \to \mathbb R$ be a twice continuously differentiable function, with $f(0)=f(1)=f'(0) = 0$. Then $f^{"}$ is the zero function. $f^{"}(0)$ is zero. $f^{"}(x)=0$ for some $x \in $ (0,...
1
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1answer
51 views

what is the n times differentiation of $e^{3x}\cdot \sin^2x$ [closed]

I have an equation y = $e^{3x}\cdot \sin^2x$ and I want to calculate n times differentiation of it. As it is in the form of uv so I am lost here because at every iteration more term will generate and ...
0
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2answers
45 views

A derivative of an integral question

Hi I have some question about a derivative of an integral. $\frac{d}{dx}\left(\int_{\sqrt{x}}^{x}e^{-u^{2}}du\right) $ My try to solve $\int_{\sqrt{x}}^{x}e^{-u^{2}}du=\frac{d}{dx}\left(\int_{0}^...
0
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1answer
31 views

Derivative of an integral function

Say $$\varphi(\theta) = \int_{0}^{2\pi} \frac{f((1-\theta)z+\theta(z_0+re^{it}))}{z_0+re^{it}-z}ire^{it}dt.$$ Why do we have $$\varphi'(\theta) = \int_{0}^{2\pi} f'((1-\theta)z+\theta(z_0+re^{it}))...
1
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1answer
52 views

Show that $f$ is differentiable at $x_0$

Problem: $f: \mathbb{R^{+}} \to \mathbb{R^{+}}$ and strictly positive. Given that: $$\lim_{h \to 0} \left(\frac{f(x_0+h)}{f(x_0)}\right)^{\frac{1}{h}}= L,$$ where $L$ is not $0$ and is finite. Show ...
0
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1answer
20 views

Simplifying second order chain rule

In these notes: http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx We have $$\text{If } y = f(x(t)), x = g(t)$$ $$\text{Then } \dot y = \dot{f}(x(t)) =\dfrac{ \partial f}{\partial x}\...
3
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2answers
32 views

$\mu$ is a $\sigma-$finite measure on and $\{E_n\}$ measurable sets. When $\nu(E)=\sum \mu(E\cap E_n)$, is $\nu$ is $\sigma$-finite?

My question stems from the following problem. Suppose $\mu$ is a $\sigma-$finite measure on $(X,M)$ and $\{E_n\}$ a sequence of measurable sets. Define $\nu$ on $M$ by $\nu(E)=\sum \mu(E\cap E_n)$. ...
3
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1answer
74 views

Suppose $f(0) = f(1) = 0$ and $f(x_0) = 1$. Show that there is $\rho$ with $\lvert f'(\rho) \rvert \geq 2$.

Suppose that $f : [0; 1] \rightarrow \mathbb{R}$ is continous and differentiable on $(0,1)$, that $f(0) = f(1) = 0$, and that $\exists_{x_0 \in (0; 1)} f(x_0) = 1$. Prove that $\exists_{\rho \in (0;1)...
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0answers
41 views

How to find the common denominator with multiple variables

Find $\frac{zf^{'}(z)}{f(z)}$, where $-1 \leq \alpha \leq 0 $ and $0< v < 1$ Given: $f(z)= \frac{1}{\pi}(-\log (1-vz)+ \alpha \log(1-vz^{-1}))$ and $f^{'}(z)= \frac{1}{\pi}\left(\frac{v}{1-vz} ...
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2answers
81 views

Examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$

How to examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$ for $a, b, c> 0$ using Taylor's theorem?
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2answers
55 views

Prove that $\exists_{x_0}f''(x_0)=0$

$f \in C[0;2]$, $f(0) = 0$, $f(1) = 1$, $f(2) = 2$. $f''$ exists on $(0;2)$. Prove that $\exists_{x_0 \in (0;2)}f''(x_0)=0$. Mayme it's possible to use Rolle's theorem, but I don't see how to apply it ...
2
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1answer
20 views

Numerical force due to Lennard Jones potential

I am stuck with a problem related to simulating a Lennard-Jones system. The Lennard Jones potential is $U(r) = 4\epsilon [ \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6} ]$. Hence the force will ...
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2answers
45 views

Problem with differential equation,someone to give hint?

I tried to switch all without $y'$ on the right side, but I don't get anything. Does someone have a hint to do this equation $\left(y-x^2\right)y'-x=0 $?
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2answers
44 views

If we have $0\leq f'(x)\leq f(x)$, show that $f\equiv0$ if it vanishes at some point

Question: Let $f : \Bbb R \to \Bbb R$ be differentiable. Assume that for all $x \in \Bbb R$, $\ 0\le f'(x) \le f(x)$. Show that $f\equiv 0$, if $f$ vanishes at some point. I think it ...
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1answer
43 views

Finding $f^{(12)}(0)$ with $f(x)=\log(e^{x^4}-2x^8)$

Here's how I proceeded: We have $f(x)=x^4+\log\left(1-2x^8e^{-x^4}\right),$ hence for all $x$ such that $-1\le2x^8e^{-x^4}<1$ the following holds: \begin{align} f(x)=x^4+\sum_{n=1}^\infty\frac{(-...
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0answers
15 views

Proof verification: $\frac{d(\nu_1\times \nu_2)}{d(\nu_1 \times \nu_2)}(x_1,x_2)=\frac{d\nu_1}{d\mu_1)}(x_1)\frac{d\nu_2}{d\mu_2}(x_2).$

This is exercise 3.12 from Folland's Real Analysis. It took me a long times to come up with a solution to this problem, and I'd appreciate it if anyone could verify if my answer is correct. For $j=1,...
2
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0answers
23 views

Differentiation and integration of power series

I'm learning calculus and my textbook states that: A power series can be differentiated or integrated term by term over an interval contained entirely within its interval of convergence. In ...
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1answer
37 views

Given a Lipschitz continuous function, prove there is some $M > 0$ such that $|f'(x)| < M$

Given $f: I \rightarrow \mathbb{R}$ is Lipschitz continuous, where $I \in \mathbb{R}$ is an interval, prove there exists some $M > 0$ such that if $f$ is differentiable at a point $x \in (a,b)$, ...
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0answers
20 views

Modelling the path of a boat crossing a river

Alright so I am completely stumped! Assume a boat leaves a point A on the left bank of a river and heads upstream at an angle with velocity 2i-j relative to the river. The boats velocity relative to ...
0
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1answer
38 views

Dependence of the derivative of a pseudo-Boolean function on its variables

I am going through Pseudo-Boolean optimization by Boros et al. In the section 2, the paper introduces the idea of derivative and residual of a peudo-Boolean function. It is claimed that both $\...
1
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1answer
45 views

What is the derivative of $\frac{d}{d\alpha}Tr(A^T(\alpha)BA(\alpha))$?

Is the derivative of $Tr(A^T(\alpha)BA(\alpha))$ given by $\frac{d}{d\alpha}A(\alpha).2BA^T(\alpha)$, using the chain rule, or is it something else? I was expecting it to be a scalar valued function. ...
0
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0answers
27 views

Using chain rule to evaluate differentiability of product of functions $fg$.

For my maths course I have to solve the following problem: Suppose that functions $f,g:E\rightarrow\mathbb{R}$ are given, and the set $E$ is open in $\mathbb{R}^n$. Prove using the chain rule that $...
2
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0answers
55 views

Derivative of $(\lambda I - A)^{-1}$ with respect to $\lambda$

Is need to work with $\frac{d}{d\lambda} (1 - v^{T}(\lambda I - A)^{-1}u)$. Is it true that: $$\frac{d}{d\lambda} (1 - v^{T}(\lambda I - A)^{-1}u) = -v^{T}\frac{d}{d\lambda}(\lambda I - A)^{-1}u$$ ...