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

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23 views

Given $g \in C^2(\mathbb{R})$ Study Derivability of $f(x) = \frac{g(x)}{x} $

Given $g \in C^2(\mathbb{R})$ ($g$ is a function twice derivable), consider the function $f: \mathbb{R}\rightarrow\mathbb{R}$ defined by $$ f(x) = \frac{g(x)}{x}\:\:\:\:\forall x \in \mathbb{R}^*, ...
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1answer
41 views

Proof for Mean of Geometric Distribution

I am studying the proof for the mean of the Geometric Distribution http://www.math.uah.edu/stat/bernoulli/Geometric.html (The first arrow on Point No. 8 on the first page). It seems to be an arithco ...
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1answer
47 views

Tangent map of orthogonal projection

Given a tangent bundle $TM$ and its natural projection $\pi:TM\to M$, I want to compute tangent map $T\pi:TTM\to TM$. Here is my method. Suppose a curve $c:\mathbb{R}\to TM$ with $c(0)=(x,y)$ and ...
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2answers
48 views

Cauchy-Riemann equations Complex Numbers [closed]

I have used the theorem if f'(z) = 0 then f(z) is a constant. I have proved it by using Cauchy Riemann's theorem. b
3
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2answers
35 views

Complex Differentiation

Can anyone give a hint to how to approach this question?
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1answer
19 views

Maximum volume of a cuboid with constraints

Find maximum volume of cuboid for which sum of three dimensions (x,y,z) is not greater then 108. I am looking for the most straightforward approach to the question. Thus the volume will be $xyz$ and ...
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2answers
18 views

Using substitution to make an equation into a separable differentiable equation

I have the question: By making the substitution $y = t^nz$ and making a cunning choice of n, show that the following equations can be reduced to separable equations and solve them. $$\dfrac{dy}{dt} = ...
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4answers
69 views

What is the result of $\frac{d\dot x}{dx}$?

I have a problem with a step in assignment. What is the result of $\frac{d\dot x}{dx}$ ? $x$ is the displacement, and $\dot x$ is the speed. I'm not sure if this equation itself is right. Thank you! ...
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1answer
21 views

Computing the Frechet derivative

How does a computation involving the Frechet derivative typically proceed? The definition, $$\lim_{h \to 0} \frac{\| f(x + h) - f(x) - f'(h)\|}{\|h\|} = 0$$ seems somewhat cumbersome to use in order ...
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3answers
36 views

$\sqrt{y}+\sqrt{x}=\sqrt{A}$ … prove that x-intercept + y-intercept of any tangent = constant [closed]

This is equation of a curve $\sqrt{y}+\sqrt{x}=\sqrt{A}$ $A$ is constant $T$ is a tangent of the curve from any point on it $B$ is y-intercept of $T$ $C$ is x-intercept of $T$ ...
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4answers
43 views

Gradient of a curve $y=\ln \sqrt{x+y}$

Find the gradient of the curve $y=\ln \sqrt{x+y}$ at the point when its y-coordinate is 1. My attempt, I differentiated and I got $\frac{dy}{dx}=\frac{1}{2x+2y-1}$. But I've problem in finding the ...
2
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1answer
46 views

Computing Fréchet derivative

I am reading Methods in Nonlinear Analysis by Kung-Ching Chang and having trouble in obataining a Fréchet derivative in the text. For those who has the book, it is on page 37, which concern Euler ...
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2answers
45 views

Is there has a smart way to compute the 1 order derivative of the circle equation? [closed]

I have encountered a compute problem. This exercise has given the circle equation and a para-curve equation with unknown parameters, the para-curve and circle has the same radius of curvature, and ...
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3answers
33 views

Logarithmic differentiation of $y=(x^2(7x-14)^{1/3})/(1+x^2)^4$

Honestly I have no clue how to rewrite then start it. I know you have to Ln both sides but how would you Ln the right side?
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0answers
15 views

Derivative of convoluted 2D image w.r.t. to its coefficients

I am creating an image with the following variables with the following dimensions: $A: (1,i)\\ X_a: (i,x,y)\\ B: (1,j)\\ X_b: (j,x,y)\\ Image=A\cdot X_a\odot B\cdot X_b $ Where $\odot$ stands for ...
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1answer
23 views

Calculate angle of inclination in NE direction [closed]

A man is on the hill in a point $(-100, -100, 430)$, the hill is given by an equation $z=500-0.003x^2-0.004y^2$. What is the angle of inclination in NE direction? (i guess ne direction on the ...
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6answers
96 views

Why $\int 2 \cdot \frac{\ln(x)}{x} dx$ is $\ln(x)^2 + C$?

Why the integral of $2 \cdot \frac{\ln(x)}{x}$ is $\ln(x)^2 + C$ (where $C$ is of course a constant) ? After some years of my high school math classes, I am again doing derivatives and integrals, but ...
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2answers
18 views

Derivative of: $x\log_2(x)$

Can someone please help me with the derivative of this function: $$y = x\log_2(x)$$ This is the answer: $$1+\ln(x)\over\ln2$$ When I try to solve it I get stuck here: $$\log_2(x) + {x\over x\ln2} ...
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0answers
24 views

Differentiate $g\circ f$ transformation

Differentiate $g \circ f$ of the following functions: $$f: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$f(x,y)=(x-y,x+y)$$ $$g: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$g(x_1,x_2)=(e^{x_1} \cos ...
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3answers
76 views

Find the Derivative of $\frac{1}{\cos^2(2x)+\sin^2(2x)}$ [closed]

Calculate the derivative of: $$\frac{1}{\cos^2(2x)+\sin^2(2x)}.$$ How would I calculate such a derivative?
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1answer
105 views

Clarification on Implicit Derivatives steps

I have been attempting to wrap my head around this problem for a couple days now. I've attempted numerous different iterations to try and find how the answer is derived, but I just don't see the ...
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4answers
155 views

$f$ differentiable at $0\iff\lim_{x\to 0}\frac{f(2x)-f(x)}{x}$ exists

Let $f$ be a real function that is continuous at $0$. Prove that $f$ differentiable at $0\iff\lim_{x\to 0}\frac{f(2x)-f(x)}{x}$ exists The $\Rightarrow$ part is trivial, and $\lim_{x\to ...
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0answers
17 views

(Partial) derivatives exist vs. are finite?

Is there a difference between the following two statements or do they mean the same? The (partial) derivatives of $f$ exist. The (partial) derivatives of $f$ are finite. I believe that it ...
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1answer
29 views

How to prove that a $\phi \in C^{\infty}(\mathbb{R})$.

I would like to prove that the function, defined as: \begin{equation} \phi(x)=\begin{cases} e^{-1/x}, & x>0 \\ 0 , & x \leq 0\end{cases} \end{equation} is a $C^{\infty}(\mathbb{R})$. So ...
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0answers
21 views

Cannot find horizontal tangent of the curve

Does the curve represented by the equation $y= \cos x + 5x$ have any horizontal tangent? I calculated $y\prime=0$ and i got $\sin x=5$ which is false so what should i do ? Here is what i did : ...
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4answers
75 views

If $f'(c) \neq \frac{f(b)-f(a)}{b-a}$, then find number of such $c$.

Let $f(x)=x^3+3x+2$ and $x=c$ is a point such that $$f'(c) \neq \frac{f(b)-f(a)}{b-a}$$ for any two values of $a$ and $b$, where $a,b$ and $c \in \mathbb R$. Then find the number of ...
0
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2answers
33 views

Linear Hamiltonian System

Suppose the linear system: $\dot{z} = J \frac{\partial{H}}{\partial{z}} = J S(t) z = A(t) z$, with Hamiltonian $H=H(t,z)=\frac{1}{2} z^T S(t)z$. How can I prove that: $$\frac{d}{dt}H(t,\xi(t)) = ...
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1answer
8 views

$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty\Rightarrow\lim_{k\to\infty}\nabla f(x^k)=0$

Let $f\in C^1(\mathbb{R}^n)$ and $(x^k)_{k\in\mathbb{N}_0}\subseteq\mathbb{R}^n$ with $$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty$$ Why can we conclude that $$\lim_{k\to\infty}\nabla ...
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1answer
20 views

Proving $f(x)=2x+|x|$ is not differentiable at (0,0)

Prove $f(x)=2x+|x|$ is not differentiable at (0,0) I know that the limit as x tends towards 0 from positive and negative of the derivative must be equal for it to be differentiable. The mark scheme ...
0
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3answers
69 views

Taylors Series for Limits

For the equation: $$\lim_{ x\to 1} \frac{1−x+\ln x}{1+\cos(\pi x)}$$ How can you evaluate this limit using a Taylor Series for both the numerator and deminator? Would I need to create a taylor ...
0
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1answer
15 views

Dervative in $\mathbb{R}^1$ v/s derivative in$\mathbb{R}^n$ $n \geq 2$

There is a fundamental reason of this? : In $\mathbb{R}^1$, any smooth function can be expressed as the derivative of some other function. In $\mathbb{R}^n$, however, not every vector-valued function ...
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1answer
56 views

Fundamental Theorem of Calculus With Function Containing Limit Variable

I'm trying to solve the following question: Evaluate $$\frac{\mathrm{d} }{\mathrm{d} s} \int^s_0 e^{st^2} dt $$ My thinking was that by the fundamental theorem of calculus, we have $ F(s) = ...
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1answer
44 views

How to find the general solution of the following differential equation [closed]

Could someone please explain to me how to solve the differential equation below: \begin{equation*} 2y\cot x\frac{dy}{dx} = (4+y^2)\cos x? \end{equation*} Thank you very much :)
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1answer
28 views

How does differentiability affect the extremum of a function?

I have this function $$f(x)= \begin{cases} (x+1)^3 & -2< x\le-1\\ x^{2/3}-1 &-1<x\le1\\ -(x-1)^2 &1<x<2 \end{cases}$$ I'm supposed to find the total number of maxima and ...
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0answers
6 views

How do I find the 2nd order Taylor expansion of this function of matrices?

I am looking to form the 2nd order Taylor approximation of the following function of matrices: $$f(W_1,W_2,W_3) = \left\lVert y - g_3(W_3g_2(W_2g_1(W_1x))) \right\rVert_2^2$$ Where: $x \in ...
3
votes
4answers
260 views

Proving a function is not differentiable

Given the function $f(x) = |8x^3 − 1|$ in the set $A = [0, 1].$ Prove that the function is not differentiable at $x = \frac12.$ The answer in my book is as follows: $$\lim_{x \to \frac12-} ...
2
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1answer
64 views

The series of function $f(x)=\sum_{n\geq 1}\frac{1}{n}\ln(1+\frac{x}{n})$; the convergence and the differentiability.

Consider the series of function $f(x)=\sum_{n\geq 1}\frac{1}{n}\ln(1+\frac{x}{n})$ for $x>-1$. a) Show that the series is pointwise convergent. Answer: I actually don't know how to show ...
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1answer
54 views

Prove a function has a maximum and minimum along a domain

Given the function $f:[13,132] \to R$ defined by $f(x)=sinx+x^3-$2 $e^x $ prove that the function has a maximum and minimum along the domain. I understand that a function has a maximum and minimum ...
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1answer
25 views

Derivative of a polar coordinate equation

I was trying to plot the polar curve: $r=\cos(2n\theta)$ ($0\leq\theta\leq 2\pi$) and tried differentiating with respect to $\theta$ to get some information about where the petals would be. My ...
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1answer
26 views

How to find a Newton-Cotes formula with weights?

I want to build a Newton-Cotes formula with weights $\int_0^1f(x)x^\alpha dx = a_0f(0) + a_1f(1) + R(f), \alpha > -1$ But, I cannot find any example, moreover I don't really know where to ...
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1answer
24 views

$f(x)= sin(x)^{3}+cos(x)^{3}$ prove ${f}''(x)= \frac{3}{2}(cos(x)+sin(x))(3sin(2x)-2)$

$f(x)= \sin(x)^{3}+\cos(x)^{3}$ prove that ${f}''(x)= \frac{3}{2}(cos(x)+sin(x))\, (3sin(2x)-2)$ I tried to solve it but I can't complete it.
2
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1answer
35 views

How do I prove this function is differentiable at 0?

Define $f:\mathbb{R}\longrightarrow \mathbb{R}$ by $$f(x) =\begin{cases} x^{4/3}\cos \left(\frac1x\right) & \text{if } x \neq 0, \\\\ 0 & \text{if } x =0. \end{cases}$$ Prove that ...
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1answer
42 views

Gradient and Hessian of Abs(Non-Repeated Eigenvalue) of Non-Symmetric Matrix

I would like to compute in MATLAB, without resort to automatic differentiation), the gradient, and ideally also the Hessian, of the absolute value of a non-repeated eigenvalue of a non-symmetric ...
3
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1answer
100 views

Integral Inequality Proof Using Hölder's inequality

I'm working on the extra credit for my Calculus 1 class and the last problem is a proof. We have done proofs before, but I'm unsure of how to approach this problem. Any help would be much appreciated, ...
2
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1answer
42 views

Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
0
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1answer
26 views

Interchanging total derivative and partial derivative

Say I have a function $F(x,y)$, where $x = f(t)$ and $y = g(t)$. $$\frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial F}{\partial x} \tag{1}$$ $$\frac{\partial }{\partial x} \frac{\mathrm{d} ...
0
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2answers
65 views

Using l'Hopital's rule to find the limit .

I need a hint to evaluate the following limit: $$\lim_{x \to 0} \frac{x^3\sin\left(\frac{1}{x^2}\right)}{\cos x}$$
2
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3answers
61 views

When is a continuous function differentiable? [duplicate]

I have been doing a lot of problems regarding calculus. An utmost basic question I stumble upon is "when is a continuous function differentiable?" (irrespective of whether its in an open or closed ...
2
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4answers
54 views

Limit of a sum of two variables

Recently at my calculus course we are doing derivatives and integrals. I've stumbled upon a sum that seems to have nothing in common with our current objectives, though I'm sure it does have, but ...
3
votes
1answer
27 views

Integral and derivative

Let $g(x) = \int_{[0;2^x]}{\sin(t^2)} dt$ for $x \in \mathbb{R}$. I have to calculate $g'(0)$. So, $g'(0) = \lim_{h \to 0}{\frac{g(h) - \int_{[0;1]}{\sin(t^2)} dt}{h}}$. Maybe I should apply the ...