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

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1answer
17 views

finding angle between two curves using knowledge of derivative

The curves $y=\sin 2x$ and $y=\cos 2x$ intersect at $x=\frac{π}{8}$. Find angle between the curves at this point. Extend your solution to find the angle between the curves $y=\sin 5x$ and $y=\cos ...
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0answers
24 views

derivative of periodic function (find the greatest angle)

The centre of the light cast by a light suspended by a chain 3 m from a chthedral roof moves backwards and forwards across the floor with a speed of $2\sin(5.1t)$ m/s. The light is 2 m above the ...
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0answers
11 views

applying derivatives of periodic functions

A railway 'bumper' spring is ttached to a fixed barrier at the end of a shunting track. The spring is compressed a distance of 1.5cm when a carriage runs into the bumper and first returns to its ...
1
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1answer
34 views

help with wrong result for $v(x) = \frac{\sqrt[3]{x-1}}{(x+2)^2}$

I need to differentiate this: $$v(x) = \frac{\sqrt[3]{x-1}}{(x+2)^2}$$ I used this formula: $$ \frac{f'(x)g(x) - f(x)g'(x)}{g^2(x)}$$ Where: $$ f'(x) = \frac{1}{3\sqrt[3]{(x-1)^2}}$$ and $$ g'(x) ...
2
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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 ...
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2answers
48 views

Finding the derivative of an integral with variable limits: ${\mathrm{d} \over \mathrm{d}x}\int_{x}^{x^2}{1 \over -2y}e^{-5xy^{2}}\mathrm{d}y$?

How do you compute the derivative $${\mathrm{d} \over \mathrm{d}x}\int_{x}^{x^2}{1 \over -2y}e^{-5xy^{2}}\mathrm{d}y$$ where the integral has variable limits?
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2answers
45 views

Problem with understanding a Differential in Multivariable Calculus

I have just started with Partial Differentiation and the book from where I'm learning (Mathematical Methods in the Physical Sciences) had the following problem on approximations using differentials ...
3
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1answer
35 views

An effective way of finding the order of the zero $z=0$ of $e^{\sin z}-e^{\tan z}$

An effective way of finding the order of the zero $z=0$ of $f(z)=e^{\sin z}-e^{\tan z}$? What I tried is developing both exponentials by their Taylor series around $z=\sin z$ and $z=\cos z$, getting ...
-1
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1answer
43 views

ode and area of triangle

Question: find a curve $x$ so that the area bounded between it's tangent at some point $t$ and the time axis on the interval between the point of contact of $x$ and it's tangent ( $t$ ), and the ...
1
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3answers
69 views

If a function is defined on the interval $(a, b)$, is the derivative necessarily defined at $a$ and $b$?

I am asked to prove something that assumes this. But is it true that derivative is necessarily defined at the "edges" of the domain of the definition of its function? Does it matter if the original ...
2
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1answer
27 views

Why not an Absolute maximum in an open interval?

The function $x^3+x^2\: \text{has a maximun value at}\: x=-\frac{2}{3} \text{in (-1, 0) }.$ My question is why call it a Local Maximun and not an Absolute Maximum when it is the highest value in that ...
2
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0answers
17 views

Closed representation of Ladder operators in One Dimensional Second Order Homogeneous Differential Equations

(1) Has anyone published the closed representation of ladder operators for second order differential equations? More specifically the second order differential equation $$ -\partial_x^2\Psi_m(x) + ...
1
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1answer
38 views

Function dominated by convex function is eventually convex

Suppose that we have a twice-differentiable function $f$ on $x\in [0,\infty)$ such that $f(x)>0$ on $x\in [0,\infty)$ (i.e. strictly positive) $f'(x)<0$ on $x\in [0,\infty)$ (i.e. strictly ...
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1answer
34 views

Why is the following true? functions

$$x , x_0 \in [a,b]$$ $x_0$-fixed $f \in D(a,b)$- differentiable on [a,b] $$\triangle (x)=f(x)-f(x_0)-f'(x_0)(x-x_0)$$ $$\triangle '(x)=f'(x)-f'(x_0)$$
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2answers
23 views

Comparing derivatives of two decreasing functions

Suppose we have two differantiable functions $f$ and $g$ on $x \in[0,\infty)$ such that $f(x)>0$ and $g(x)>0$, $f'(x)<0$ and $g'(x)<0$, $f(0)=g(0)$ and $f(x)>g(x)$ Observe all ...
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0answers
21 views

Derivative with respect to a 2-norm

Given $f = f(u(t), t)$, is it possible to find the following derivative? ($f$,$u \in \mathbb{R}^n$, $t \in \mathbb{R}$) $$ \frac{\partial f}{\partial||u||^2_2} $$ I am aware of the following ...
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3answers
43 views

Implicit derivative of a rational function.

I have this equation: $$x^2={{(x+2y)} \over {(x-2y)}}$$ I want to differentiate with respect to $x$. How do I do that without multiply the denominator by $x^2$? I tried to apply the quotient rule, ...
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0answers
40 views

How to compute the derivative of $ (f^{-1})'(a) $ for some $a?$

We have a function $f:\mathbb{R} \to \mathbb{R}$ defined as $$f(x)= \begin{cases} e^{\frac{1}{x-1}} \ ; \ \ \ x < 1 \\ (x-1) \ln \frac{1}{x} \ ; \ \ \ x \ge 1 \end{cases} $$ Let $g=f^{-1}.$ ...
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0answers
39 views

A Simple Question About Directional Derivatives

I am stuck with this one question in our worksheets. The question is : Let $f:\mathbb{R}^n\mapsto \mathbb{R}$ and $x\in\mathbb{R}^n$. For all $v∈\mathbb{R}^n$ the directional derivative exists and ...
3
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0answers
62 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 ...
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2answers
56 views

Is this Function differentiable and continuous at x=0? [closed]

Is $f(x)$ continuous and differentiable at $x = 0$ ? $$f(x) = x(\sqrt{x} - \sqrt{x+1})$$
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0answers
35 views

Is this function differentiable in $(1,-1)$?

I have this function: $$f(x,y)=\begin{cases}\displaystyle\frac{x \sqrt{x^2+y^2-1}}{x^2+y^2} &\text{if $x^2+y^2-1\geq0$}\\0 &\text{if $x^2+y^2-1\leq0$} \end{cases}$$ I have to say if it ...
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1answer
40 views

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$ where $x=(x_1, x_2, x_3) \in \mathbb{R}^3$ I was trying to look at Hessian matrix and use Sylwester theorem, but I see that ...
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0answers
14 views

Calculation of second derivative of Rayleigh quotient

I have an eigenvalue problem of the form [(A-kB)V=0] and I calculate the eigenvalues k and left (Vl) and right (Vr) eigenvectors using the qz command in matlab. For verification reasons, I also ...
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0answers
23 views

proving differentiability at $x_0$ using mean value theorem [duplicate]

if $f$ is a continuous function and $\lim_{x->x_0} f'(x)$ existed how to prove the differentiability of $f$ in $x_0$ ? is this using a mean value property? thank you xx
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2answers
37 views

How to prove this product rule?

If $f,g:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}$ are differentiable in $x_0\in\Omega$ ($\Omega$ is open), then the function $(f*g)$ is differentiable in $x_0$ and: $(f\cdot ...
2
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3answers
42 views

Derivative of a lemniscate at the left hand side

How does $${d \over{dx}}(3(x^2+y^2)^2)$$ turn into $$12y(x^2+y^2){{dy} \over{dx}}+12x(x^2+y^2)$$? I'm having a hard time solving it algebraically without it turning into a huge polynomial.
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3answers
87 views

Locally Lipschitz and Gâteaux Derivative if and only if Frechet Derivative

Consider $f$ locally Lipschitz. So $f$ is Gâteaux Derivative if and only if $f$ is Frechet Derivative. PS.: the converse is trivial.
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1answer
36 views

How to calculate a Fréchet derivative?

What is the standard algorithm for calculating a Fréchet derivative? i.e. $f(x,y)=x^2y$ for $(x_0,y_0)\in\mathbb{R}^2$
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0answers
12 views

Find a derivative of equation that contains Fourier series

I need to find a derivative of follow equation $$ \left(r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] \right)({\sin\phi-k\cos\phi}) - b = 0 $$ I know the derivative of $\left(r_{0} + \sum ...
1
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2answers
31 views

Show that $f(z)=x^3+ i(1-y)^3$ is differentiable only at $z=i$.

Here's the exact phrasing of the question: Show that when $f(z)=x^3+i(1-y)^3$, where $z=x+iy$. it is legitimate to write: $$f'(z)=u_x+iv_x=3x^2$$ only when $z=i$ Here's my best attempt We have ...
14
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3answers
1k views

Find the value of a function whose derivative is zero

The initial function is $$h(x)=\arcsin x + \arccos x$$ The derivative of this function is $0$ since $$h'(x)=\frac{1}{\sqrt{1-x^2}}-\frac{1}{\sqrt{1-x^2}}\equiv0$$ This means that $h(x)$ is a ...
11
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2answers
241 views

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, ...
1
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2answers
43 views

The derivative of matrice's multiplication

Given $F=A^TA$, with $A$ is a $m\times n$ matrix. Then what is the derivative w.r.t. $A$ ? I know when $A$ is a $m\times 1$ vector, the derivative is $$\frac{\partial F}{\partial A} = 2A$$. Does ...
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0answers
31 views

How to derive first, second and third derivatives from a curve of data points?

I have a set of year-long time-series of satellite data points, 23 in total, each logged at 16-day intervals. It usually exhibits a curve showing vegetation productivity - increasing and subsequently ...
0
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1answer
29 views

Find the two compass directions of a level path on a surface, $z=2000 - 0.02x^2 - 0.04y^2$

Suppose the elevation $z$ above a point $(x,y)$ in an $xy-$plane at sea level is given by $ \ z \ = \ 2000 - 0.02x^2 - 0.04y^2 \ $. Assume that the positive $x-$axis is oriented due east and the ...
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2answers
78 views

When does $f\sim g$ implies $f'\sim g'$?

Given two $C^1$ functions $f,g:[0,+\infty)\to [0,+\infty)$ such that $f(x)\sim g(x)$ as $x\to\infty$, which good conditions guarantee that $f'(x)\sim g'(x)$? I thought that monotonicity of the ...
4
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3answers
710 views

Derivative of a function w.r.t. another function.

How is this? I'm getting $(-\tan(x))$. Here's my attempt: Let $u=\sin(x)$ and $v=\cos(x)$. Then, the derivative we seek is, $$\frac{\mathrm dv}{\mathrm du}$$ Using chain rule, we have, ...
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2answers
33 views

Maximun vertical distance beween$ \frac{1}{\sqrt{x}} \:and\:\frac{1}{x\sqrt{x}}$

To calculate the maximun vertical distance beween$ \frac{1}{\sqrt{x}} \:and\:\frac{1}{x\sqrt{x}}$ at a point x=a, where a>1 I proceeded as follows: ...
3
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6answers
504 views

Why is the second derivative of an inflection point zero?

I wanted to know why is that so. I am just lost and want a detailed explanation.
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2answers
28 views

Derivative and graph mismatch

Using the implicit function $(x^2+y^2-1)^3=x^2y^3$ it can be shown that $y'=\frac{2xy^3-6x(x^2+y^2-1)^2}{6y(x^2+y^2-1)^2-3x^2y^2}$ but when I evaluate it for the point (1,0) I get ...
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5answers
71 views

$f(x)=2x-e^x<0, \forall x \in \mathbb{R}$

The question is quite simple, but I'm finding some trouble doing it... Prove that the function $f(x)=2x-e^x$ is negative, i.e., $f(x)<0, \forall x \in \mathbb{R}$. Thanks for the help.
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4answers
93 views

Derivative of$ \sqrt{x^2}$

I found the Derivative of $\sqrt{x^2}$ to be 1 using simplifications and the power rule, but when I checked the answer, it was in fact $=\frac{x}{\left|x\right|}$ and not $=1$. What could have been ...
2
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6answers
103 views

If $f'$ is increasing and $f(0)=0$, then $f(x)/x$ is increasing

Let $a>0$ and $f:[0,a] \to \mathbb{R}$ continuous function that is twice differentiable on $(0,a).$ Also $f(0)=0$ and $f'$ is strictly increasing function on $(0,a).$ I have to show that the ...
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0answers
14 views

Speed and velocity in x-direction of a point in polar coordinates

I have a list of values that describe the angle (a) of the polar coordinates to a time (t). The radius is 1. I was asked to estimate the speed of the point in the x-direction at time t(3)=2.4° My ...
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1answer
15 views

How to prove that a function is convex.

Let $h:[0,1] \to (0,\infty)$ be a continuously differentiable function such that the following inequality is true: $$\frac{h'(t)}{h(t)} > -\frac{1}{2} \ \ \ \ \text{for all $t\in (0,1)$} .$$ Let ...
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0answers
17 views

Functions with 0 as an interior point of the domain

We have $f,g,h : \mathbb{R} \rightarrow \mathbb{R}$, with $0$ an interior point of the domain. Let $k \in \mathbb{N}$. We use the following notation $h = \mathbb{O}(x^k)$ if the open interval $ I ...
4
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0answers
61 views

Trouble differentiating $\int_1^{x^3}\arcsin(t)dt$

I'm having trouble with an integral problem which goes like this: Differentiate $$\int_1^{x^3}\arcsin(t)dt$$ The rule I know would be that you make $t$ equal to $x^3$ and then use the chain rule to ...
0
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2answers
37 views

A question on a proof of $C^1$ implies locally Lipschitz

I stumbled upon this answer here while studying the proposition that if $f: \mathbb R^n \to \mathbb R^n$ is $C^1$ then $f$ is locally Lipschitz. The answer in the link applies Taylor's theorem. And ...
1
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3answers
117 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 ...