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

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68 views

On proving the total differential.

I am following an open-course on multi variable calculus provided by MIT taught by Denis Auroux. The question I am about to ask is from this lecture. In the lecture Denis Arnoux gives a sketch proof ...
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2answers
26 views

Domain of derivative on open interval is open

Let $f : (a, b) \to \mathbb{R}$. Suppose that the derivative $f'$ exists at every point of a set $E \subseteq (a,b)$. Is it true that the domain $E$ of $f'$ is open? And if it is not true, is it true ...
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1answer
47 views

Can ∂x and ∂y in a derivate be seen as ∂ times x or ∂ times y?

I'm watching some tutorials on machine learning and know just enough calculus to have an intuition on what a derivative is, but that's it. But this question is bugging me so much that now I'm pretty ...
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1answer
24 views

What is $\nabla\cdot A\nabla u$ for $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$?

Let $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$. How can we compute $\nabla\cdot A\nabla u$? I assume we need to apply some kind of product rule, but I ...
2
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1answer
28 views

What is $\nabla Au$ for $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$?

Let $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$. How can we compute $\nabla Au$? I assume we need to apply some kind of product rule, but I wasn't able to figure out ...
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1answer
85 views

If $f(0)=0$ and $f''\ge 0$, then $f(a+b)\ge f(a)+f(b)$

Given $\ f$ so $\ f''(x) \ge 0$ for every $\ x \ge 0$, also $\ f(0)=0$. Trying to show that if $\ a,b \ge 0 \Rightarrow f(a+b) \ge f(a) + f(b)$ Using Taylor I used $\ f(0)=0$ and got ...
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2answers
34 views

Uniqueness of $c$ in mean value theorem

The mean value theorem says that If $f(x)$ is continuous on the closed interval $x\in [a,b]$ and differentiable on open interval $(a,b)$ then there exists $c\in (a,b)$ such that ...
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2answers
41 views

Find gradient for a 2D slice of a 3D function

I've got a mathematical problem that should have a general solution, but trying to solve it with mathematical software tools like Wolfram/Mathematica/Matlab etc. gave either complex or no solutions, ...
2
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1answer
29 views

Given that $f(1) = f'(1) = 1$, use Taylor polynomials to show that $\lvert f(x) - x \rvert \leq A(x - 1)^2$

Given that $\ f$ has continuous second derivatives in$\ [0,2]$ and $\ f(1)=f'(1)=1$, I'm trying to prove that for every $\ x \in [0,2]$ exists an A so that: $$ |f(x)-x| \le A(x-1)^2 $$ The second ...
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2answers
97 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 ...
33
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14answers
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 ...
1
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1answer
54 views

Find the derivative of an integral with respect to the upper limit

Let $f(t,y): \mathbb R^2 \to \mathbb R$ be a continuous function of two variables and let $\phi:\mathbb R\to\mathbb R$ be a continuous function of one variable. Fix $a\in\mathbb R$. Compute ...
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0answers
20 views

$Df(x,y) = \nabla4(x^3-y,y^3-x)$

In a textbook I found this: $$ f(x,y) = x^4+y^4-4xy $$ $$ Df(x,y) = \nabla4(x^3-y,y^3-x)$$ Shouldnt it be: $$ Df(x,y) = 4(x^3-y,y^3-x)$$ Am I missing something here?
5
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4answers
222 views

Is it possible to work out the derivative of $e^x$ using the summation definition of $e = \sum_n 1/n!$?

So I know this question is a bit obtuse because usually we define $e$ in terms of the $\lim_{n \to \infty} (1 + 1/n)^n$ definition, and then compute derivatives of $e^x$ from there appealing to the ...
1
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1answer
16 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
23 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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0answers
30 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 ...
0
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2answers
75 views

Find $\frac{\mathrm dy}{\mathrm dx}$ if $x^y +y^x =1$ [closed]

Find $\frac{\mathrm dy}{\mathrm dx}$ if $x^y +y^x =1$ I am not able to achieve the solution for this. my approach seems to be flawed since I am getting dy/dx as 0
-1
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1answer
42 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
26 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 ...
0
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1answer
33 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 ...
0
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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
38 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
57 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 ...
2
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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 ...
0
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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 ...
0
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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 ...
0
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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
0
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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
41 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
85 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.
0
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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 $(x_0,y_0)\in\mathbb{R}^2$
0
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0answers
11 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
29 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 ...
9
votes
2answers
140 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, ...
1
vote
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 ...
0
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0answers
29 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 ...