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

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Let $f:[a,b]\to R$ is continuous and $G(x,t)=t(x-1)$ when $t\leq x$ and $x(t-1)$ when $t\geq x$.

Let $f:[a,b]\to R$ is continuous and $G(x,t)=t(x-1)$ when $t\leq x$ and $x(t-1)$ when $t\geq x$. Let $g(x)=\int_0^1f(t)G(x,t)dt$. Show that $g''(x)$ exists and eqals $f(x)$ for $x \in (0,1)$. I ...
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2answers
43 views

Matrix Derivative d(AXA)^(-1)/dX

I am having trouble figuring out the following matrix derivative $\frac{\partial(B X A')(AX A')^{-1}}{\partial X}$, where $X$ is square $n\times n$, A is $m\times n$, with $m<n$. and B is dimension ...
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3answers
29 views

A problem of Schwarz derivative

I need help with the following problem analysis: Suppose $f$ is defined on an interval around $x$. The limit $$\lim_{h\to0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2},$$ if it exists, is called the Schwarz ...
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0answers
12 views

Consider the Plane Curve?

Consider the plane curve $$\gamma(t) = \left( \cosh(t) \cos(t), \cosh(t) \sin(t) \right), \;\; t \in \mathbb R.$$ Is $\gamma$ regular? If $\gamma$ is not regular, can you restrict the parameter ...
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2answers
9 views

Time dependence of velocity from position dependece of velocity

I know dependence of velocity on position $v(x)$ and I wan't to know dependence of velocity on time $v(t)$ I was thinking that using some chain rules or derivative of inverse it would be possible to ...
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0answers
18 views

Find global minimum of the function

I need to find the global minimum of the function $$f ( x) = \langle Ax,x \rangle + 2\langle b ,x\rangle+c$$ where $c \in \mathbb{R}$ is constant, $b \in \mathbb {R}^n$, and $A$ is a positive ...
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3answers
150 views

Finding the derivative of the integral using the Fundamental Theorem of Calculus.

I'm still not entirely solid on the concept of the Fundamental Theorem of Calculus, but I believe that the first step of the theorem will give us $$2x-1$$ which is the derivative of F(x). Usually, ...
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2answers
32 views

Find when the population is growing the fastest, under the logistic model

The population $P$ of an island $y$ years after colonization is given by the function: $\displaystyle P = \frac{250}{1 + 4e^{-0.01y}}$. After how many years was the population growing the fastest? ...
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30 views

How to show that each tangent vector to $\mathbb{R}^n$ at a point $a$ is of the form $\xi(f) = \sum_{i} c_i \frac{\partial f}{\partial x_i}(a)$? [on hold]

How to show that each tangent vector to $\mathbb{R}^n$ at a point $a = (a_1, \ldots, a_n) $ is of the form $\xi(f) = \sum_{i=1}^n c_i \frac{\partial f}{\partial x_i}(a)$? Thank you very much. Edit: ...
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1answer
14 views

Problem with real differentiable function involving both Mean Value Theorem and Intermediate Value Theorem

Problem: Let $a,b \in \Bbb R$, $a<b$, and let $f$ be a differentiable real-valued function on an open subset of $\Bbb R$ that contains $[a,b]$. Show that if $\gamma$ is any real number between ...
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3answers
53 views

What does the 2nd degree derivative of a cubic Bezier curve actually represent?

I have a $3D$ Bezier curve. Each co-ordinate along its path is defined by the equation: $$ f(t) = t^3 \bigl(a_2+3(c_1-c_2)-a_1\bigr) + 3t^2 (a_1-2c_1+c_2) + 3t(c_1-a_1) + a_1 $$ where $a_1, a_2$ are ...
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2answers
99 views

Derivative of the power tower

May somebody help me to correctly calculate the dervative of the $n$-th power tower function? $$ \begin{align} f_1(x)&=x\\ f_n(x)&=x^{f_{n-1}(x)}\\ &=x^{x^{x^{...^x}}}\text{ where ...
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1answer
20 views

Continuity and Directional Derivatives

Does every absolutely continuous function on a compact set possess a left and right hand derivative everywhere on its interior? Although the two need not be equal of course.
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128 views

Derivative of $\int_0^1 e^{\sqrt{x^2+t^2}}\,\mathrm{d}x$ at $t = 0$

Let the real-valued function $\phi:\mathbb{R}\to\mathbb{R}$ be defined by $$\phi(t)=\int_0^1e^{\sqrt{x^2+t^2}}\,\mathrm{d}x,$$ it can then be shown that $\phi$ is continuous and differentiable. I ...
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29 views

Why can't we differentiate one variable with respect to another variable?

This sounds a little immature, but why can't we differentiate $y$ with respect to $x$ ? Why does $y$ have to be written in terms of $x$ to differentiation? Why cant $\frac{dy}{dx} = 3a^2$ where ...
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0answers
13 views

Representation of Rate of change

My question is from Irrigation Engineering. A Canal Outlet has a property called Flexibility (F). It is defined as the "Ratio of rate of change of discharge of an outlet to the rate of change of the ...
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1answer
36 views

Derivatives - Show equality

Let $y(x)$ be defined implicitly by $G(x,y(x))=0$, where $G$ is a given two-variable function. Show that if $y(x)$ and $G$ are differentiable, then ...
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1answer
458 views

Derivative (or differential) of symmetric square root of a matrix

Let A be a square, symmetric, positive-definite matrix. Let S be its symmetric square root found by a singular value decomposition. Let vech() be the half-vectorization operator. Is there a ...
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1answer
30 views

Show that the function is differentiable

I have to prove that the following function is differentiable and to find its derivatives at any point. $$f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x,y) \rightarrow x^2+y^2$$ In my book there is a ...
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3answers
57 views

If $ f'(c) > 0 $, then there is an $ x $ such that $ f(x) > f(c) $.

Here is the homework question that I have: If $ f: [a,b] \to \Bbb{R} $ is differentiable at $ c $, where $ a < c < b $ and $ f^{\prime}(c) > 0 $, prove that there exists an $ x $ such ...
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1answer
2k views

Derivative of Softmax loss function

I am trying to wrap my head around back-propagation in a neural network with a Softmax classifier, which uses the Softmax function: \begin{equation} p_j = \frac{e^{o_j}}{\sum_k e^{o_k}} ...
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0answers
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Function intersecting 3 points & deriviate is positive for a range of x values

Thank you for taking the time to help out on this question. I'm looking for a function that intersects 3 points, and a derivative for every value of x between x=0 and x = 365 where dy/dx >= 0. My ...
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0answers
9 views

How to get the Riesz representative of the derivative of $L(K):=\text{tr}(\Lambda^* K A)$

$\DeclareMathOperator{\tr}{tr}K,\Lambda, A$ here are appropriate matrices. The question is not completely accurate as I can differentiate it, but I would prefer it to be in the form $⟨DL,h⟩$ for some ...
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1answer
20 views

Nonnegative Dini derivative implies nondecreasing function

This was posed as one of the proposition in my lecture note: If $f$ is continuous on $[a,b]$ and one of its Dini derivative is everywhere nonnegative on $(a,b)$, then $f$ is nondecreasing on ...
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4answers
12k views

What exactly is the difference between a derivative and a total derivative?

I am not too grounded in differentiation but today, I was posed with a supposedly easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to ...
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2answers
25 views

Let $f:[0,\infty]\to R$ be differentiable on $(0, \infty)$, and $f'(x)\to b$ as $x \to \infty$. Show that $\lim_{x \to \infty}\frac{f(x)}{x}=b$

This is actually part (c) of the original question. Part (a) asks to prove for any $h>0$, we have $\lim_{x\to\infty}\frac{f(x+h)-f(x)}{h}=b$. Part (b) asks to prove if $f(x) \to a$ as $x\to\infty$, ...
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38 views

Trig and derivatives: If condition holds for derivative, does it hold for the original equation?

Let's say I have some trigonometric identity such as $\sin(x) + 1 = -\cos(y)$. As we can see, the derivative of this identity gives $\cos(x) = \sin(y)$, which implies that $x + y = \pi/2$. Does that ...
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0answers
13 views

Fixed point of a differentiable function on a closed interval

Given a differentiable function $h:[0,3]\to [0,3]$ such that $h(0)=1 h(1)=2, h(3)=2$. (a) Argue that there exists a point $d ∈ [0,3]$ where $h(d)=d$. (b) Argue that at some point c we have ...
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0answers
22 views

Solution to a path problem with maximum values of derivatives

I want to minimize the travel time from known position A to known position B while the derivatives of path are below their maximum value. I have: ...
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1answer
491 views

The normal line intersects a curve at two points. What is the other point?

The line that is normal to the curve $\displaystyle x^2 + xy - 2y^2 = 0 $ at $\displaystyle (4,4)$ intersects the curve at what other point? I can not find an example of how to do this equation. Can ...
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13 views

How to prove first derivative test for an inflection point?

If $f'(x_0)=0$ and $x\in (x_0-\delta;x_0+\delta)\setminus\{x_0\}\implies f'(x)> 0$ for some $\delta>0$, prove that $x_0$ is an inflection point. That's what Wolfram MathWorld says. How ...
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1answer
66 views

How to take the derivative of this

I got this problem in school, but do not know how to answer it, does anyone know how? How would you take the derivative of $$\frac{a}{\cos\ \theta} + \frac{b}{\sin\ \theta}$$ and then set the answer ...
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1answer
34 views

Function that determines angular velocity?

I see that someone posted the same problem a year ago, but the answer didn't quite give enough info. Here's the question: A movie crew is working on a scene that involves filming a car moving at a ...
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1answer
43 views

A debate over the limit of $ \frac{f(x + a h) - f(x + b h)}{h} $ as $ h $ approaches $ 0 $.

This may seem like an easy question, but a few of us are having a debate over it. We are looking at the following limit below, where $ f $ is a real-valued function on an open subset $ U $ of $ ...
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1answer
1k views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
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1answer
27 views

Angle between slopes of a curve

I am trying to understand what the change in angle of the slope of a curve means. It is hard to explain with words so here's an image that should help. The red curve has had its derivative ...
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2answers
34 views

Differentiate problem with respect to x [on hold]

Really basic problem I'm stuck on. $y=\frac{4}{3 \sqrt{x}}+\frac{1}{3x^2}$ with respect to $x$ any help would be appreciated.
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5answers
54 views

Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable. Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$ Consider $g(x):= ...
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0answers
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This should be an easy derivatives problem. Composite functions. [on hold]

The number of locusts (given by $\ell$) after $d$ days of an infestation is given by the equation $$ \ell = 5d^2 + 10d + 100 $$ The area of grass left (given by $g$) in square meters is given by ...
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2answers
18 views

Proving that a positive derivative means the function is smaller “to the left” and larger “to the right” for certain values

I was trying to prove that if $g$ is differentiable on an open interval $I$ with $a\in I$ and $g'(a)>0$ then we can find $x<a$ for which $g(x)<g(a)$ and $y>a$ for which $g(y)>g(a)$, I ...
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1answer
37 views

Is function $f(x,y)=\begin{cases}(x^{2}+y^{2})(\sin(x^{2}+y^{2}))^{-1/2}, (x,y)\neq (0,0)\\0,(x,y)=(0,0)\end{cases}$ differentiable?

Is this function differentiable at (0,0)? . $f(x,y)=\begin{cases}\frac{x^{2}+y^{2}}{\sqrt{\sin(x^{2}+y^{2})}}, (x,y)\neq (0,0)\\0,(x,y)=(0,0)\end{cases}$ \begin{align*} \lim_{h\mapsto 0} ...
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1answer
192 views

Matrix Derivations-Research

$DT_3(R)$ is the upper triangular matrix with the diagonal being the same element over the real field R, for example $\begin{bmatrix}a&b&c\\0&a&d\\0&0&a\end{bmatrix}$ where ...
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2answers
47 views

Differentiation of a Vector with respect to a vector

I am studying a paper and I am going crazy about one differentiation which it is written on it but not explained. I think I am missing something and probably something easy. I would love if someone ...
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1answer
23 views

Differentiating a purely imaginary function

If a function is defined as being the imaginary part of some expression, how do I take the derivative of the function? Do I: (a) Take the imaginary part of the expression, and then differentiate? ...
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65 views

Does $\lim_{x\rightarrow\infty}[ f(x)+f'(x)]=0$ imply $\lim_{x\rightarrow\infty}f(x)=0$? [duplicate]

Let $f$ be a continuous function with continuous derivative such that the $\lim_{x \to \infty}[f(x)+f'(x)]=0$. Is it true that the $\lim_{x \to \infty}f(x) = 0$? Thanks for your help.
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1answer
74 views

derivative of the integral

I am working on a few problems, just need some help to see if I'm working them correctly, $$(1)\;\;\;\;\;g(x)=\int_0^x(x-u)e^{u^2}du$$ find $g'(x),g''(x)$ $$(2)\;\;\;\;\;\psi(x,y)=\int_1^xe^{ty}dt$$ ...
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0answers
16 views

Leibniz rule under double intergral

I have looked through other posts about Leibniz rule under integral, and I have done with my problem. But I am not sure if it's correct or not, please check it for me if you don't mind. ...
3
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3answers
76 views

Prove that $g(x):=|f(x)|$ is differentiable iff $f'(a)=0$

Suppose that $f(a)=0$. Prove that $g(x):=|f(x)|$ is differentiable iff $f'(a)=0$ Not sure how to go about this at all. The limit definition that I am working with is $$ g'(a)=\lim_{x \rightarrow ...
2
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2answers
59 views

How to solve the derivative of $b^x$ using the defintion

I know that the derivative of $b^x$ is just $b^x \log{(b)}$, and I've seen it being derived using chain rule and such (not that I understand how it's done, I just learned about $e$ today so using the ...
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0answers
12 views

Minimising the surface area of a rectangular prism [Solution Verification]

A packaging company is going to make open topped boxes, with square bases that hold $100$ centimetres$^3$. What are the dimensions of the box that can be built with the least material?