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

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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
48 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
28 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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2answers
87 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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3answers
48 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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1answer
16 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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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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12 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
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
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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0answers
8 views

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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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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0answers
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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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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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3answers
36 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
21 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
37 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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1answer
18 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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0answers
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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
32 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
65 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
42 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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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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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
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
36 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
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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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. ...
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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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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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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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0answers
11 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?
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26 views

Derivatives of semi-elasticities

I want to find the sign of $\partial_z (\partial_z \log F(T1) - \partial_z \log F(T2) )$ where $T1<T2$ and $F(T) = \int_0^T e^{g(t) + z\, h(t)} dt$ All we know is that $h(t)>0\forall t$ ...
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60 views

Why can't we differentiate constants like variables?

I understand why we the derivative of $x$ is $1$. Similarly, the derivative of a constant function $y=a$ is $0$, because it's slope is flat, since for every $x$, $y=a$. But can you please tell me why ...
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0answers
55 views

Div, grad, curl in curvilinear coordinates

I've a lot of different formulas for div, grad, curl, and laplacian in different coordinate systems. How are these formulas derived? What's the general procedure for finding the formula of say the ...
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1answer
29 views

Numerical approximation of differentiation

I have the following task to solve: Let $b>x$ be defined, determine $w_0,w_1$ and $w_2$ in dependency of $b$ such that the approximation $f''(x) \approx w_0 f(x-h) + w_1 f(x) ...
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0answers
21 views

derivative of a tensor function

If e is a second order tensor and it's symmetrical, assuming abs() is a tensor function such that returns all the components of e be positive, I am interested the derivatives of the function abs(e) ...
2
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1answer
40 views

Why this particular definition for the derivative on a group ring?

I am reading Introduction to Knot Theory by Crowell and Fox, and I am a bit confused at the way they define a derivative on a group ring (and on a group). I understand a derivative (or derivation ...
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1answer
17 views

Maximize yield given two products and price

Just started learning to use derivatives to apply to real world, and this homework problem is really tripping me up, and getting a formula to start is even giving me a bunch of trouble. If fertilizer ...
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1answer
29 views

Difference Between Differentiation on Time vs Position

I'm hoping somebody could help me understand the difference between the following: $$∂_tc(x,t)$$ $$∂_xc(x,t)$$ My understanding is the the top derivative would be something like velocity but what ...
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2answers
27 views

A question involving Frechet differentiability

Let $X, Y$ be real normed spaces and $U \subset X$ open subset. In "Nonlinear functional analysis and applications" edited by Louis B. Rall, we have the followint definition (page 115) A map $F : U ...
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2answers
60 views

Differentiating the function $\arcsin(3x-4x^3)$

When I have to differentiate the function $\arcsin(3x-4x^3)$ which of the following methods is more appropriate ? Putting $x=\sin θ$,simplifying and then differentiating for certain ranges of $x$. ...
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1answer
34 views

Derivative of an expodential function

Let the function: \begin{equation} \phi(x)=\begin{cases} e^{-\frac{1}{1-x^2}}, & x\in (-1,1)\\ 0, &x \not \in (-1,1)\end{cases} \end{equation} How can I show that $\phi(x)$ is differentiable ...
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0answers
14 views

Gateaux derivative vs Frechet derivative - what is the essential difference?

To a non-mathematician with a background in calculus, what is the primary difference between the two derivatives? What is happening in the infinitesimal?
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1answer
81 views

The importance of being real

Let $\Sigma$ be a collection of holomorphic, one-to-one function from some simply connected region $\Omega$, which map $\Omega$ into the open unit disc $U$. Fix $z_0 \in \Omega$ and put $$\eta = ...