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

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0
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1answer
19 views

f '(x) = -f(x) and f(1) = 1, Solve for f(2)

I am honestly not even sure how to start this problem... My first thought was that f(2) = 2 ... But now I don't even know where to go from there.
-1
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1answer
15 views

Find the line normal to the curve at a specified point [on hold]

Find the line normal to the curve $xy^2 + 2xy = 8$ at the point $(1,2)$.
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0answers
10 views

Mean Value Inequalities for vector-valued functions

Let $X$ and $Y$ be Banach spaces, and let $U\subset X$ be open. If $f\colon U\to X$ is continuously differentiable and $x,v\in X$ are such that the line segment $\ell=\{x+tv\mid t\in[0,1]\}$ lies ...
0
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1answer
27 views

Why is the chain rule applied to derivatives of trigonometric functions?

I'm having trouble to understand why is the Chain rule applied to trigonometric functions, like: $$\frac{dy}{dx}cos 2x=[2x]'*[cos 2x]'=-2 sin 2x$$ Why isn't it like in other variable derivatives? Like ...
3
votes
1answer
26 views

Stochastic calculus - Ito confusion

We have $W(t) = f(t)X(t)$. My textbook says that $dW = fdX + X\dfrac{df}{dt} dt$. I don't get how they arrived at this conclusion. I get the first part, because $\dfrac{dW}{dX}dX = fdX$. But for the ...
2
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1answer
32 views

Understanding the Definition of a derivative as slope of a tangent line

I'm trying to understand the derivative and am wondering why the derivative is described as the slope of the tangent line and not the slope of a function itself. Say $f(x) = 2x+5$ where ...
0
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1answer
9 views

What is the mean value theorem for the Fréchet (total) derivative?

What is the mean value theorem for the Fréchet (total) derivative? Off the top of my head, it's something like $$ \|F(x+h)-F(x)\|\leq \sup_{c\in[0,1]} \|F^\prime(x+ch)\|\|h\| $$ but the double ...
-1
votes
3answers
44 views

Maximum & minimum area of rectangle outside another.

Find the maximum & minimum area of an outer rotated rectangle when the inner rectangle has the side lengths $a$ and $b$. Here's an image: I have already tried to relate the side of ...
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0answers
18 views

Grand canonical derivative.

I've been trying to work out how to find the density in the thermodynamic limit of a nearest neighbour magnetic lattice gas in the grand canonical ensemble. I'll with hold the Hamiltonian for the ...
2
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0answers
8 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
4
votes
1answer
26 views

Can I interpret the exponential of the derivative operator, $e^D$, as infinite shift operators each shifting “infinitesimally”?

To better explain what I mean, an example can be very useful. Consider $e^{i\theta}$. We could express this using the series definition or the limit definition of $e^x$ instead: $$e^{i\theta} = ...
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1answer
40 views

Real Analysis Question- Differentiability on an interval

So this is the question I am trying to answer... At $(-2,0)$ and $(0,2)$ we just differentiate using normal rules of calculus, yes? Here is my attempt for at $x=0$. Is this correct? For d)I think ...
3
votes
3answers
103 views

Easy question : $\int (xdy+ydx)$

I am ashamed to ask such an easy question but, well: Lets say I got a function $$ f(x,y)=xy $$ Now let's compute the total differential of the function $$ d(f(x,y))=xdy+ydx $$ Now if I do $$ \int ...
1
vote
1answer
27 views

Find derivative using the definition of the derivative as a limit

$$f(x) = \frac{1} {\sqrt{x}}$$ find $f'(x)$ using the definition of the derivative as a limit. I know that $$ f'(x) = \frac{(x + \delta)^{-1/2} - (x)^{-1/2}}{\delta} $$ as $\delta$ goes to $0$. ...
2
votes
4answers
57 views

Real Analysis - differentiable

$f:[0,\infty]\rightarrow \mathbb{R}$ is twice differentiable. If $f''$ is bounded and exists the limit of $f(x)$ at infinity, then $\lim_{x\rightarrow \infty}f'(x)=0$. I tried to use the Taylor's ...
4
votes
7answers
379 views

Understanding derivatives

I don't know if this is written somewhere else. I've looked all over the internet so apologies if this has already been covered. I'm doing Year 12 Maths in Australia for what it's worth. In our ...
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0answers
13 views

selecting points on a domain which represent the derivative of a function

I'm working on some algorithm part of which entails me to subdivide a domain based on the derivative of a function. Let's just consider the 1D case with a closed and bounded domain $[a,b]$ for a ...
1
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2answers
73 views

derivative of $\ln(4)$

what is the derivative of $\ln(4)$? I am trying to find the derivative of this equation: $h(x)=\ln(\frac{x^3\cdot e^x}{4})$ by rules of logs I simplified the $h(x)$ to the following: ...
0
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0answers
19 views

Generalized “softmax”

I'm looking for a function $f$ from $\mathbb{R}^n$ to $[0,1]^n$, similar to softmax in the sense that is satisfies these properties: $\sum_i f(x)_i = c$, where $c$ is a chosen constant (i.e., c=1 ...
6
votes
2answers
87 views

Limit of $\frac{f\left(x+g\left(x\right)\right)-f\left(g\left(x\right)\right)}{x}$ as $x\to 0$

I'm trying to answer the following question: Let $f$ be continuously differentiable in all of $\mathbb{R}$ and let $g:\mathbb{R}\to\mathbb{R}$ be a function satisfying ...
1
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2answers
32 views

Using the same limit for a second derivative

I've been trying to answer the same question answered here: Second derivative "formula derivation" And I'm stuck in a step that is not addressed both in the answer and in the comments of ...
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0answers
16 views

diffeomorphism on intervals on R

I came across a line in a proof which involved a diffeomorphism $f:I_1 \rightarrow I_2$ (with $f$ a homeomorphism, $f,f^{-1}\in C^{\infty}$) mapping open intervals in R, which claimed that ...
1
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1answer
36 views

Solving second order nonhomogeneous linear equation

So i have the equation $$\frac{d^2y}{dt^2} + y = \sin(t)$$ I know the first step is to find the corresponding homogeneous equation, which i think would be: $$r^2+1=0$$ giving real roots and therefore ...
-1
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1answer
24 views

Check differentiablity of $f$ [on hold]

Consider a function \begin{equation*} f(x)=|\cos x|+|\sin (2-x)|. \end{equation*} At which of the following points f is not differentiable? a)$\{(2n+1)\frac\pi2:n\in \Bbb Z\}$ b)$\{n\pi:n\in \Bbb ...
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0answers
24 views

A “prove or disprove question” on absolutely continuous functions [on hold]

Let $f:\left[a,b\right]\rightarrow\mathbb{R}_{+}$ be an absolutely continuous function. ($a<b$). Prove or disprove that the right ( respectively left) derivative of f exists at each point of the ...
2
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1answer
38 views

Are the extrema of this function global or local?

Last question about this function, I promise. The function $f: \mathbb R \rightarrow \mathbb R$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; x < -3 \\ 0 & ...
0
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1answer
20 views

Derivatives of real part of function

My physics textbook gives me the complex form equation of simple harmonic motion as: $$z = Ae^{i(\omega _{o}t+\phi )}$$ and then defines $$ x = Re (z) $$ From there they argue that $$ \frac{\partial ...
1
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1answer
56 views

Chain rule and a function of a function of a function

Suppose we have a composite function: $f(g(h(x)))$, and we want $\frac{\partial f}{\partial h}$. By the chain rule $\frac{\partial f}{\partial h} = \frac{\partial f}{\partial g}\frac{\partial ...
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3answers
51 views

Derivative of a Rational function $f(x)=\sqrt{2x-5\over3x+1}$

I'm trying to find the derivative of, $$f(x)=\sqrt{2x-5\over3x+1}$$ I think I can change this into $$f(x)= \left({2x-5 \over 3x+1}\right)^{1\over2} \\ =[(2x-5)(3x+1)^{-1}]^{1 \over 2}$$ Am I not ...
1
vote
1answer
27 views

Deciding whether $(ax+b)/(cx+d)$ is increasing

In order of studying usual function such us : 1) $f(x)=ax^2$ 2) $g(x)=\frac{a}{x}$ 3) $h(x)=\frac{ax+b}{cx+d}$ 4) $p(x)=ax^2+bx+c$ To know if a function is increasing or decreasing we can ...
1
vote
1answer
81 views

Feynman Integration Problem

$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 ...
1
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1answer
35 views

Critical Point of $\mathbb{R}^2 \to \mathbb{R}^2$ function

Given a function $f:\mathbb{R}^2 \to \mathbb{R}$ I can find critical points by finding the $1\times 2$ Jacobian matrix, setting each partial derivative equal to zero and solving the equations. I can ...
8
votes
1answer
80 views

If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$

Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that $f(0)=f(1)=f(2)=0$ Prove that $\forall x\in[0,2], \exists c\in[0,2], f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ This problem got me stuck. I ...
1
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1answer
27 views

Can someone please help me in proving this?

Let $k_{2}>k_{1}>0$, prove that for any $x>0$, $f(x)$ is a monotonically increasing function. $$ f(x)=\frac{1-e^{-k_{1} x}}{1-e^{-k_{2} x}}. $$ We can have ...
0
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2answers
27 views

With $f(x)= 32 \cosh(x) \sinh(2x) $, determine the slope of its tangent at $( \ln 2 , \, 75)$

With $f(x)= 32 \cosh(x) \sinh(2x)$, determine the slope of its tangent at $( \ln 2 ,\, 75)$. My work $$\sinh x \cosh y = \frac{1}{2}(\sinh (x + y) + \sinh (x - y))$$ $$\cosh(x) \sinh(2x)= ...
0
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2answers
25 views

$y = 3\sin^{-1}(\sqrt{x})/x$ find $y'(1/4)$

$y = 3\sin^{-1}(\sqrt{x})/x$ find $y'(1/4)$ my work is that y'= (x*$ ('\sin^{-1}(\sqrt{x}))+\sin^{-1}(\sqrt{x}) *1$)/(x)^2 my problem how to Derivative $ \sin^{-1}(\sqrt{x})$ ...
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vote
2answers
22 views

Derivatives problem

Given the equation $f(x)=\frac{2x+4}{\sqrt{x}}$, evaluate $f(0.5)$ and $f'(0.5)$. I am having a problem understanding the problem. The first part is straight forward, but it's the second part I'm ...
1
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0answers
32 views

Prove that $g$ is differentiable and that $g'$ is not differentiable at $0$

For part ($c$) I assume I have to show that the limits from above and below are equal? I am having trouble doing this though... I get limit as $h$ tends to $0$ of ( $(4-h^2)^{0.5}$ - ($4^{0.5}$) ...
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3answers
42 views

A really basic integration question concerning differentials

I'm really, really confused with this. Please, please help me. $$$$ My Calculus teacher taught me that the integral symbol and the differential with respect to which we are integrating are like ...
0
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1answer
33 views

if g is not constant zero, $f\circ g$ has a local minimum at zero

Consider $f:\mathbb{R}^2\to\mathbb{R}\; f(x,y)=(x^2-y)(x^2-3y)$ and a linear function $g:\mathbb{R}\to\mathbb{R}^2,\; x\mapsto \begin{pmatrix} g_x(x)\\ g_y(x) \end{pmatrix} $. The claim is: If $g$ ...
0
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1answer
67 views

How to prove this limit of derivative to zero [on hold]

This is a test question in real analysis and I need help to prove it. Let $f:\mathbb{R}\to\mathbb{R} \in C^{\infty}$ be periodic of period $1$ and nonnegative. Show that $$ ...
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0answers
15 views

Can we calculate the derivative of a distribution function with respect to its parameters?

I am asking a very basic question. Can we calculate the derivative of a density function with respect to its parameters, mean and variance? Can we calculate the derivative of a distribution function ...
1
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1answer
45 views

Alternate formula for $Γ(t + 1)$

I believe that: $$\frac{d^{n}}{dx^{n}}[x^{n}] \equiv Γ(n + 1) \equiv n!$$ Would this have any application, if it has not already been discovered, which I am almost certain that it has?
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1answer
26 views

Differentiating expression involving summation

My problem seemed very simple at glance but I keep missing one term from the answer. Any suggestions? This is the problem: We have $$x_i^* + \xi_i + \frac{\alpha_i}{p_i} \left[ y - \sum_{j=1}^n ...
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0answers
24 views

Call spread derivative [on hold]

delete this tbh lads, this is not the site for finance questions apparently Using the notation $V(E)$ to mean the value of a European call option with strike $E$, what can you say about ...
0
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1answer
21 views

derivative of infinite sum of terms

If we have $\sum_{i=1}^\infty f_i(x)$ and assume this is a convergent sum and asumme all the $f_i$ are differentiable in every point. Is the derivative of the infinite sum equal to the sum of the ...
0
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1answer
27 views

Differentiating a sum involving logs

I was doing the problem provided in the picture but I do not understand how do they obtain the answer. I am not sure how to differentiate the sum. I end up getting: alpha - 1 - 1/K. I believe I need ...
0
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2answers
28 views

The Fourier transform of functions with compact support is differentiable.

1) How can I prove that if $f(x)$ is a continous function with compact support (let's say $f(x)=0$ $\forall x\in B(0,R)^c$), then its Fourier transform $\hat{f}(\xi)$ is differentiable? 2) Is there ...
2
votes
1answer
40 views

Can critical point that $f''$ has different sign in its every neighborhood be a local extreme point?

Suppose that $f$ is a second order derivable function on $[0,1)$, and $f'(0)=0$. It is true that: If there exits $\delta>0$ such that $f''(x)\geq0$ for all $x\in[0,\delta)$, then $0$ is a local ...
2
votes
1answer
41 views

Can a function be differentiable at the end points of its (closed interval) domain?

Assume $f$ has a domain of $[a,b]$. Is it possible that $f$ is differentiable on the closed interval $[a,b]$, or must the maximal domain for $f'$ be $(a,b)$?