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

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1answer
26 views

A function $g \in C^{1} (\mathbb{R})$ with $g(0)=0$ verifies $|g(s)| \leq C|s|$ for $s \in [-M,M]$

I need to show that a function $g \in C^{1} (\mathbb{R})$ with $g(0)=0$ verifies $|g(s)| \leq C|s|$ for $s \in [-M,M]$. This is what I have done: Using Mean Value Theorem, we have: ...
3
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5answers
251 views

Check if a Function is Differentiable at a Point

Let $$f(x)= \begin{cases} x+1 & x\leq0 \\ 3^{-x} & x>0 \end{cases}$$ Is the function differentiable at $x=0$? I should look at the limits $$\lim_{h\to 0-} ...
0
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1answer
38 views

Calculation of $DF$ of $F(x,y,z)=(x+y+z,x^2+y^2+z^2,xyz)$

Calculate the differential, the Jacobian matrix $JF(1,0,2)$ and $[DF(1,0,2)](2,3,0)$ of $$F(x,y,z)=(x+y+z,x^2+y^2+z^2,xyz)$$ My thoughts so far: the differential is the linear mapping ...
2
votes
1answer
59 views

when is it safe to swap the order of differentiation with integration?

Let $f(u)=\cos u$ and $$g(u)=1-u^2,-1\leq u \leq 1;g(u)=0, otherwise\tag{0}$$. Thus $$\partial_u^2g(u)=-2,-1\leq u \leq 1;\partial_u^2g(u)=0, otherwise\tag{1}$$. Let $h(u)$ be the convolution of ...
3
votes
3answers
109 views

minimum point of $x^2e^{-x}$

I have looking for the minimum point of $$f(x)=x^2e^{-x}$$ I differentiated once and got $f'(x)=-e^{-x}(-2+x)x$ so $x=2$ and $x=0$ can be min/max points. I have differentiated again and got ...
3
votes
1answer
78 views

Derivative of functions of the form $f(x)^{g(x)}$

I am trying to find the derivative of the function $h(x)=f(x)^{g(x)}$. I just wanted to be sure my derivation was correct: We proceed by using logarithmic differentiation. $h(x)=f(x)^{g(x)}$ $\log ...
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1answer
42 views

Matrix and derivations rules

I am studying control systems and I noticed that i had problems specially when it comes to matrix derivations . I am looking for a memento to remember rules about this subject. Does anyone have ...
1
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0answers
40 views

Meaning of a theorem regarding the limits of derivatives?

I'm confused about this theorem which is sometimes associated to the Darboux theorem for real functions. Let $f: dom(f)\subseteq\mathbb{R} \rightarrow \mathbb{R}$ be a function continuous in a point ...
5
votes
3answers
79 views

Differentiable function such that $f(x+y),f(x)f(y),f(x-y)$ are an arithmetic progression for all $x,y$

If $f$ is a differentiable function on $\mathbb{R}$ such that $f(x+y),f(x)f(y),f(x-y)$(taken in that order) are in arithmetic progression for all $x,y\in \mathbb{R}$ and $f(0)\neq0,$then ...
0
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1answer
21 views

Getting the derivative of this time dependent vector.

This vector is time dependent with the following formula: $$ \begin{align} \vec b(\color{blue}{t}) = & \ ( \color{#e69900}{r} \cos(\color{#e69900}{\omega} \color{blue}{t})+\color{#e69900}{l}\sin ...
-2
votes
1answer
55 views

Derivative of $y=\sqrt{e^{6t}}$? [closed]

How to differentiate $y=\sqrt{e^{6t}}$? I know the answer is $3\sqrt{e^{6t}}$. But I couldn't get it.
4
votes
0answers
89 views

Exterior Differential (and its Equivalent Differential Operator) of an Integral 0-Form

I am reading Witten's 1982 paper "Supersymmetry and Morse Theory," and while I am slowly learning the material as I read through the paper, I have come across an equivalence that, while it should be ...
0
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2answers
36 views

Some unsettling technical issue when solving probability expectation problems

I'm suddenly not sure about something I used a few times. Say I get to the expectation of some descrete random variable with parameter $p$, say $p$ is the probability of success: $\sum_{k=0}^\infty ...
0
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0answers
29 views

Derivative of the integral of a vector function

Consider the following function: $E({\bf y(x)}) = \int\|{\bf y(x) - t}\|^2~p({\bf x})~d{\bf x}$ I want to compute $\frac{d E}{d {\bf y(x)}}$. My main concern is not the vector derivative of the ...
2
votes
0answers
40 views

Matrix inverse series expansion

I want to prove that when $I+K$ is invertible, $$(I+K)^{-1}=I-K+o(K)$$ to establish that the matrix inverse function has derivative $-I$ at $I$. My hope is that this identity carries over from ...
3
votes
1answer
59 views

What does a power of $|\nabla|$ mean in the context of PDE?

I've seen the notation $|\nabla|^\alpha f$ used in a PDE setting, where $f$ is some function on $\mathbb R^n$. Could someone tell me what that means? For example in this discussion on Math Overflow ...
0
votes
1answer
25 views

Which of the following options are true in case of a differentiable function [duplicate]

Let $f:\Bbb R\to \Bbb R$ be a differentiable function such that $\sup _{x\in \Bbb R} |f^{'}(x)|<\infty$. Then which are true: $f$ maps bounded sequences to bounded sequences. $f$ maps Cauchy ...
1
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2answers
51 views

how to find the slope of a curve at origin

What is the slope of the curve $x^3 + y^3 = 3axy$ at origin and how to find it because after following the process of implicit differentiation and plugging in $x=0$ and $y=0$ in the derivative we get ...
0
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1answer
29 views

Examine the function $f(x)=\tan^2(\arccos x)$

How to find the maximum domain of the function $f(x)=\tan^2(\arccos x)$. Show that $f(x)$ is a rational function. Calculate $f'(x)$ and discuss the monotony of $f$. Draw the graph of $\arccos$ and ...
1
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2answers
57 views

Let $f(x) = x^n$. Show that $f^{(n)} = n!$ and $f^{(m)} (x) = 0$ for all $m > n$.

I'm supposed to use mathematical induction to solve this Show that $P(1)$ is true Assume $P(K)$ is true Show that $P(K+1)$ is true How do I approach this problem?
1
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1answer
52 views

Related Rates (Pyramid)

This is the question → I'm having trouble forming an equation for all of this question. Also do I have only have to use one chain rule or 2 chain rules to find $\frac{\mathrm{d}SA}{\mathrm{d}t}$, I ...
3
votes
1answer
48 views

Fréchét derivative, its inverse and existense of a solution to $X^2 = B$ in an open subset of $M_2 (\mathbb{R})$

Let $A = \operatorname{diag}(\lambda_1, \lambda_2) $ and let $f:M_2 (\mathbb{R}) \to M_2 (\mathbb{R}), f(X)= X^2$. We consider an arbitrary matrix norm. If we look at the Fréchét derivative at $A$ of ...
0
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1answer
74 views

First and second derivatives of max function

I have two functions $f(x)=(x-K)_+$ and $g(x) = \max\{x,K\}, x \geq 0, K = const \geq 0$. I was told that $$f'(x) = \mathbb{I}_{[K,+\infty)}(x)$$ and $$f''(x)=\delta_K(x),$$ because ...
0
votes
1answer
67 views

How do I calculate the derivativee of the function $x\mapsto \sqrt{x}$ in two ways?

I need to calculate the derivative of the following function in two ways: $$ f\colon\mathbb{R}^+\to\mathbb{R}^+, \quad x\mapsto \sqrt{x} $$ a) by means of differential quotient b) using the ...
3
votes
3answers
139 views

Can one find a line that is tangent to a cubic polynomial more than once?

I know that any line cannot be tangent to the graph of $y=Ax^3+Bx^2+Cx+D$ at more than one point. Question: how can one show this, or even prove it?
0
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1answer
26 views

Does $f''(0^+)=f''(0^-)$?

Consider the function, which is the join of two semicircles $$ f(x) = \left\{ \begin{array}{cc} \sqrt{1 - x^2} & x > 0 \\ 1 & x = 0 \\ \sqrt{ 2 - (x-1)^2} & x < 0\end{array} ...
-1
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1answer
21 views

Differentiability of a function of two variables

Is the function \begin{equation} f(x, y) = \begin{cases} \frac{2xy}{\sqrt{x^2 + y^2}} & (x, y) \ne (0, 0) \\ 0 & (x, y) = (0, 0) \end{cases} \end{equation} differentiable at the origin? ...
1
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0answers
15 views

How to adjust finite differencing method for mapping from $\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ where $n = m^{2}$?

So I'm supposing $F:\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is differentiable, and I have MatLab code that evaluates $F$ at an arbitrary $x$ in $q$ flops. I know that given $F(\bar{x})$ where ...
0
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4answers
69 views

Show that limit of $g(x) = 0$ when $x$ goes to infinty

$f(x)$ is differentiable for $x > 0$, and $$\lim_{x\to\infty} f'(x)= 0\,.$$ We have a new function $g(x) = f(x+1) - f(x)$. Prove that: $$\lim_{x\to\infty}g(x) = 0\,.$$ I thought about using ...
2
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0answers
37 views

If number of points of discontinuity of the function $f(x)=\lfloor{2+10 \sin x\rfloor}$,in $[0,\frac{\pi}{2}]$ is same as

If number of points of discontinuity of the function $f(x)=\lfloor{2+10 \sin x\rfloor}$,in $[0,\frac{\pi}{2}]$ is same as number of points of non-differentiability of the function ...
3
votes
1answer
115 views

Population dynamics

I don't understand why we make the three assumptions underlined above.
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2answers
42 views

The nth derivative of $x^a$

I've three cases with $f(x)=x^a$: $$\frac{\text{d}^n}{\text{d}x^n}\left(x^a\right)=\begin{cases}a!\space\space\space\space\space\space\space\text{when}\space n=a\\ ...
0
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1answer
22 views

Parametric Equations (Concavity)

The question is: A curve is defined by the parametric equations $$ x = t^2 + a $$ $$ y = t(t-a)^2 $$ Find the range of values for t in terms of a where the function is concave up? What I have ...
1
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1answer
78 views

Multivariable chain rule: how to take this derivative with respect to a matrix?

I have a simple model and I want to update the parameters using a gradient descent algorithm. Thus I must find derivative with respect to my parameters. Here is my model: $$s = Wx + b$$ $$a = max(0, ...
3
votes
3answers
79 views

Work differential: Why $\mathrm{d}W = f \mathrm{d}s$ not $\mathrm{d}W = f \mathrm{d}s + s\space \mathrm{d}f$

Starting from the formula for work given a constant force $W = f s$, if you take the differential of both sides you would expect to get: $$\mathrm{d}W = f \mathrm{d}s + s \space \mathrm{d}f$$ by an ...
1
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2answers
72 views

Differentiating this integral,

I want to show that $u_{xx} + u_{yy} = 0$ for the integral given below, so I think I want to differentiate under the integral with respect to both $x$ and $y$. The goal is to show that $u$ is ...
0
votes
1answer
28 views

Vector by vector derivative of linear expression

I have to take the derivative of $y$ with respect to $x$ of: $$y = Ax + b$$ I am unsure if the answer is supposed to be $A$ or $A^T$. Here is my working of the problem: $$y_i = \Sigma A_{ij}x_j + ...
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1answer
81 views

Slope of a curve at origin

Suppose there is a general curve f(x,y)=0.We need to find the equation of tangent at origin....Suppose I wish to use the trivial method....i find the derivative of the given function using implicit ...
0
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1answer
37 views

How to prove that a function is continuous in functional equation?

I was wondering about different methods or properties to prove that a function in a functional equation is continuous or differentiable. Can somebody give me some examples of such problems or methods, ...
4
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2answers
70 views

Given a function $f(x)$ defined for all real $x$,and is such that $f(x+h)-f(x)<6h^2$ for all real $h$ and $x.$Show that $f(x)$ is constant

Given a function $f(x)$ defined for all real $x$,and is such that $$f(x+h)-f(x)<6h^2$$ for all real $h$ and $x.$Show that $f(x)$ is constant. To prove $f(x)$ as constant i need to prove that ...
0
votes
1answer
50 views

Finding the derivative using Fermat's Method

I have been having trouble understanding how to calculate the derivative using Fermat's Method when it comes to various variables. For example, when given, $$ y^2 = x(8-x), $$ is there a general ...
0
votes
1answer
35 views

How do I apply ∂/∂x in these examples?

If we have a function $ f(x,y(x)) $ it is clear to me that $$ \frac{df}{dx}=\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}\frac{dy}{dx} \tag{A} $$ so $ \frac{df}{dx} \neq ...
1
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1answer
29 views

Implications of continuous differentiability at a point

Consider a function $f:\mathbb{R}^l\rightarrow \mathbb{R}$ continuously differentiable at $x_0$. This implies that (1) the function is differentiable on a neighbourhood of $x_0$ which means that the ...
1
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2answers
39 views

Prove that if $f$ is a real valued function on a open connected subset of $\mathbb{R}^n$ and $f_i^{'}=0$ then $f$ is constant

Prove that if $f$ is a real valued function on a open connected subset $U$ of $\mathbb{R}^n$ and $f_1^{'},\ldots, f_n^{'}=0$ then $f$ is constant. If $U$ is an open ball then it is easy. Choose any ...
0
votes
3answers
54 views

differentiating $\frac{5x}{1 + x^2}$ using the power rule

I am trying to differentiate $s=\frac{5x}{1+x^2}$. I have decided to tackle this by using the power rule if $y=x^n$ $$\dfrac {dy}{dx}=nx^{n-1}$$ This gives me $$\dfrac {dy}{dx}=\frac{5}{2x}$$ I got ...
1
vote
1answer
38 views

Determine point of maxima and minima of the function

Determine point of maxima and minima of the function $f(x)=\frac 1 8 \log x -bx +x^2$, $x> 0$, where $b \geq0$ and is a constant. I found out $f'(x)$ and equated it to $0$ and got it as $$8 ...
1
vote
2answers
64 views

is the Derivative's power rule wrong?

I'm studying about the proof of Derivative's power rule and confuse in the algebra of this limit: consider : $ f(x) = x^n $ , $n = 0, 1, 2, 3, ...$ $n=0 : f(x) = x^0 = 1 $ (where x not equal 0) $ ...
5
votes
3answers
121 views

Differentiating $x=1$ with respect to $x$

Sorry this may sound like a silly question and I know that this does't meet the quality standards of Math S.E, I found this in one of the Math-Jokes websites and found it interesting, $$x=1$$ ...
0
votes
3answers
38 views

Find the slope of the tangent to the curve for $y=3+4x^2-2x^3$

I am completing a question which asks me to find the slope of the tangent to the curve for $y=3+4x^2-2x^3$ at the point $x=a$. In Klein's book on calculus, he shows that you can find the rate of ...
0
votes
1answer
67 views

Optimisation Problem on butter melting

The time in minutes ($t$) taken to melt 100 g of butter depends upon the percentage of the butter which is made of saturated fats ($p$) as in the following function: $$ t = \frac{p^2}{10000} + ...