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

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1answer
42 views

Differentiating $\int_{0}^{\tan(x)} \sin(t^2) \, dt$

My attempt: $$f(x)=\int_{0}^{\tan(x)} \sin(t^2) \, dt$$ $$\frac{d}{dx}f(x)= \sec^2(x)\sin(\tan(x)) \, $$ I don't know if this is correct. I have a test tomorrow and I have nowhere else to check. ...
0
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0answers
29 views

Deriviative of Triangular Moving Average

Maths is not my strong point and I would like to know the formula to the first derivative of a triangular moving average. The source formula can be found here ...
1
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1answer
24 views

Which changes faster $ f(m) = {n \choose m}$ or $ g(m) = n^{-2c \frac{mn - m^2}{n-1}}$ wrt $m$

I have two functions of $m$. $m$ and $n$ are both integers with $m <n$. The functions are :- $$ f(m) = {n \choose m}$$ and $$ g(m) = n^{-2c \frac{mn - m^2}{n-1}}$$ $c$ is a constant which you ...
1
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2answers
40 views

Show |f(b)-f(a)| ≤ (1/2)|b-a|

So I'm trying to show that for: $f(x)=\ln(\sqrt{1+x^2})$ $|f(b)-f(a)|\le \frac{1}{2}|b-a|$ for all $a, b \in \mathbb{R}$. Honestly. I'm pretty lost here. How would I even start this off?
1
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4answers
76 views

Show that $\dfrac{ -1 }{ 2 } \le \dfrac{ x }{ 1+x^2 } \le \dfrac{ 1 }{ 2 }$

So I'm trying to show that: $\dfrac{ -1 }{ 2 } \le \dfrac{ x }{ 1+x^2 } \le \dfrac{ 1 }{ 2 }$ for every value of x. I know I have to use mean value theorem so I tried to show it with cases. First I ...
-1
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0answers
23 views

Extrema of multivariable functions with Lagrange multipliers and bounded domains

I understand how to do Question 3 but I don't quite know how to do Question 4b. If anyone can offer any help I would really appreciate it :)
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2answers
23 views

What type of initial value problem is this?

So I'm trying to solve this initial value problem: $x^2 dy/dx + xy = 1$, $y(-1)=1$ Now I think that it's some sort of Linear Equation and I know how to solve linear equations like $dA/dt+1/100A=6$, ...
3
votes
1answer
71 views

Frechet Derivatives of a nonlinear integral operator

The nonlinear integral operator $P:C[0,1]\to C[0,1]$ is defined as follow: $$P(f)(x)=1+kxf(x)\int_0^1\frac{f(s)}{x+s}ds$$ In order to obtain the Frechet derivative of the operator, I start with: ...
2
votes
1answer
40 views

Proving that $Df(\vec{0})=\vec 0$ for differentiable and even function $f : \mathbb{R}^n\to \mathbb{R}$

Let $f : \mathbb{R}^n\to \mathbb{R}$ a function so that $f(\vec x)=f(-\vec{x})$ for all $\vec x\in \mathbb{R}^n$ ($f$ is even function). If $f$ is differentiable in all point $\vec ...
0
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1answer
55 views

Nonnegative upper derivative implies continuous function nondecreasing. Why is this not obvious?

I know I'm underthinking this. Can someone tell me why I'm wrong? I'm looking at Royden's Chatper 5, Proposition 2. It states: If $f$ is continuous on $[a,b]$ and one of its derivatives $(D^{+})$ ...
2
votes
1answer
43 views

Proving that a function is not differentiable using a certain definition

I have been given the following function: $$f(x,y) = \begin{cases} \dfrac{xy(2x^2 - y^2)}{2x^2 + y^2} & \text{if $(x, y) \ne (0, 0)$} \\ 0 & \text{if $(x, y) = (0, 0)$} \end{cases}$$ Now we ...
0
votes
1answer
20 views

Tangent of Cubic Hermite curve

I have created cubic curve using CatmullRom Spline or Akima spline. From those, I obtain $a, b, c, d$ parameters. To get point on the curve, I do this $f(t) = a + bt + ct^2 + dt^3$ How to get ...
1
vote
2answers
36 views

Finding the derivative of $(\frac{a+x}{a-x})^{\frac{3}{2}}$

This is a very simple problem, but I am stuck on one step: Differentiate $(\frac{a+x}{a-x})^{\frac{3}{2}}$ Now, this is what I have done: $$ (\frac{a+x}{a-x})^{\frac{3}{2}} \\ \implies ...
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0answers
26 views

Jacobian of Rotation composition

I need to compute Jacobians from compositions of rotations. E.g. $R = R_1 R_0$ \begin{align} \frac{\partial R}{\partial R_0} = ?\\ \frac{\partial R}{\partial R_1} = ?\\ \frac{\partial R_1 ...
4
votes
1answer
40 views

Finding values of a piecewise function such that it is differentiable at $x=1$

Let $$ f(x)= \begin{cases} a-x & x \leq 1, \\ \frac{1}{bx} & x>1. \end{cases} $$ Considering this piecewise defined function find values of $a,b$ such that the function is differentiable ...
0
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0answers
16 views

Uniqueness of a solution

Let $f_i(x_1, x_2, ..., x_n)$ for $i=1,...,n$, be real-valued differentiable functions with the following properties: 1) $f_i(x_1, x_2, ..., x_n)=0$ if $x_i=0$. 2) $f_i(x_1, x_2, ..., x_n)=1$ if ...
1
vote
1answer
39 views

Directional derivative of $\frac{xy}{x^2+y^2}$ exists only for $t(1,0)$ or $t(0,1)$?

This is a question related to the directional derivative. I believe $$f(x,y)=\frac{xy}{x^2+y^2}\space\text{for }(x,y)≠(0,0)$$$$f(0,0)=0$$ has a directional derivative at the point $(0,0)$ only for ...
3
votes
1answer
61 views

All functions such that $f'(x) = f(x+1)-f(x) = \frac{f(x+2)-f(x)}{2}$ for all $x \in \mathbb{R}$

I would like to find all (differentiable) functions $\mathbb{R} \to \mathbb{R}$ satisfying $$f'(x) = f(x+1)-f(x) = \frac{f(x+2)-f(x)}{2}$$ for all $x \in \mathbb{R}$. I claim that the only functions ...
0
votes
2answers
44 views

Determining the rate of change of the x-intercept

So my teacher gave me this question on a quiz. Let the red line represent $ y = \frac{1}{2} x $, point P represents the intersection of the red and green line which moves along the red line at a rate ...
1
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2answers
88 views

Find a polynomial of lowest possible degree

Find a polynomial $f$ of lowest possible degree such that $f(x_{1})=a_{1}$, $f(x_{2})=a_{2}$, $f'(x_{1})=b_{1}$, $f'(x_{2})=b_{2}$ where $x_{1} \neq x_{2}$ and $a_{1}, a_{2}, b_{1}, b_{2}$ are given ...
-1
votes
1answer
46 views

Local maximum implies derivative is $0$

If $f$ is a differentiable real function in an open set $E \subset \Bbb R^n$ and $f$ has a local maximum at a point $\textbf{x} = (x_1, x_2, \cdots , x_n) \in E$, show $f'(\textbf{x}) = 0$. I ...
-2
votes
2answers
50 views

how do i solve this differentiation question?

please help me with the question.I have tried numerous times but I cant seem to find the answer. Differentiate the given function and simplify your answer. $ y  =  (4x^4 + 7x^2 − 4)^3 $
0
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1answer
37 views

How to find the minimum of $c|1+x|^n+|1-x|^n$

How to find the minimum of the \begin{align} f(x)=c|1+x|^n+|1-x|^n \end{align} for $n \ge 1$ and $c > 0$. If we take the derivative of $f(x)$ we get \begin{align} f'(x)=-c {\rm sign}(1+x) ...
0
votes
1answer
21 views

Find the differential (if exist) of the function $h(\vec{x}) = \frac{f^3(\vec{x})+f(\vec{x})g^2(\vec{x})}{f^2(\vec{x})+g(\vec{x})}.$

Let $A\subset\mathbb{R}^n$ a noempty open set and $\vec{x}_0\in A$. Let $f,g:A\to\mathbb{R}$ two differentiable function in $\vec{x}_0$ so that, $g(\vec{x})>0$, $\forall \vec{x}\in A$. Consider ...
1
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1answer
46 views

What are the exceptions to the 2nd Derivative Test?

Today in class my professor went over the 2nd derivative rule and was mentioning how there are some exceptions. I wanted to know if anyone I had a more in depth analysis of it. An explanation of what ...
0
votes
1answer
23 views

Splitting a smooth bump function into a product

Given a smooth bump function $\phi$ which vanishes outside of a compact set $K$, can I split it into a product $\phi = \phi_1 \phi_2$ where both $\phi_1$ and $\phi_2$ are smooth bump functions ...
1
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5answers
59 views

Derivative of $1-5^{-x}$

What is the derivative of $y=1-5^{-x}$. Any help is greatly appreciated! I have tried using logs, but I don't think it is correct; $$y=1-5^{-x}$$ $$\ln(y)=\ln(1) +x\ln(5)$$ $$y =x\ln(5)$$ and hence ...
0
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1answer
24 views

Differentiability class

In what differentiable class of functions does $f(x)$ fall knowing that all its derivatives are undefined at $x=0$, given by: $$ f(x)=e^{-a|x|} $$ Where $a>0$ and $x\in \mathbb{R}.$
1
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1answer
21 views

Integral of derivative notation/terminology

In a problem solution I am given that $$\int^\infty_0 \left(e^{{-M^T}t}\Lambda e^{-Mt}M + M^Te^{{-M^T}t}\Lambda e^{-Mt}\right)dt$$ $$= - \int^{\infty}_0 d \left( e^{{-M^T}t}\Lambda e^{-Mt} \right)$$ ...
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0answers
26 views

Derivative of the norm of a path

Let's say $\mathbf c : \mathbb R \rightarrow \mathbb R^n$ is a smooth ($C^\infty$) path where $\mathbf c(t) = \left(c_1(t), c_2(t), \ldots, c_n(t) \right)$. I think this is how to calculate the ...
0
votes
2answers
58 views

Limit as $x \to 64$ of $\frac{\sqrt[6] x - 2}{\sqrt x - 8}$ [closed]

The question below is a question my teacher suggested for contest worthy of Grade 12 Calculus students. I don't need this answered right away but it would be nice if someone gave it a try $$\lim_{x ...
0
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2answers
37 views

Calculate the derivative of the product of three functions $e^x\cdot \ln(x) \cdot \cot x$

I am trying to compute the derivative of $$e^x\cdot \ln(x) \cdot \cot x$$ It's a product of three functions. I imagine I should first calculate the derivative of the first pair: ...
1
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2answers
34 views

When does the limit of derivatives coincide with the derivative of the limit function?

Thinking about the (probably) well-known fallacy about approaching a unit square diagonal with staircase functions and thus concluding the diagonal length be $2$ instead of $\sqrt 2$ led me to an ...
1
vote
2answers
30 views

The derivative of $-e^{\sqrt{2}\cdot x}\cdot 5x^3$

Calculate the derivative of $$-e^{\sqrt{2}\cdot x}\cdot 5x^3$$ Well, we use the product rule. Which is like "the derivative of the first by the second, plus the derivative of the second by ...
1
vote
1answer
31 views

derivative of an objective function with trace of the matrix

How can you derive the gradient of $$f_{\mu}(U) = \mu \log (\mathbf{Tr}\exp(A+U)/\mu) -\mu \log n$$ as $$f_{\mu}(U) = (\mathbf{Tr}(A+U)/\mu)^{-1} \exp(A+U)/\mu$$ where $A,U$ are symmetric matrices ...
2
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1answer
41 views

term-by-term differentiation of function series $\frac{d}{dx}( \sum_{k=0}^{+\infty} f_{k}(x))=\sum_{k=0}^{+\infty}( \frac{d}{dx}f_{k}(x) ) $

I wonder How do I write that equality legally? $$\frac{d}{dx}\left( \sum_{k=0}^{+\infty} f_{k}(x)\right)=\sum_{k=0}^{+\infty}\left( \dfrac{d}{dx}f_{k}(x) \right) $$ indeed, In case of ...
0
votes
1answer
11 views

Find derivative values when knowing anther function value

I have a problem as follow: Suppose $f(-2)=0$, $f(0)=\pi$ and $f(2)=2\pi$ and $f'(x)={\sqrt{4-x^2}}$ A. If $h(x)=f(2\sin(x))$, what is $h'(0)$? Show clearly and completely how to find. B. If ...
3
votes
1answer
92 views

Why are these two definitions of differentiability identical?

Recently, I have learned the following as the definition of multivariable differentiability. Assume that one can express $f(x, y)$ in the following form: $$f\left(x, y\right) = f\left(x_0, ...
0
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0answers
18 views

Derivative of an operator

I am trying to understand a few things about the following problem. I am given an operator $A(s)$, time dependent, positive definite and bounded (uniformly in time), boundedly invertible with compact ...
2
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1answer
25 views

Where does $(a^2+2ab\cos(\theta)+b^2)^{p/2}+(a^2-2ab\cos(\theta)+b^2)^{p/2}$ have a Maximum?

Consider the real-valued function $$ \phi : [0,2 \pi) \rightarrow \mathbb{R} \\ \phi (\theta)=(a^2+2ab\cos(\theta)+b^2)^{p/2}+(a^2-2ab\cos(\theta)+b^2)^{p/2}$$ for $1<p$ and $a,b \in \mathbb{R}$. ...
0
votes
3answers
32 views

$\frac{1}{(1-x)^{2}}=\sum_{k = 0}^{n}(k + 1)x^k+o(x^{n}).$

I would like to show that Taylor expansion of $\dfrac{1}{(1-x)^{2}} $ around $0$ is : $$\dfrac{1}{(1-x)^{2}}=\sum_{k = 0}^{n}(k + 1)x^k+o(x^{n}).$$ My Proof: note that ...
2
votes
1answer
21 views

Find all twice-differentiable functions

Find all twice- differentiable functions $f$ such that the average value of $f$ on each closed subinterval of $[a,b],$ $a < b,$ is the mean of $f$ at the endpoints of the subinterval. Please give ...
2
votes
2answers
65 views

How to write down proof that if $\lim_{x\to \infty}f(x)=\alpha$ then $\lim_{x\to \infty}f'(x)=0$?

Let $a, \alpha \in \Bbb{R}$; let $f: (a,+\infty)\to \mathbb{R}$ be differentiable; let $\lim_{x\to \infty}f(x)=\alpha$; let $\beta := \lim_{x\to \infty}f'(x)$. I want to show that $\beta = 0$. Now, ...
1
vote
0answers
36 views

Jacobians and least squares normal equations

I am trying to solve a problem with non-linear least squares (Gauss-Newton), but the question is more about a single iteration, or least squares. The Jacobian $J$ for each constraint is a 1x6 vector. ...
1
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2answers
55 views

Proving the chain rule

First see the first comment on this post how to prove the chain rule? and this post in general Chain rule proof doubt So we begin by proving the chain rule by assuming we have $f,g$ where $f$ is ...
1
vote
3answers
62 views

How do you implicitly differentiate $y$ from $y\sqrt{x^2+y^2} = 15$?

I've been working on this problem for the last 45 minutes, and I keep getting the wrong answer, no matter what I do. I tried squaring the whole equation, so that there was no radical to deal with - ...
0
votes
0answers
17 views

Wronskian for n-dimensional systems of ODEs

We define Wronskian for the case of a single 2nd-order ODE as $W(y_1, y_2)=\begin{vmatrix} y_1(x) & y_2(x) \\ y_1'(x) & y_2'(x) \end{vmatrix}$. But for the more general case, that is for the ...
1
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2answers
78 views

Finding root of $\cos(x)$ by Newton-Raphson method

The exercise asks me that if I want to find the root of $f(x) = \cos(x) = 0$ using Newton-Raphson method, does the initial value matters? I know that Newton-Raphson method is a special case of the ...
0
votes
2answers
23 views

Find the points on $y = 1/(2x-1)$ where the slope of the tangent line is $-2$

I have a homework question that I don't know how to approach. Can I get some help? Thanks Find the points on $\displaystyle y = \frac{1}{2x-1}$ where the slope of the tangent line is $-2$.
1
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2answers
39 views

$g'(x) = cg(x)$. Are there any other functions, aside from ${e^{cx}}$, that satisfy the condition?

Assume that $g:\mathbb R \longrightarrow \mathbb R$ and $g'(x) = cg(x)$, where $c\in\mathbb R$ and $\forall\ x\in\mathbb R$. Are there any other functions, aside from ${e^{cx}}$, that satisfy the ...