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

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2
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1answer
29 views

Derivatives and Linear transformations

Let G be a non-empty open connected set in $R^n$, $f$ be a differentiable function from $G$ into $R$, and $A$ be a linear transformation from $R^n$ to $R$. If $f$ '($a$)=$A$ for all $a$ in $G$, find ...
0
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2answers
31 views

Rigorous definition of the derivative of $f\left(x,p\left(x\right)\right)$

If we have $f\left(x\right)$ $x$ real and $f$ a real function. The rigorous definition of the derivative of the function is $$ \lim_{h\rightarrow 0} \frac{f\left(x+h\right)-f\left(x\right)}{h} $$ My ...
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2answers
39 views

Implicit Differentiation problem (Exponential Derivatives) Please help!

Use the process of implicit differentiation to find $dy/dx$ given that: $$x^2e^y − y^2e^x=0 $$ I am trying first to find $y$, $$y^2e^x = x^2e^y$$ $$y^2 = (x^2e^y)/e^x$$ $$y = ...
1
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0answers
50 views

eigenvalues of transformed Hessian

Let us define the vector $\mathbf y$ by $y_i := \exp(x_i)$, with $\mathbf x = (x_i)\in \mathbb{R}^N$, and $f : \mathbb{R}^N \rightarrow \mathbb{R}$, $$\displaystyle f\left(\mathbf x(\mathbf y)\right) ...
0
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2answers
36 views

How to differentiate $y = \sqrt{1-f(x)}$

I am in highschool, so forgive me if this question is considered too easy, but I was having trouble understanding how to tackle this question. Would I re-write it as in terms of $f(x)$ or perhaps use ...
0
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2answers
49 views

$f:(-1,1) \to \mathbb R$ twice differentiable, $f(0)=1$ , $f(x) \ge 0 , f'(x) \le 0 , f''(x) \le f(x) , \forall x \in [0,1)$ , to prove $f'(0) \ge -2$

Let $f:(-1,1) \to \mathbb R$ be a twice differentiable function such that $f(0)=1$ $f(x) \ge 0 , f'(x) \le 0 , f''(x) \le f(x) , \forall x \in [0,1)$ , then how to prove that $f'(0) \ge -2$ ? I am ...
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6answers
99 views

Find $f(x)$ from the differential equation $f(x)+{f}'(x) = x^{3} +5x^{2} +x +2$

find $f(x)$ from $f(x)+{f}'(x) = x^{3} +5x^{2} +x +2$. I tried to impose $f(x)$ and $f'(x)$ but i can not solve it.
0
votes
1answer
64 views

Prove that the series is continuous and differentiable [closed]

How to prove that the series $\sum e^{-nx+\cos(nx)}$ is defined, continuous and differentiable (with a continuous derivative) on $(a, \infty)$ for any $a > 0$. I am good with continuity part. But ...
3
votes
1answer
10 views

Discrete-time derivative of the sign function

How does one calculate the time derivative of $$ x_{k+1} = C_1\, \text{sign}(x_k-y_k)\sqrt{2\vert x_k-y_k\vert}, $$ with respect to $x_k$ ? I know that the right part of the equation should yield ...
0
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0answers
12 views

For what condition for $f(x)$ will $m$th derivative $d^m (f/(x-1))/dx^m$ less than $2d^m f/dx^m$?

For what condition imposed on $f: \mathbb{R} \to \mathbb{R}$, or for what range of $x$ would $\frac{d^m}{dx^m} \frac{f}{x-1}$ be less than $2\frac{d^m f}{dx^m}$, where $f(x)$ is some arbitrary ...
1
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1answer
54 views

What is the practical meaning of derivatives? [closed]

I mean practically integration means sum of all components, and the integral can be visualized as the area below a curve. Is there a similar intuition or geometric meaning of the derivative?
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0answers
15 views

Show that if f is differentiable at $x_0$, then it is continuous at $x_0$. (Weierstrass-Caratheodory formulation)

this is an argument for a question which I am unsure whether it is sufficient or not. We are asked to try show the continuity at $x_0$ given that $f$ is differentiable at $x_0$. My argument goes as ...
1
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1answer
53 views

Modulus differentiation

For a Java project, I need to find a way to compute the derivate of a modulus function like $$f(x) = g(x) \pmod{h(x)}$$ for any value of $x$. I know that the modulus function is discontinuous. If ...
2
votes
1answer
57 views

$y^5 =(x+2)^4+(e^x)(ln y)−15$ finding $\frac{dy}{dx}$ at $(0,1)$

Unsure what to do regarding the $y^5$. Should I convert it to a $y$= function and take the $5$ root of the other side. Then differentiate? Any help would be great thanks.
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0answers
18 views

getting to 1 variable

I'm working on the Skimpy Doughnut problem, and I need to get the surface area to one variable so I can differentiate it to get the min & max to answer the next 2 questions. I have: $$when\ ...
2
votes
1answer
21 views

Why do the dimensions not line up when I calculate this (directional) derivative using the chain rule?

an arbitrary, differentiable function $f : \mathbb{R}^n \to \mathbb{R}$ and the function $\gamma: \mathbb{R} \to \mathbb{R}^n$ defined as $\gamma(t) = u + tv$, where $u, v$ are fixed vectors in ...
3
votes
2answers
117 views

Help with Calculus Newton's Method

Alright So I have kind of an interesting question involving Newton's method and have been at this for quite some time, and have come up with that it is not possible. I would like some input to if this ...
1
vote
1answer
53 views

Find equation of tangent line using differential equation: dy/dx = x(y^1/3)

The expression $\displaystyle\frac{\mathrm{d}y}{\mathrm{d}x} = x\sqrt[3]{y}$ gives the slope at any point on the graph of the function $f(x)$ where $f(2) = 8$. a. Write the equation of the ...
0
votes
2answers
57 views

how to derive a function defined with a summatory

Probably I'm missing something very easy but I don't know what to do in this case: let $f(x)$ be a function defined with a summatory, for a trivial example: $$f(x)=\sum_{k=0}^x k$$ Now this kind ...
0
votes
1answer
49 views

matrix gradient

I found the gradient of an optimization problem as $$ b*I + \rho\big(-A+diag(A)+X-2diag(X)\big) = 0 $$ But my problem is, I want to find the equation for $X$. From the above equation, because of the ...
0
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1answer
55 views

How to prove funciton continuity - hard one

Let $f : (-5; 5) \rightarrow \mathbb{R}$ be given as $f(x) = \sum_{n = 5}^{n = \infty}{\frac{1}{n^2-x^2}}$.How to prove that $f$ is continous? Is $f$ differentiable? If yes, how to compute $f'(0)$? ...
2
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2answers
51 views

Prove $g(x+h) = g(x) + hg'(x) + \frac{1}{2} h^2 g''(x) + o(h^2)$ from definition of limit

I want to prove $g(x+h) = g(x) + hg'(x) + \frac{1}{2} h^2 g''(x) + o(h^2)$ from the definition of limit, where x is a 1D variable, $o(h)$ is a quantity that depends on scalar h negligible compared to ...
2
votes
0answers
27 views

Prove the part of the graph of $f$ on $I$ is never below the tangent line to the graph at $(c,f(c))$.

Let $f''(x) \geq 0, \forall x\in I$. If $c\in I$, show that the part of the graph of $f$ on $I$ is never below the tangent line to the graph at $(c,f(c))$. I do not understand the question and I ...
1
vote
2answers
64 views

Is there a function than transitions smoothly between two values, over an interval and remains constant elsewhere?

Specifically, does it exists an infinitely differentiable function $f:R \to R$ that meets the following conditions?. $f(x)=0$ if $x \le 0$. $f(x)=1$ if $x \ge 1$. Physical interpretation: Is there ...
2
votes
1answer
36 views

Total derivative for a polynomial

I refer to Rudin's (Principles of Mathematical analysis, 3rd ed.) definition of differentiability: Suppose E is an open set in $R^n$ and f maps E into $R^m$ and $x \in E$. If there exists a linear ...
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0answers
26 views

Finding global maximum

I have a program which must quickly find $x$ and $y$ where $x,y\in\mathbb{N_0}$ which correspond with maximum value of a function: $$f(x,y)=\frac{\sum_{i=0}^{|b|-1}{|b_i ...
1
vote
1answer
18 views

How can I find the values of $x$ where the tangent line is horizontal?

How can I find the values of $x$ where the tangent line is horizontal? I have done the first part of the problem. And to find the value of $x$, I think I'm just supposed to set the derivative equal ...
-1
votes
2answers
58 views

why function $f(x)=x|x|$ is differentiable ? and what is the derivative? [closed]

let $f:\Bbb R \rightarrow \Bbb R$ defined by $f(x)= x |x|$. Is this function differentiable over $\Bbb R$? If yes, then what is the derivative of $f$?
2
votes
2answers
30 views

$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\frac{d}{dy}[\Phi(\Phi^{-1}(y))]}$?

If $\Phi(y)$ is a monotonic decreasing function is true that $$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\Phi'(\Phi^{-1}(y))}$$ If so, how? It works for $y = \Phi(x) = e^{-x}, \quad \Phi^{-1}(y) = ...
2
votes
1answer
31 views

Order of differentiaton for multivariable functions with arbitrary dependence of variables

While studying Neural Networks, I was bogged with a nasty problem, for which I did not find a satisfying answer using my mathematical knowledge. Let's assume we have a complex multivariable function, ...
1
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1answer
24 views

Related rates, cone-shaped pile of sawdust

Problem: The volume of a cone-shaped pile of sawdust increases by $4.7m^3/\mathrm{min}$. The radius increases 30% faster than the height. How fast does the height increase in the moment that the ...
1
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1answer
62 views

Prove that $\exists \delta>0$ such that $0<|x-y|<\delta \Rightarrow\Big|\dfrac{f(x)-f(y)}{x-y}-f'(c)\Big|<\varepsilon$

Let $f: I \rightarrow \mathbb R$ be differentiable at $c\in I$. Prove that for every $\varepsilon>, \exists \delta>0$ such that $0<|x-y|<\delta$ and $a\leq x\leq c \leq y\leq ...
0
votes
1answer
40 views

How to take derivative of matrix inside integrate $\frac {\partial \int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx}{\partial A}$

I have a function as following $$F=\int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx+\lambda_1 A^2+\lambda_2 B^2$$ where $A^T$ is transpose of vector $A$. $A$ is a column vector such as $A= \begin{bmatrix} ...
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2answers
23 views

Differentiation of quadric function

Could someone please show the steps of differentiating the quadratic function of following form $x'Ωx$ where $Ω$ = variance covariance matrix and $x'$= vector of shares and $x$ = total portfolio of ...
0
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0answers
20 views

A property of solution of ODE $y''+p(x)y=0$

Let $f$ be a solution of the following equation $y''+q(x)y=0$, $q$ is continuous on $\mathbb{R}$ such that $q(x)\leq 0$ for all $x\in\mathbb{R}$. We have $f$ is defined on $[a,+\infty)$, ...
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2answers
21 views

Using chain rule to represent second order derivatives

Is this methodology correct $$\frac{d^{2}r}{dt^2}=\frac{d^{2}r}{dx^2}*\frac{dx^2}{d^{2}\beta}*\frac{d^2\beta}{dt^2}$$ r is interms of x $\beta$ rotates at constant velocity, and x is independent ...
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votes
5answers
1k views

does this have a meaning?

suppose we have a function and want to derive it with respect to itself e.g: $$\frac{dy}{dy} $$ does this have any meaning , and if so what will be it's value?
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1answer
35 views

If a polynomial of degree n satisfies $f(x) = f'(x).f''(x)$ such that $n$ belongs to $R$ , then $f(x)$ is?

A) an onto function B) an into function C) no such function possible D) even function I tried this question by letting a polynomial $f(x) = ax^n + bx^{n-1} \cdots$ and then derivated it but it ...
1
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1answer
53 views

Prove equation using taylor series

Given $f(x)$, knowing that $f'(x)$ and $f''(x)$ exist for every $0\leq x\leq1$, and provided I know that $f(0)=f(1)$ and that for each $0\leq x\leq1$, $|f''(x)|\leq A$, how can I prove that ...
0
votes
0answers
28 views

How to take derivative of integral of square matrix function

I have a function as following $$F=\int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx+ \int |A^TG(x)-C^TJ(x)|^2 (1-H(x)) \, dx+\lambda_1 A^2+\lambda_2 B^2+\lambda_2 C^2$$ where $A^T$ is transpose of vector $A$. $A$ ...
0
votes
1answer
35 views

If $p$ is a fixed point, does $\forall x\in U \setminus\{p\}: f(x)\neq p$ hold?

Let $f\in C^2(\mathbb R, \mathbb R), f(p)=p, f'(p)=0,$ and $f''(p)\neq 0$. I showed that there is a neighborhood $U$ of $a$ such that the fixed-point iteration $x_{k+1}:=f(x_k)$ converges to $p$. Then ...
4
votes
1answer
65 views

What did i do wrong with this derivation?

$$ \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} $$ Therefore \begin{align} \frac{1}{\cos(x)} &= \frac{1}{1-(\frac{x^2}{2} - \frac{x^4}{4!} + \frac{x^6}{6!} - \cdots)} \\ &= ...
0
votes
1answer
17 views

Taylor expansion of second order

I have to find the Taylor expansion of second order of the following functions with center the given point $(x_0, y_0)$. $f(x, y)=(x+y)^2, x_0=0, y_0=0$ $f(x, y)=e^{-x^2-y^2}\cos (xy), x_0=0, ...
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votes
1answer
45 views

How can we show that there is such $g\in C^2$?

Show that if $f\in C^2 (\mathbb R, \mathbb R)$ with $f(a)=0$ and $f'(a)\neq 0$ then there is a function $g\in C^2 (\mathbb R, \mathbb R)$ such that $f(x)=(x-a)g(x)$ for all $x \in \mathbb R$. ...
2
votes
4answers
35 views

Finding horizontal tangents to a function.

Find the points at which the line tangent to the following function is horizontal $$q(x)=(x+3)^4(2x-1)^7$$ Every time I've gotten to the point of finding $x$ the numbers are all irrationally too ...
0
votes
0answers
24 views

In what direction does the altitude increase faster?

We assume that a mountain has the shape of the elliptic paraboloid $z=c-ax^2-by^2$, where $a$, $b$ and $c$ are positive constants, $x$ and $y$ are the geographical coordinates (east-west, north-south ...
0
votes
0answers
24 views

Inequality, derivates

$f,g:[0,\infty)$ functions $f(0) \lt g(0)$ and $f'(x) \lt g'(x)$ for any $x\in [0,\infty)$. Does this means that $f(x) \lt g(x)$ for any $x\in [0,\infty)$? And can I use this without proof? I need it ...
5
votes
2answers
72 views

Directional derivative

The governor Ralph has trouble on the bright side of Mercury. The temperature in the wall of the vessel, when it is in the position $(x, y, z)$ is given by $T(x, y, z)=e^{-x^2-2y^2-3z^2}$, where $x$, ...
0
votes
3answers
19 views

General Formula of the $n$th Derivative for $f(x) = xe^{2x}$

Find the general formula for the nth derivative of $f(x)=xe^2x$ in the form: $$ f^{(n)}=A(n)e^{2x}+B(n)xe^{2x} $$ I've evaluated the first five derivatives in that for and for $A(n)$ have found ...
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votes
1answer
59 views

Verification that $\int x\sin x=\sin x- x\cos x + C$ by differentiating both sides of the equation

The original question is: Confirm that the formulae stated below are correct by differentiating both sides: $\int x\sin x=\sin x-x\cos x+C$ Where does the cancellation occur, and what is the ...