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

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1answer
571 views

deriving second order transfer function from spring mass damper system..

I am having a hard time understanding how a differential equation based on a spring mass damper system $$ m\ddot{x} + b\dot{x} + kx = 0$$ can be described as an second order transfer function for an ...
0
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1answer
26 views

computing directional derivate/ differentiability iff linear map exists

Let $\| \cdot \|$ be a norm on $\mathbb R^2$ and $S= \{ x \in \mathbb R^2 | \| x \| =1 \}$, $f: S \rightarrow \mathbb R$ a function with $f(-x) = -f(x)$ for all $x \in S$. Let $F: \mathbb R^2 ...
3
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4answers
202 views

What functions have the property that $\frac{d}{dx}f(x) = c \cdot f(x+1)$?

If we are allowed to pick any real-valued constant $c$ that helps, when does $$\frac{d}{dx}f(x) = c \cdot f(x+1)$$ In other words, when does the derivative of a function $f(x)$ equal some constant ...
2
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3answers
62 views

Limits of trig functions

How can I find the following problems using elementary trigonometry? $$\lim_{x\to 0}\frac{1−\cos x}{x^2}.$$ $$\lim_{x\to0}\frac{\tan x−\sin x}{x^3}. $$ Have attempted trig identities, didn't help. ...
0
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1answer
79 views

Help to solve this problem, the result doesn't seem right :/ silly mistake somewhere probably

Suppose that Coke and Pepsi are the only firms producing cola. Their products are not identical, but are very close substitutes. Let $P_c$ denote the price of Coke and $P_p$ the price of Pepsi. Demand ...
0
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1answer
405 views

Find the slope of the tangent line to the curve.

So I am trying to find the slope of the tangent line to the curve $$\sqrt{4x+2y} + \sqrt{xy} = \sqrt{38} + \sqrt{24}.$$ at the point $(8,3)$. I ended up implicitly differentiating and getting ...
1
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1answer
101 views

Proof around Rolle's Theorem

Let $f(x)=\exp(\sin(2\pi x))$. I'm trying to prove that there is $a\in[0,1]$ such that, for $(n_r)$ a sequence of integers tending to $\infty$ (by this I mean $n_r$ tends to $\infty$ as $r$ tends to ...
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4answers
65 views

Differentiate $y=4\,e^{\cos2x}$

I do not know what to do for this question $$ y=4\,e^{\cos2x}$$ Can anyone show me or give me an example? I think I should be using the chain rule but not $100\%$ sure how to break up the question.
0
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1answer
34 views

Mean value of second differential

I have shown the first part that $\frac{d^2y}{dx^2}$ is that, in the second part they tell find the mean value of $\frac{d^2y}{dx^2}$ , how do I do this ? I don't understand what they mean or how to ...
2
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1answer
47 views

Smoothness of $f(x)/(1+|f(x)|)$ where $f \in C^1(E)$ for $E$ an open subset of $\mathbb{R}^n$

(a) Show that if $E$ is an open subset of $\mathbb{R}$ and $f \in C^1(E)$ then the function $$F(x) = \frac{f(x)}{1+|f(x)|}$$ satisfies $F \in C^1(E)$. (b) Extend the results of part (a) to $f \in ...
2
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1answer
39 views

Zero points of derivatives

It's obvious that if $f(x)$ is a polynomial then it's derivatives $f^{(n)}$ are equal to zero for $n>\deg f$. I'm trying to prove the "inverse" statement: if for each $x\in\mathbb{R}$ there ...
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1answer
20 views

Oscillating Spring & Rates of change

How to solve? Are they asking for: instantaneous rate of change: $\frac{d}{dt}h(t)=2.5$ and solve for value of $t$ or when $\frac{d}{dt}h(t_1)$ where $t_1$ is when $h(t)=2.5$ but both methods ...
2
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3answers
609 views

Differentiate the following function

$$y = \sqrt {\sin x} = (\sin x)^{\frac 12}$$ \begin{aligned} {dy \over dx} & = \frac 12 (\sin x)^{-\frac {1}2}{d\over dx} \sin x \\ & = \frac 12 (\sin x)^{-\frac 12} \cos x \\ & = ...
0
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1answer
44 views

Differentiate the following functions

Let $$y(x)= 4 x^3 e^{2x},$$ then $$y'(x) = 4 \times 3 \, x^2 e^{2x} + 4 \, x^3 \times 2 e^{2x} = 12 \, x^2 e^{2x} + 8 \, x^3 e^{2x}$$ Does this look correct?
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1answer
27 views

Simple partial derivatives and Chain Rule.

Shouldn't the function $g(x,y(x))=x^³ + y(x)$ satisfy $\frac{dy}{dx}=-\frac{g_x}{g_y}$? I get $g_x=3x^2 + \frac{dy}{dx}$ and $g_y=-1$ which does not satisfy it... Where do you think the mistake is? ...
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0answers
23 views

How to find the minima for $y = x^2 + a.x - \lfloor\sqrt{x^2+a.x - b}\rfloor^2$?

Please guide in how to find the value of $x$ for which $y = x^2 + a.x - \lfloor\sqrt{x^2+a.x - b}\rfloor^2$ will be minimum. I know this involves differentiation but am not sure on how to ...
0
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1answer
79 views

I can't seem to find this derivative any help would be great.

A rocket of mass m = 1000 kg is traveling in a straight line for a short time. The distance in meters covered by the rocket during this time is described by the function $r(t)=t^3 −3t^2 +6t$ where ...
1
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1answer
24 views

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. [duplicate]

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.
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1answer
57 views

Implicit differentation with chain rule

Problem Find the derivative, using implicit differentiation: $$2x^3=(3xy+1)^2$$ Progress Used the chain rule for the derivative $(3xy+1)^2$. Do I move the $2x^3$ over once I get its ...
2
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1answer
51 views

Newton's binomial for matrices that don't commute?

I'll give a bit of background info as to why I'm asking. I need to find the directional derivative of $f(A)=A^m$ where $m>0$ and $A$ is an $n$ by $n$ matrix with real entries. I want to do this ...
2
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1answer
40 views

How to find the differential of this function

we are given the function $f: \mathbb R^n \setminus \{0\} \to \mathbb R^n$ defined by: $f(x) = \frac{x}{|x|}$ Find $Df(a)$. What I did: I tried working this out from the definition. the ...
3
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2answers
83 views

Passing a derivative through a limit.

After searching around on the net and on SE I have not found a satisfactory answer. Let $f_n: D \to \mathbb R$ be a sequence of functions. What assumptions, aside from $f$ being differentiable, do we ...
0
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4answers
70 views

The sum of two variable positive numbers is $200$. Find the maximum value of their product.

The sum of two variable positive numbers is $200$. Let $x$ be one of the numbers and let the product of these two numbers be $y$. Find the maximum value of $y$. NB: I'm currently on the ...
6
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6answers
205 views

Prove that $f(x)\equiv0$ on $\left[0,1\right]$

Let $f(x)$ differentiable on $\left[0,1\right]$ such that $f(0) = 0$. Also, assuming that $\forall x\in \left[0,1\right]:\left|f'(x)\right| \le \left|f(x)\right|$. Prove that $f(x)\equiv 0$ What I ...
2
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1answer
26 views

partial derivative of function with a matrix

Let $A$ be a $n\times n$ matrix. Let $f\in C^1(\mathbb R^n)$ and $g:\mathbb R^n\rightarrow\mathbb R, g(x)=f(Ax)$. What is the partial derivative $\partial_{x_i} g(x)$? So $Ax=(\sum_{l=1}^n ...
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0answers
24 views

What is the derivative of $\frac{f^{(3)}(\xi(x))}{6}$ at $x=x_0$

The error of interpolating polynomial is $$ E_n(x)=\frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x)) $$ The derivative of $E_n(x)$ is $$ ...
2
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3answers
116 views

Differentiate the function into the simplest form

My question: $y=\sin^{2}(x)$ My attempt: Is $\sin^{2}x$ the same as $(\sin(x))^2$? By rearranging the function I came up with the following. $$ \begin{align} u = \sin(x), \ & y=u^2 \\ ...
2
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0answers
65 views

How “ugly” can a derivative get?

There are plenty of examples of differentiable functions $\Bbb R\to\Bbb R$ with derivatives that are not everywhere continuous. However, as stated here, it is impossible for the derivative to be ...
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2answers
78 views

Complex Exponential

how would you use the complex exponential to evaluate: Thank you.
1
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1answer
68 views

Residue of this function for $z_0=0$

I have this function $$\frac{\sin (2z)-2z}{(1-\cos z)^2}$$ I want to find its residue around $z_0=0$, however I've been battling it for hours but I get nowhere. I've tried finding its Laurent series, ...
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4answers
67 views

Second Implicit Derivative [closed]

There is a large argument over the answer to this question: Find the second implicit derivative of $x^2 + xy + y^2=2$. Can someone please answer it and explain their answer?
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2answers
75 views

Why does the derivative and integral of the funcion exist (doesn't exist)?

Given $ f(x) = \dfrac {2\sec(x)(\cos^2(x)+3x^4\cos(3x^2))}{3x^3(1+3x^2)} $, I know that $f(x)$ is not defined in $x=0$. And it is not defined in $ {k\fracπ2:k∈Z}$ either (thanks Git Gud) And i ...
2
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1answer
61 views

Derivative of determinant, which is correct?

I've seen two different results on the derivatives of determinants of matrices: $$\frac{\partial |X|}{\partial X_{ij}}=X_{ij}.\tag1$$ $$\frac{\partial\det(X)}{\partial X}=|X|(X^{-1})^{T}.\tag2$$ ...
0
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2answers
122 views

Is there a general formula for estimating the step size h in numerical differentiation formulas?

Using three-point central-difference formula $$ f^{\prime}(x_0)\approx \frac{f(x_0+h)-f(x_0-h)}{2h} $$ and for $f(x)=\exp(x)$ at $x_0=0$ we have $$ \begin{array}{c, l, r} h & f^{\prime}(0) ...
0
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1answer
37 views

How to verify the gradient of a symbolic function using numerical gradient?

I have a function $f$, which takes as inputs a three arrays and returns an array. I have written a symbolic function $g$ to calculate the gradient of this function and I want to verify that it ...
0
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2answers
28 views

Can I change determinant and partial derivation?

Let $f(t,x)$ be a function whereat $x\in\mathbb{R}^n$ and $t\in\mathbb{R}$ fixed. Furthermore both $\frac{\partial}{\partial t}f(t,x)$ and $\frac{\partial}{\partial x}f(t,x)$ exists and are ...
0
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1answer
52 views

Given that $|f(x,y)| \le x^{2}y^{2}$, prove that $f$ is differentiable at (0, 0).

Given that $f : \mathbb{R}^{2}\rightarrow \mathbb{R}$ is a function such that $|f(x,y)| \le x^{2}y^{2}$ for all $(x,y) ∈ R^{2}$, prove that $f$ is differentiable at $(0, 0)$. I know that I should ...
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1answer
57 views

Write this ODE without any square roots

Given the function $$u(t):=\sqrt{\sum_{i=0}^n \alpha_i t^{2i}}$$ is it possible to plug this into the ODE $$(t^2-1)u''(t)+tu'(t)(1-8a+8at^2)-4(a+a^2-2at^2+n(-a+2at^2)-C)u(t)=0 $$ such that I get a ...
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0answers
67 views

Proof regarding derivatives and Mean Value Theorem.

Original question: $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Show that for any $c \in (a,b)$ that is not a point of maximum or minimum for $f'$, there exist $x_1, x_2 \in (a,b)$ ...
3
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2answers
73 views

Inverse Function Theroem in $R^1$

I have a question about the inverse function theorem in R1. The version of the theorem that I know says: Let $y = f(x)$ be a continuously differentiable function defined on an open interval $I$ in ...
4
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2answers
79 views

Is $\dot u/\dot \phi$ is the same as $\mathrm du / \mathrm d\phi$?

I have two functions $u$ and $\phi$ given. I am not sure what they depend on, but I think that it is a common variable $\tau$. So $u(\tau)$ and $\phi(\tau)$. Then $\dot u$ is the derivative of $u$ ...
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0answers
123 views

Using Lagrange polynomial to obtain the Second Derivative Midpoint formula

The Second Derivative Midpoint/Central Formula is $$ f^{\prime\prime}(x_0)=\frac{f(x_0-h)-2f(x_0)+f(x_0+h)}{h^2}-\frac{h^2}{12}f^{(4)}(\xi) $$ I tried to get this formula using Lagrange polynomial. ...
0
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1answer
45 views

Show uniformly convergence

I would like to show, that $\frac{f(h,x)-f(0,x)}{h}\to \left. \frac{\partial}{\partial t}f(t,x)\right|_{t=0}$ converges uniformly for $h\to 0$ on a compact set $K$. The tutor gave the ...
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1answer
26 views

derivative of sum of vectors

suppose i need to make the partial derivative of this vector function $f(\vec{a},\vec{b})=\frac{1}{| \vec{a}+\vec{b}|}$ respect to $\vec{a}$: $\frac{\partial }{\partial \vec{a}} f(\vec{a},\vec{b})$, ...
2
votes
1answer
69 views

The only differentiable function $f \colon \mathbb R \to \mathbb R$ such that $f^\prime(x)=f(x)$ is $f(x)=ce^{x}$

Prove that the only differentiable function $f \colon \mathbb R \to \mathbb R$ such that $f^\prime(x)=f(x) \mspace{1ex} \forall x\in \mathbb R$ is $f(x)=ce^{x}, \forall x\in \mathbb R$, and for some ...
0
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4answers
58 views

Differentiating fraction $\phi(\theta) = \frac{\theta}{(1+\theta^2)}.$

I graduate high school many years ago, and I have no idea where to find how to differentiate this formulae. $$\phi(\theta) = \frac{\theta}{(1+\theta^2)}.$$ I have a solution of it, but I know not ...
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0answers
14 views

How do you determine such derivatives can be simplified?

Background When working out the Laplacian in spherical coordinate system, either via chain rule or differentiating the basis vectors, one arrives at the following: $\nabla^2 = ...
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3answers
73 views

How to find $\frac{dy}{dx}$ at the point $(-2,0)$?

Could someone please demonstrate this problem? I can't seem to get it right. Find $\dfrac{dy}{dx}$ at the point $(-2,0)$, if $y^3 = x^3 + e^x \sin y +8$.
1
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1answer
22 views

Function that got positive derivative for all $x\neq 0$ And zero derivative for $x=0$ must be increasing on $\mathbb{R}$

I got this statement that I am trying to prove Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that for all $x\neq 0$, $f'(x)>0$ and for $x=0$, $f'(x)=0$, how do I prove that $f$ ...
2
votes
2answers
77 views

Finding the absolute minimum and maximum of a function

The function is $$f(x)=x+\sin(2x)$$ I need to find the absolute maxima and minima of several different domains using this function. I have found that the derivative of this function is ...