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

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1answer
197 views

Cosine law derivative assistance needed

I read the wiki, but I would really appreciate it if someone can explain it to me and help me solve it; I don't know how to do it. By no means is this HW. Derivative Cosine law Given a planar ...
0
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3answers
42 views

What is the derivative of $\ln\left(x^2 + (3/4)x\right)$?

What is the derivative of $\ln\left(x^2 + (3/4)x\right)$? I did $\dfrac{1}{\left(x^2 + (3/4)x\right)}$ and multiplied it by $(2x + 3/4)$ Is this right? Should I have flipped $3/4$ instead ...
3
votes
1answer
59 views

Derivative of sum of Schwartz functions

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Schwartz-class function. Let $F(x)=\sum_{n\in\mathbb{Z}}f(x-2\pi n)$. I understand that because Schwartz-class functions are nice, this series converges ...
2
votes
1answer
56 views

Differentiability of function defined as integral form

Let $H(t)=\int_{\Bbb R}|f(x)+tg(x)|^p\mathrm dx$ and $f,g\in L^p(\Bbb R)$. Then, how to prove that $H$ is differentiable and find its derivative? I think it's impossible to find it by ...
0
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1answer
55 views

How should I define the limit definition of a derivative using negative numbers?

Typically the derivative is defined at a point $x$, assuming it is differentiable at it, by \begin{equation} \lim_{n \rightarrow \infty} \frac{f(x + \frac{1}{n}) - f(x)}{\frac{1}{n}} \end{equation} ...
5
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0answers
106 views

When does Gâteaux imply Fréchet? [duplicate]

Speaking of the relation between Gâteaux and Fréchet, authors usually point out that $$\text{Fréchet} \implies \text{Gâteaux}$$ and then give a counterexample to illustrate that the converse doesn't ...
1
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2answers
68 views

1st derivatives of $f(\alpha) = \frac{\sin(2\alpha)}{\sin(\alpha+1)}$

Could someone help me out with the following? I have to get a maximum using the derivative $$f(\alpha) = \frac{\sin(2\alpha)}{\sin(\alpha+1)}$$ $$f(\alpha) = \sin(2\alpha) \cdot ...
1
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1answer
24 views

Equality between a fonction and an infinite serie.

I want to prove this equality : -ln(x+1) = S(from 1 to infinite) ((-x)^n/n) //for x between -1 and 1 (not included of course). I wrote on my notebook : show the derivates are equals and the two ...
1
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0answers
29 views

Algorithm for determining the differentiability of a single variable real function

What would be an appropriate algorithm, for students of first year calculus, to determine where a given function is differentiable ? When a function is defined by cases, as in $$f(x) = \begin{cases} ...
0
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0answers
56 views

What's wrong with this derivative calculation

For the following function: $$f(x)=\frac{2}{2x^2}-\frac{x}{3}+\frac{4}{5}+\frac{x+1}{x}$$ I got the individual derivatives below: $$\frac{d}{dx}(\frac{2}{2x^2}) = \frac{d}{dx}(\frac{1}{x^2}) = ...
0
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2answers
37 views

Derivative Proof using Basic Defintion

I have to use the basic definition of the derivative to find the derivative of $$f(x)=\frac{1}{\sqrt{x}}$$ for x>0 I need to use the limit... $$\lim_{x \to c}\frac{f(x)-f(c)}{x-c}$$ So I have ...
1
vote
1answer
65 views

Derivative of Trig. Function

if $f(x)=\tan(3x)$, then $f'(\pi/9)=$? I thought the answer was $4$ but my teacher marked it wrong. Work: $f'(x) = \sec^2(3x)\cdot 3 = \frac 3{\cos^2(3x)} = \frac{3}{\cos^2(\pi/3)} = 3/(3/4) = 4$.
2
votes
3answers
465 views

Intersection of normal to the curve

The line that is normal to the curve $x^2+3xy-4y^2=0$ at $(6,6)$ intersects the curve at what other point? If I implicitly differentiate this curve, I will get the equation of the slope: ...
1
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2answers
57 views

Closed form for repeated theta operator applied to $x\cos(x)$

let $\theta_{x}$ be the operator : $$\theta_{x}=x\frac{d}{dx}$$ What is the closed form for : $$\theta_{x}^{n}\left[x\cos(x)\right]$$ $n$ being an positive integer.
1
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2answers
238 views

Point on the graph of $y=\sqrt{4x+13}$ closest to $(5,0)$?

Just did this question on an exam earlier today, I'm curious to see if I'm correct. What point on the graph of $y=\sqrt{4x+13}$ is closest to $(5,0)$? My answer: $(-1,3)$
2
votes
1answer
1k views

Finding the taylor series of $f(z) = 1/(1+z^2)$.

I am working on the following exercise: Find the Taylor expansion of the function $f(z) = \frac{1}{1+z^2}$ about $z = 3i$. We had the Taylor Series Theorem in the lecture: Let $D \subset ...
0
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2answers
75 views

AP Calculus Derivative

So I have this question for AP Calculus. It doesn't seem to be that difficult, but I just can't wrap my head around it. I believe that I have answered (a) and (b) correctly, but I have no clue as to ...
0
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1answer
109 views

If f is n-times differentiable, and $f^n$ is never 0, then f has at most n zeros in R

Let $n \ge 0$, let $f:\mathbb{R} \rightarrow \mathbb{R}$ be n-times diff erentiable on $\mathbb{R}$, and assume that $f^{(n)}(x) \neq 0$ for all $x \in \mathbb{R}$. Show that $f$ has at most $n$ zeros ...
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2answers
87 views

1st and 2nd derivatives of $x\sqrt{9-x}$

Do I start with the power rule? I know to rewrite it as $x(9-x)^{1/2}$ and if I use the power rule I get $\sqrt{9-x}/2(9-x)^{3/2}$ and I have no idea if that's right, and then I need to find the ...
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3answers
97 views

Help understand chain rule derivative

I was verifying a larger function derivative on wolfram alpha and came across this derivative: $\frac{d}{dx} (1-x)^2 = 2(x -1)$ Using the chain rule, I was expecting to get: $2(1 - x)$ Instead. I ...
0
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1answer
85 views

Using definition of derivative to find $\sqrt{11\theta}$

$$\lim_{h \to 0} \frac{f(\theta+h)-f(\theta)}{h} \;\;\;\;\;\;\;\;\;\;\; \textbf{(1)}$$ $$\lim_{z \to \theta} \frac{p(z)-p(\theta)}{z-\theta} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \textbf{(2)}$$ ...
2
votes
1answer
45 views

Derivative of function that includes norm

I was solving the problem: find the derivative of a function f : H → R, $f (x) = \sin ||x||^3$ (H is Hilbert space). I got the answer $f'(x)=3\cos||x||^3 x||x||$. Is this correct or I am doing ...
1
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1answer
43 views

Partial derivative at (0,0).

Let $f: \Bbb R^2 \to \Bbb R$ defined by $f(x,y) = 0$ if $xy = 0$ and $f(x,y) = 1$ if $xy \not = 0 $. Show that both partial derivatives exist at $(0,0)$. The way I have ...
0
votes
1answer
50 views

Vector functions and motion along a curve

A particle moves along the curve $x=\ln y$ with a constant speed of $4$ units per second. Find the normal scalar component of acceleration as a function of $x$. Honestly, what I don't understand ...
2
votes
1answer
94 views

Derivative of antipodal map between $n$-spheres

Let $S^{n-1}\subseteq \mathbb{R}^n$ denote the $(n-1)$-sphere $x_1^2+\ldots+x_n^2=1$. Let $f:S^{n-1}\rightarrow S^{n-1}$ be the map $f(x_1,\ldots,x_n)=(-x_1,\ldots,-x_n)$. What is the derivative of ...
2
votes
2answers
80 views

Is this an accepted/given form of the definition of derivative?

$$\lim_{h\to 0}\frac{\int_x^{x+h}f(x)dx}{h}=f'(x)$$ Is this an accepted/given/understood form of the definition of derivative? Or does one need to work this into the traditional difference quotient ...
4
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1answer
64 views

Is $\dot{f}(0)$ a function or a point?

Say \begin{align} g: & \mathbb R^m \to \mathbb R^n \\ \implies g': & \mathbb R^m \to \mathcal{L}(\mathbb R^m,\mathbb R^n) \\ \implies g'(0): & \mathbb R^m \to \mathbb R^n \end{align} ...
4
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1answer
86 views

Application of Mean Value Theorem and Interval

Using the mean value theorem establish the inequality $$7\frac{1}{4}<\sqrt{53}<7\frac{2}{7}$$ This is obviously a true statement but can you help me form the interval and what function I should ...
1
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1answer
77 views

A differentiable function whose derivative is not elementary.

Do we know of any differentiable function whose derivative is not an elementary function? This may be a silly question, but in the light of this answer, as pointed in the comments, finding an example ...
1
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1answer
87 views

Related Rates Triangle Problem

Part of a homework problem I have for my Calc class. In a right triangle with base = 10, height = h, and the angle across from the hight = theta, theta is increasing at a rate of 15/26 rad/min At ...
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2answers
14 views

Derivative with two varaibles

Find $y'$: $$yx^2+y^3 = x-y$$ I tried using $\frac{dy}{dx}$ and have gotten $\frac{1}{x}+3y^2$, which isn't right. Any ideas on what I'm doing wrong?
1
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3answers
29 views

Relative Minimum, value of $a$

For what value of a does the function $f(x) = x^2 + ax$ have a relative minimum at $x = 1$? I don't know where to begin on this problem.
0
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1answer
51 views

Tangent Line is horizontal when???

If $f(x)=2x^3+3x^2-36x+5$, then the tangent line to the graph $y=f(x)$ is horizontal when?? I need help with this problem, I'm studying for finals so can anybody help me how to figure out this ...
1
vote
2answers
114 views

For a differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ what does $\lim_{x\rightarrow +\infty} f'(x)=1$ imply? (TIFR GS $2014$)

Question is : For a differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ what does $\lim_{x\rightarrow +\infty} f'(x)=1$ imply? Options: $f$ is bounded $f$ is increasing $f$ is unbounded ...
1
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2answers
32 views

Find $\displaystyle \frac{dy}{dx}$ when, $\displaystyle y \arcsin x - x \arctan y = 1$

yesteryday was my class test and I found this question. Find $\displaystyle \frac{dy}{dx}$ when, $\displaystyle y \arcsin x - x \arctan y = 1$ I have read the question for arctanx as $1/1 + x^2$. ...
0
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1answer
43 views

Push-forward vector field for constant vector field

Let $v$ be a constant vector field $v=\sum_{i=1}^n c_i\dfrac{\partial}{\partial x_i}$, and let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a bijective linear map. What is the vector field $f_*v$? ...
1
vote
2answers
44 views

Definition of derivative

Well, I know that the derivative of a function $f(x)$is defined this way: $$\frac{df(x)}{dx} = \lim_{\Delta x\to 0}\frac{f(x+\Delta x) - f(x)}{\Delta x}$$ And it's pretty clear that the expression ...
0
votes
1answer
26 views

derivation of a differential Eq

Look at $F(u) = \frac{\partial u}{\partial t}-\nabla \cdot (a(u)\nabla u)$. My question is, what $F'(u)$ is. I need this for the linearization of a PDE. The idea is to use the newton-approximation. ...
1
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1answer
107 views

Question on derivatives

Let f be a function such that f is equal to the limit as h approaches 0 of [f(7+h) - f(7)]/h = 4. Which of the following must be true i. f is continuous at x=7 ii. f is differentiable at x=7 iii. The ...
1
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1answer
37 views

Mapping one integral curve onto another

Let $v$ be the vector field $\sum_{i=1}^n x_i\dfrac{\partial}{\partial x_i}$, and let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be the projection map $(x_1,\ldots,x_n)\rightarrow x_1$. Show that $v$ ...
0
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1answer
52 views

Condition for $f$-related vector fields

Let $U\subseteq\mathbb{R}^n, V\subseteq\mathbb{R}^m$ be open subsets, and $f:U\rightarrow V$ a $C^1$ map. Let $u,v$ be vector fields on $U,V$ respectively. Show that $u,v$ are $f$-related if and ...
1
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1answer
27 views

Multiplication formula for Lie derivative

Let $U\in\mathbb{R}^n$ be an open set, and let $f_1,f_2\in C^1(U)$. Prove that $$L_v(f_1f_2)=f_1L_vf_2+f_2L_vf_1$$ Suppose $f_1,f_2:U\rightarrow\mathbb{R}$. Let $p\in U$. We have ...
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2answers
173 views

Definition of Lie differentiation for vector fields

I am trying to track the definition of Lie differentiation. According to my notes: Given $p\in\mathbb{R}^n$, the tangent space to $\mathbb{R}^n$ at $p$ is the set of pairs ...
0
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3answers
89 views

Decompose integral of derivative and $e^{st}$ (laplace transform)

I'm reading on Laplace transform and stumbled upon the transform of a derived function. Could someone explain me this? $$ \begin{equation} \int_{0^{-}}^\infty \frac{d}{dt}f(t)e^{-st} dt = ...
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2answers
51 views

Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ for the curve $x=1+t^2$, $y=t^3-3t$

Is this question simply asking me to find the first and second derivative for the two given equations? I really don't know how to get started with this one and would appreciate any hints. Find ...
0
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3answers
43 views

second derivative of exponential $e^{x^2}+3x-2$

I have to find the first and second derivative of $e^{x^2}+3x-2$, the first one i can do ok but can someone please help me with the second. thanks
1
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2answers
99 views

Are the differential and derivative of a single-variable function exactly the same thing?

I just started taking a calculus class but I got in late and it had already started like weeks ago, so I'm completely lost. I believe the teacher uses this same formula in order to get the ...
3
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2answers
145 views

Matrix derivatives

Can someone please help me with this problem? ?I've already searched for similar examples in some linear algebra textbooks, but I couldn't find any... Thanks a lot! where $x\in \Bbb R^{n\times 1}$
0
votes
1answer
74 views

If a function has nonzero derivative for all $x\in\mathbb R$, it is a homeomorphism of the real line onto its range

Let $f : \mathbb R \to \mathbb R$ such that $f '(x) \neq 0$ for every $x$ in $\mathbb R$. Prove that $f$ is a homeomorphism from $\mathbb R$ onto $f(\mathbb R)$. I know that homeomorphism can be ...
2
votes
2answers
69 views

find the derivative $\displaystyle \frac{d}{dx}\int^a_x \tan(\tan(t))\,dt =$ [closed]

Find the derivative: $\displaystyle \frac{d}{dx}\int^a_x \tan(\tan(t))\,dt = $ I tried to take the derivative but I am getting the wrong answer every time.