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

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1answer
76 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by finding a linear function T

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
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2answers
53 views

Why does $d$ mean?

What do the $d$'s mean? I've seen them in other formulas as well.
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0answers
42 views

Calculus and Matrices

Suppose I have a linear operator $T: \mathbb{R} \rightarrow \mathbb{R}$, and also suppose that it's a composition of elementary functions, so its derivative, $T'$, is reasonable easy to find. I can ...
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2answers
50 views

How to differentiate $f(x) = 1-xe^{1-x}$ w.r.t. $x$?

I would like to differentiate the following with respect to $x$: $$f(x) = 1-xe^{1-x} \tag 1$$ How would I do this please? I can see that the 1 would disappear, then I am left with ...
1
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4answers
91 views

Several (advanced) L'Hospital problems

Problems : $$ \begin{align} &\text{A}.\ \lim_{x\rightarrow1}(2-x)^{\tan\left(\frac{\pi x}{2}\right)}\\ &\text{B}.\ \lim_{x\rightarrow 0}\left(\cot x-\frac{1}{x}\right)\\ &\text{C}.\ ...
2
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1answer
61 views

Find the $dy/dx$ of $y=y=x\int \limits_2^{x^2}\sin\left(t^3\right){d}t$

Need to find $\frac{dy}{dx}$ for this: $$y=x\int \limits_2^{x^2}\sin\left(t^3\right){d}t$$ I tried using the chain rule and I am still left with $\int \limits_2^{x^2}\sin\left(t^3\right){d}t$ in my ...
2
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1answer
45 views

Integral of $e^{(a+ib)x}$

Given the function $f:\mathbb{R}\rightarrow \mathbb{C}$, such that $f(x)=e^{(a+ib)x}$, how can I compute $f'(x)$ and $\int f(x)dx$ ? Certanly, one can use the identity $e^{ibx}=\cos(bx)+i\sin(bx)$ and ...
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0answers
60 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by producing a linear function

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
0
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1answer
14 views

Derivatives with ln Issues

I got 3x^2/x^3-7 but I'm not sure where to go from there. Also I ran into this problem and haven't been able to figure it out. Thanks for your time, I really appreciate it.
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1answer
25 views

Derivatives from the left and right

I need to give a definition of the derivative from right and from left. (I know it doesn't make a whole lot of sense, but it is supposed to be similar to the same thing as the limits from the right ...
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0answers
38 views

Convergence in $L^2$ of difference quotients to derivative of function in $H^1$

Is it true that if $u\in H^1({\mathbb R})$, then $(u(x+h)-u(x))/h$ converges to $u'(x)$ in $L^2({\mathbb R})$, as $h\to 0$? It's hard for me to get a handle on this, since $u'$ doesn't have to be ...
0
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1answer
26 views

How to find dy/dx by logarithmic differentiation

The question says find dy/dx by logarithmic differentiation 2 Definite integral. E^-1/x divided by x^2 dx 1 Answer choices are A 1-sqrt(e)/e B 1-e C sqrt(e)-1/e D sqrt(e)-e/e E sqrt(e) If you ...
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0answers
46 views

Leibniz's Derivative Rule for Integral in Measure Theory

I saw the extension of Leibniz rule for integrals for measure theory on Wiki, although I am not sure if the proposition there is correct. Besides there is no proof for it. Can anybody please introduce ...
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0answers
20 views

Calculate $h'(0)$

Define $h_n(x) = \left\{ \begin{array}{ll} -K_n(-x+\frac{3}{2^{n+1}})^{\frac{2^n}{3}}+\frac{5}{2^{2n+1}} & \mbox{if } \frac{1}{2^n} \leq x \leq \frac{3}{2^{n+1}} \\ ...
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2answers
27 views

Help understanding question regarding 3rd derivative and “smallest uniform bound”?

I'm a big user of Stack Overflow, however, a first time user here. I'm working on a problem for a math class that's pretty easy (I'm sure), I just don't understand the question really. Here it is ...
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2answers
72 views

Derivative of $x^2\sin(\frac{1}{x})$

I was reading an article in American Mathematical Monthly and came across this example.It says that derivative of $x^2\sin(\frac{1}{x})$ takes on all values in $[-1,1]$ in any interval ...
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0answers
91 views

Formula for nth derivative of $\arcsin^k(x/2)$

I need to find formula for $n$-th derivative of $\arcsin^k(\frac{x}{2})$. I have found formula for ...
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1answer
39 views

Reference request: Limits and Derivatives

Could you recommend/suggest a good E-book about Limits and Derivatives with exercise solutions What do you think about that book Limits and Derivatives Made Easy ? looks good but it's not available ...
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0answers
48 views

Can we make sense of the expression $(f^{-1})'$ even if $f^{-1}$ does not exist?

Given a function $f : \mathbb{R} \rightarrow \mathbb{R}$ like $f(x) = \exp(-x^2),$ is it possible to make sense of the expression $(f^{-1})'$ despite that $f^{-1}$ does not exist? I would like to know ...
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4answers
63 views

Integral of a function $f:\mathbb{R}\rightarrow \mathbb{C}$

My real analysis book defines derivatives and integrals only for a function $f:A\rightarrow \mathbb{R}$, where $A\subset \mathbb{R}$. But, when talking about Fourier series, it comes out an integral ...
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0answers
60 views

Show that $y/x$ tends to a finite limit as $x \to + \infty$ and determine this limit.

Let $y=f(x)$ be that solution of the differential equation $$y' = \frac{2y^2+x}{3y^2+5}$$ which satisfies the initial condition $f(0)=0$. (Do not attempt to solve this differential equation.) (a) ...
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2answers
45 views

Is it possible to find a function if we know its differential?

Not something we were taught at uni yet, just something that peaked my curiosity. If I was given a derivative of a scalar function, for example $f'(x)=x$ then I know that $f(x)=\frac{x^2}{2}$ (let's ...
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4answers
205 views

How to prove the quotient rule?

How do we prove the quotient rule for differentiation? The proof in my book from the defintion is very long. Are there some elegant proofs?
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1answer
18 views

Differentiability and basic definitions

If $f+g$ is differentiable at $a$, must $f$ and $g$ be differentiable at $a$? If " and $f$ is differentiable at $a$, must $g$ be differentiable at $a$? If $f*g$ is differentiable at $a$ and $f$ is ...
0
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1answer
37 views

What is the constant $c$ in $\frac{d}{dx} \frac{d}{dy} c^{xy} = c^{xy}$?

In the manner in which $\frac{d}{dx} e^{x} = e^{x}$. What is the value of the constant $c$ for which $\frac{d}{dx} \frac{d}{dy} c^{xy} = c^{xy} $ ?
2
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1answer
58 views

Isn't partial differentiation implied by function context?

This has been bothering me for some time, so I thought I'd finally ask it here. If we are given a function, say, $$f(x,y)=x^2+y^2,$$ and are asked to differentiate it w.r.t. $x$, i.e. ...
2
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1answer
66 views

Given a differentiable function for every $x \geq 0$, define a differentiable function for every $x$

Given $f(x)$: $f(0)=1$ Positive for every $x \geq 0$ Differentiable for every $x \geq 0$ Let $g(x)= \begin{cases} f(x) & \text{$x \geq 0$}\\ 1/f(-x) & \text{$x \leq 0$} ...
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0answers
36 views

Showing where complex function is analytic and differentiable.

I've been asked to show where the following function is analytic and differentiable; $$f(z) = x^4 + i(1-y)^4$$ for $z = x + iy$ First, I noted that $u(x,y) = x^4$ and $v(x,y) = (1-y)^4$. Then, I ...
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2answers
68 views

Hyperbolic Functions (derivative of tanh x)

sinh x = 1/2(e^x - e^-x) and cosh x = 1/2(e^x + e^-x). Prove that d(tanh x)/dx = 1/(cosh x)^2 if tanh x = sinh x/cosh x. I got the derivative for tanh x as: [1/2(e^x + e^-x)]^2 - [1/2(e^x - ...
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2answers
49 views

Examples of Functions

Alright so I am trying to find examples of functions that are differentiable at a point, but not continuous there. Also a function continuous at no point; a function continuous only at one point. ...
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2answers
41 views

Derivatives of exponential functions

For what values of m does the function y = $Ae^{mt}$ satisfy the following equation? $\frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 0$ I tried taking the first and second derivative of the function, but I ...
1
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1answer
44 views

A question on Holder spaces

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. ...
1
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1answer
34 views

Derivative of a minimum

The expression, $e=\left(x(t,w)-c_x\right){}^2+\left(y(t,w)-c_y\right){}^2$, has a local minimum with respect to $t$ at some $t_0(w)$. Now what does $t_0'(w)$ look like?! $x,y\in C^2$ with respect to ...
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0answers
42 views

Multiplying partial derivatives

I am trying to understand what happens when I have a continuous differentiable function $f$ on $\mathbb{R}^n$ such that $f$($x_1,x_2,...x_n$) = 0. What is the significance of the product: ...
4
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2answers
136 views

Multivariable calculus - Implicit function theorem

we are given the function $F: \mathbb R^3 \to \mathbb R^2$, $F(x,y,z)=\begin{pmatrix} x+yz-z^3-1 \\ x^3-xz+y^3\end{pmatrix}$ Show that around $(1,-1,0)$ we can represent $x$ and $y$ as functions of ...
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2answers
132 views

Find the Taylor series of $f(x) = e ^{- 1 / x^2}$

Find the Taylor series about 0, the function defined as: $f(x) = e ^{- 1 / x^2}$ if $x \ne 0$ and $f(x) = 0$ if $x=0$ and What can i conclude of the resulting? First i note that the function f is ...
3
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3answers
332 views

Differentiation of the term x^n n times.

Kindly verify the proof i couldn't find this anywhere. I am fairly new to differentiation so i apologize for mistakes if any... $$\frac{d}{dx}\left(x^n\right)=nx^{n-1}$$ ...
1
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2answers
58 views

Help with separable differential equation? $\frac{dy}{dx} =2y^2$

I'm new to separable differential equations, and I'm stuck on this question: $\frac{dy}{dx} =2y^2$ Using the initial condition $y(2)=3$, find $y(1)$. So far I've integrated to get $\frac{dy}{dx} ...
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2answers
100 views

Is this the correct differentiation of $r (t) = r\cos(\omega t +\phi) - r\sin(\omega t + \phi)$?

I always see questions of this style around because they are so common in physics. Often we're asked to differentiate them so is this correct? $$r'(t) = -\omega r\sin(\omega t +\phi) - \omega ...
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2answers
111 views

Curl, $\vec\nabla \times (\hat{a}\times \vec{b})$

EDIT: FIXED TYPOS & Deleted most of my wrong work pointed out by others. Calculate the curl of $f(\vec r,t)$ where the function is given by $$ f(\vec r,t)=- (\hat{a}\times \vec{b}) \frac{e^{i(c ...
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3answers
310 views

first, second and third derivative of a vector function

I can define the following vector function $f(x,y,z) = [x^2 - 1, x^3 + y^2, z]$ in MatLab or in Maple. I want to find (and evaluate) the first, second and third derivatives of it. How to find : ...
0
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4answers
122 views

How to find the minimum value of $x^2+y^2+xy-4$ where $x+y=2$. [closed]

How to find the minimum value of the expression: $x^2+y^2+xy-4$ where $x+y=2$
0
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1answer
57 views

What is the answer to this derivative?

This derivative just showed up in a past paper as part of a question, i don't know what to do with it because of the summation etc?? Please help $$\frac{\partial}{\partial h} ...
0
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1answer
13 views

Differentiability proof for an interval close to a point

$f$ and $g$ are differentiable with $fg'-f'g=0$. Prove if $f(a)=0, g(a)\neq 0$ then $f(x)=0$ $\forall x$ in an interval around $a$. I think I am very close to solving this, but I am missing the last ...
6
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1answer
82 views

On derivatives that are not Riemann integrable

Let $f:\;[a,b]\to\mathbb{R}$ be differentiable on $[a,b]$. It is not a mystery that $f'$ need not be Riemann integrable. In fact even if we require $f'$ to be bounded the implication is still false. ...
3
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1answer
50 views

Derivative and lipschitz

If I have a real-valued continuous function defined on a compact subset of real line, such that its derivative(wherever it exists) is bounded. Is such a function necessarily Lipschitz? Additionally, ...
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3answers
49 views

The derivative of $2^{x+1}$ in the point $-3$

They ask me to compute the derivative of $2^{x+1}$ in the point $-3$. Since the function is continous in that point, all I have to do is to compute the lateral derivatives. The left one first: ...
0
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0answers
29 views

Summation of a function with the variable both in the function amd in the upper limit

E is defined as : E = c1 ( a$\rho$ + b$\rho ^{2}$ ) + c2 $\rho$ ( c + d $\sum_{j=0}^{n} (\log{ \frac{R\rho}{j} } ) $ ) + c3 $\rho ^{2}$ a, b, c, d, c1, c2, c3, R are known constants. $\rho$ is the ...
1
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1answer
44 views

General formula for $\dfrac{\partial^k}{\partial x^k} \left(\frac{f(x)}{g(x)}\right)$

I would like to know the general formula for expressing $\dfrac{\partial^k}{\partial x^k} \left(\dfrac{f(x)}{g(x)}\right)$ in terms of derivatives of $f(x)$ and $g(x)$. I am stuck when trying to ...
0
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3answers
40 views

Calculating $\displaystyle \lim_{x \to 0+} \frac{\log(\cos(x))}{x}$ where the domain of the quotient is $(0, \pi/2)$

Calculating: $$\displaystyle \lim_{x \to 0+} \frac{\log(\cos(x))}{x}$$ where the domain of the quotient is $(0, \pi/2)$ The fist step is setting $f(x)=\log(\cos(x))$ and $g(x)=x$, and verifying ...