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

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63 views

Geometric Interpretation of a “Near”-MVT

Going through Larson's Problem Solving Through Problems, I am asked to give a geometric interpretation of the result below. I have been sketching it, and only got so far as to note that there must be ...
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0answers
16 views

Partial derivatives of a function with conditions dependent on parameters

Not sure if that question title makes any sense, but here's my problem. I have a function $$ f(x,\alpha,\beta) = \begin{cases} {\frac{x-\alpha}{\beta-\alpha}} & {\alpha \leq x \leq \beta}\\ {0} ...
3
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1answer
119 views

Degree two homogeneous differentiable function is a quadratic form.

Let $f: \Bbb R^n \to \Bbb R^k$ be a ${\cal C}^2$ function such that $f(tx) = t^2f(x)$ for all $t \in \Bbb R$ and all $x \in \Bbb R^n$. Then there is a bilinear map $B: \Bbb R^n \times \Bbb R^n\to ...
4
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1answer
68 views

meaning of the notation f'(-x)

What does $f'(-x)$ essentially mean? $\frac{df(-x)}{dx}$, or $\frac{df(x)}{d(-x)}$, or $\frac{df(x)}{dx}|_{x=-x}$ ? I am not sure if all the options are different, though! :) EDIT 1: Let me ...
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1answer
42 views

Vector space of a sequence of scalars

Struggling slightly with the following question, not sure how to proceed: Let $c_{0}$ be the vector space of sequences of scalars $(a_{n})_{n\in\Bbb{N}}$ such that $a_{n} \to 0 $ as $ n \to \infty $. ...
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0answers
28 views

$f$ is differentiable on $[a,b]$, show that $f'([a,b])$ is an interval

Function $f$ is differentiable on $[a,b], a< b, a,b \in \mathbb R$. I need to show that $f'([a,b])$ is an interval. The standard approach would be to take two points from the interval (I assume ...
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0answers
19 views

Compute a derivative $\frac{df (a, c)}{dg(a, b)}$

I tried to compute the derivative $\dfrac{df(a, b)}{dg (a, c)}$ and wanted to check if what I did was legal. $\dfrac{df(a, b)}{dg (a, c)} = \dfrac{df}{da}*\dfrac{da}{dg} + ...
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1answer
33 views

Directional derivative of a piecewise defined function

Given $f(x,y)=\left\{\begin{matrix} \frac {x^2y}{x^4+y^2} & (x,y)\neq(0,0)\\ 0& (x,y)=(0,0) \end{matrix}\right.$ I need to calculate the directional derivative at the point (0,0) in the ...
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0answers
28 views

partial deravtive of $\sin xyz - { 1 \over z-xy} = 1$

Given z(x,y) and $\sin xyz - { 1 \over z-xy} = 1$. How to calculate $z_x(0,1)$ ? Let $$F=\sin xyz - { 1 \over z-xy} - 1 = 0 $$ $$z_x= - \frac{F_x}{F_z}= -{y(z+xz_x) \cos xyz + \frac{z_x - y}{(z-xy)^2} ...
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0answers
19 views

Derivation of the hypergeometric function $\frac{\partial {}_{3}F_{2}(a_{1}, a_{2}, a_{3}; b_{1}, b_{2}; \frac{1}{z})}{\partial z}$

We know that the first order derivative of the generalized hypergeometric function ${}_{3}F_{2}(a_{1}, a_{2}, a_{3}; b_{1}, b_{2}; z)$ is expressed as follows: \begin{equation} \frac{\partial ...
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0answers
19 views

Computing derivative of a composition.

I have to compute the derivatve $f(g(u,v))$, where $$f(x,y)=x^2+y^3+3xy^2+5,\; \; g(u,v)=(u-v,u^2+v^2)$$ in the point $(u',v')=(1,1)$. Can anyone explain to me how do I find such derivatives, a ...
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2answers
320 views

Derivative by definition

I'm trying to find the derivative by definition of the following function: $f(x)=\sqrt{|x|}\sin(x)$ I know that by definition: $$ f'(x)=\lim_{h\to0}\frac{\sqrt{|x+h|}\sin(x+h)-\sqrt{|x|}\sin(x)}{h} ...
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1answer
55 views

Why does this partial derivative of a summation work?

I'm trying to take the partial derivative of $-\sum\limits_{i=1}^n \frac{(x_i-\mu)^2}{2\sigma^2}$ with respect to $\mu$. The correct answer is $\sum\limits_{i=1}^n \frac{x_i-\mu}{\sigma^2}$. It ...
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2answers
33 views

Verifying differential equations (without substitution or integration)

I am aware that several similar threads exist on this forum, however, my particular query is different from any previous question I've seen here. All other answers have a 'y'term on both sides, and ...
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3answers
45 views

Textual explanation of a derivative

In the book Structure and Interpretation of Computer Programs, there is an interesting example on how one might explore symbolic data in programming. They used the differentiation algorithm. That is, ...
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0answers
19 views

Minimising logistic loss function to find optimal matrix

Please take a look at this paper on classifying triples (re link prediction): http://arxiv.org/pdf/1510.04935v2.pdf The question is about how to solve equation 2 using stochastic gradient descent. It ...
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1answer
30 views

Find the minima of $I(a, b) = \int_0^1 (ax + b-x^2)^2 dx$

$I(a, b) = \int_0^1 (ax + b-x^2)^2 dx$ How to find the minima of $I(a,b)$? My idea is like this, $ I(a,b) = \frac13(ax + b -x^2)^3 \cdot (\frac12 ax^2 + bx - \frac13 x^3) |_0^1$ This is the first ...
3
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1answer
45 views

Limit as $x$ goes to infinity

What is the limit of $\,x^a\big/2^{\log x}$ as $x$ goes to infinity? L'Hopital's rule doesn't help and I can't think of anything that can be done algebraically to make this expression more ...
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1answer
47 views

Check the differentiablity of $\theta :\mathbb R^2 \setminus\{(0,0)\}\to \mathbb R$

For , $(x,y)\in \mathbb R^2$ with $(x,y)\not =(0,0)$ , let $\theta=\theta(x,y)$ be the unique real number such that $-\pi<\theta \le\pi$ and $(x,y)=(r\cos \theta , r\sin \theta)$ , where ...
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4answers
87 views

What's stopping the derivative of $f(x)g(x)$ from equaling $f(x)g'(x)$?

What am I not understanding about how limits work? Please help me understand what's wrong with this proof. $f$ and $g$ are differentiable functions of $x$, and are therefore continuous. ...
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2answers
37 views

Define Derivative of Product of Polynomials

I have a a problem with defining a certain term... The derivative of a product of polynomials is the sum of derivatives of the products of the summands of the polynomials of the original product. ...
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1answer
25 views

function has a solution : Derivative at a point has rank $n$

Let $f:\mathbb{R}^{k+n}\rightarrow \mathbb{R}^n$ be class $C^1$; suppose that $f(a)=0$ and that $Df(a)$ has rank $n$. Show that if $c$ is a point of $\mathbb{R}^n$ sufficiently close to $0$, then the ...
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1answer
36 views

Differentiation of a real-valued function which depends on a vector

I am unable to derive the differentiation of this function $f$: $$f(\rho(\mathbf{r}),\mathbf{g}(\mathbf{r}))= \rho(\mathbf{r}) \times E(\rho(\mathbf{r}),\mathbf{g}(\mathbf{r})) $$ Here ...
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2answers
43 views

Find a unit vector in which the directional derivative equals zero

Find a unit vector u in which the directional derivative of $f(x,y)=\ln(1-x^2-y^2)$ at ($\frac{1}{2} ,\frac{1}{2}$) is zero $f_x=\frac{-2x}{1-x^2-y^2}$ and $f_y=\frac{-2y}{1-x^2-y^2}$ Giving ...
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4answers
62 views

Evaluate $\lim_{x\rightarrow 0}\frac{1}{\sqrt{x}}\exp\left[-\frac{a^2}{x}\right]$

I am interested in the following limit: $$\lim_{x\rightarrow 0}\frac{1}{\sqrt{x}}\exp\left[-\frac{a^2}{x}\right]$$ Does this limit exist for real $a$? Edit: I am only interested in the case when $x$ ...
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1answer
41 views

Is this strange function differentiable?

I have to demonstrate that this function is differentiable in (0,0). The function is: $ f(x,y) = \begin{cases} (x^2+y^2)[cos(\frac{1}{x})-1] & \quad \text{if } x\neq 0\\ 0 & ...
1
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1answer
59 views

How to show f is a linear transformation?

Let $f: R^n \to R$ be a continuous function on $R^n$ that is homogeneous of degree 1. Suppose that $f$ is differentiable at the origin $(0, 0)$. Prove that $f$ is a linear transformation. This is ...
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2answers
33 views

Continuity of a 2 variable function - Munkres exercise

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined as $f(0,0)=0$ and $f(x,y)=\frac{xy(x^2-y^2)}{x^2+y^2}$ for $(x,y)\neq (0,0)$ Then question asks to prove that $f$ is differentiable. Hint that ...
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2answers
38 views

Prove that there is such A, such that for each $x \in [0,2]:$ $|f(x)-x| \le A(x-1)^2 $.

Let $f$ have continuous derivatives until second order in the interval $[0,2]$, s.t. $f(1)=f^\prime (1)=1$. Prove that there is such A, such that for each $x \in [0,2]:$ $$|f(x)-x| \le A(x-1)^2 $$ ...
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1answer
20 views

Operating methods in resolving derivatives - Compound functions and Quotients

I have two questions to ask about derivatives of Compound functions and Quotients : 1) My professor suggested to start from the inside and go outwards while solving compound functions derivatives, ...
1
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1answer
48 views

Finding order and degree of a differential equation

The question was Find the sum of degree and order of the given DE (differential equation)$$ \frac{d}{dx} \left(\frac{dy}{dx}\right)^3=0 $$ So we have that $$ \left(\frac{dy}{dx}\right)^3=c $$ ...
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0answers
39 views

More intuition on the curl formula

I have a question regarding this quesiton. It says that $3$ simple fields that describe rotations around $x,y,z$ axis are: $$H_1(x,y,z)=(0,−z,y)\\ H_2(x,y,z)=(z,0,−x)\\ H_3(x,y,z)=(−y,x,0)$$ but why? ...
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3answers
51 views

what is the derivative of $f(x)$ when $x=-3$? [closed]

What is the derivative of $f(x)$ when $x=-3$? I think there is a relation between derivative and slope?
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2answers
116 views

Calculus of Variations Problem involving mixed constraints

Motivation Let $X$ be $\mathcal{N}\Big(-\frac{\sigma^2}{2},\sigma^2\Big)$ random variable, i.e. probability density function $f(x)$ is given by \begin{equation} f(x)=\frac{1}{ \sqrt{2\pi\sigma^2} } ...
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2answers
49 views

Suppose $f$ is differentiable and $f^\prime(x) = 0$, then $f(x) = o(x)$ at $x\rightarrow 0$?

I know there are two ways to understand $o(x)$, in this case we only discuss in the situation of $\lim_{x\rightarrow 0} \frac {f(x)}{g(x)}=0 \Longleftrightarrow f(x) = o(g)$ So I wonder whether it is ...
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2answers
27 views

How can I derive basic derivative identities?

I haven't taken a calculus class in a long time but last I recall you memorize things like $(\log x)' = \frac{1}{x} ~ dx$ or $(\log_a x)' = \frac{1}{x \log a} dx$ or $(a^x)' = a^x \log a ~ dx$ or ...
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5answers
83 views

Proving that $x-\frac{x^{3}}{6} < \sin(x)$ for $0<x<\pi $

I have to prove that $$x-\frac{x^{3}}{6} < \sin(x) \quad\text{ for }\quad 0<x<\pi $$ I tried to define $f(x) = \sin(x) - (x-\frac{x^{3}}{6})$, and to differentiate it, but $f'(x) = \cos(x) ...
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1answer
30 views

Does the condition of this theorem need continuous differentiability?

I'm reading the book, Analysis from its History, from E. Hairer and G. Wanner. Precisely, I'm reading the french version from 2001, L'analyse au fil de l'histoire. The following theorem is given: ...
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1answer
23 views

Why do we only consider $x\to 1$ and not $x\to 0$ as $t\to\infty$ for the solution to $\dot x=k(1-x)x$ with $x=x_0$ at $t=0$?

Q. Solve the differential equation $\dot{x}=k(1-x)x~,~k\gt 0$ with the initial value $x=x_0$ at $t=0$. Hence, show that $x\to 1$ as $t\to +\infty$ Solution: $$\frac{\mathrm dx}{x(1-x)}=k\,\mathrm ...
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0answers
23 views

Variation of $f(k)=(k-k^2c)\frac{\Gamma(L-k,ak+b)}{\Gamma(L-k)}$

I have the following function: $f(k)=(k-k^2c)\frac{\Gamma(L-k,ak+b)}{\Gamma(L-k)}$, with $kc <1$, and $a, b$ are some positive constants. We assume that $1\le k\le L-1$. Note that ...
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0answers
52 views

The function integrated over the two-sphere

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a smooth function. Define the function $h: \mathbb{R} \rightarrow \mathbb{R}$ $$ h(w) = \int_{S^2} d\Omega(\hat{n})\,f(\sqrt{|w|}\, \hat{n}) $$ where ...
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1answer
46 views

Evaluate $\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^nf'\left(\frac{k}{n}\right)$

Left $f:\mathbb{R}\to\mathbb{R}$ be continuously differentiable. We are to evaluate $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^nf'\left(\frac{k}{n}\right)$$. One thing I know is that $$\lim_{x\to ...
2
votes
4answers
92 views

Second derivative of $x^3+y^3=1$ using implicit differentiation

I need to find the $D_x^2y$ of $x^3+y^3=1$ using implicit differentiation So, $$ x^3 + y^3 =1 \\ 3x^2+3y^2 \cdot D_xy = 0 \\ 3y^2 \cdot D_xy= -3x^2 \\ D_xy = - {x^2 \over y^2} $$ Now I need to ...
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2answers
36 views

The function e - where increasing and decreasing

I have to find where $(x+\frac{1}{x})^{x}$ is increasing and decreasing. I tried to derivate it , and I have now - $f'(x) = (x+\frac{1}{x})^{x} \left (ln(x+\frac{1}{x}) + (\frac{x-1}{x}) \right )$ ...
0
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2answers
29 views

Reverse Power Rule integration.

Ok, so I am confused about the following; When we have a polynomial, say $P(x)$, and we want to solve an integral where $P(x)$ is raised to a certain power, for example; $$\int (P(x))^adx$$Why can we ...
30
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5answers
2k views

Notation of the second derivative - Where does the d go?

In school I was taught that we use $\frac{du}{dx}$ as a notation for the first derivative of a function $u(x)$. I was also told that we could use the $d$ just like any variable. After some time we ...
1
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3answers
58 views

Hundredth Derivative From Taylor's polynomial for $\frac{x^2}{1+x^4}$

I'm trying to solve this problem: Find the hundredth derivative (at $x=0$) from Taylor's polynomial for $\dfrac{x^2}{1+x^4}$. I keep getting the wrong answer; can someone help? I have tried ...
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2answers
14 views

Normal derivative when $x_0\in \partial\Omega$

When I read the Normal derivative of Wiki and this question, I am confuse with them. Assume $\Omega$ is bounded open subset of $R^n$ and $\partial\Omega$ is smooth, and some smooth function $f$ ...
6
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2answers
73 views

How to find $\lim_{x \to a}\frac{ a^nf(x)-x^nf(a)}{x-a}$

f:$\mathbb {R} \to \mathbb{R}$ which is differentiable at $x=a$ the we are to evaluate the following:- $$\lim_{x\to a}\frac{a^nf(x)-x^nf(a)}{x-a}$$ My approach:- ...
0
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1answer
27 views

Evaluating derivatives with Let f(x) = 3, f'(x) = 1 [closed]

Let f(x) = 3, f'(x) = 1 g(x) = 10 g'(x) = 7 then evaluate $$\lim_{x\to x_o}[{\frac{f(x)}{g(x)} -\frac{f(x_o)}{g(x_o)}\over x-x_o} ]$$ What does this mean? And hpw to do it in full workout please