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

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8 views

How to compute the gradient of the weighted kernel

Let's say $f(X) = \sum_{i,j}A_{i,j}x_i'\cdot x_j $ where $x_i,x_j$ are the i-th, j-th columns of $X$. So what is the gradient $\frac{\partial(f(X))}{\partial{X}}$ ?
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24 views

Find $C^1$ class function such that

Given: $$g: \mathbb{R}^3 \rightarrow \mathbb{R}, g(x,y,z)=z^3-3xyz-x-8$$ Decide whether in the neighbourhood of the point $(x,y)=(0,0)$ there exist $C^1$ class function $z=z(x,y)$, such that ...
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0answers
23 views

Differentiating the definite integral of x*f(x)

I'm trying to differentiate the integral below. I was wondering how I could approach it. z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} z \cdot \Phi ^{\prime} (z) dz$ ...
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30 views

How to derive: div ( s * [ grad(U) ]^T )

Having some questions on the derivation of the following equation (seen in someone's PhD thesis): where s is a scalar valued function, ...
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42 views

tricky derivative with logarithm of sum

I'm having trouble understanding the solution of a limit. It involves a formula for measuring certainty of a discrete probability distribution. Given a set of values $p_j$ which sum up to 1, find the ...
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0answers
28 views

Differentiating CDF

I'm trying to differentiate the cdf of z with respect to x where the upper bound is a function of x and z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} \Phi ^{\prime} (z) ...
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0answers
11 views

Minimum directional derivative

I know that maximum of $(D_uf)= ||grad f||$ Because $(D_uf)= grad f. u = ||grad f||.||u||.Cos\theta$ So how about the minimum value? I believe it must be $-||gradf||$ since $min(cos\theta = -1 ) ...
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3answers
41 views

Calculus Lab Problem Differentiation [on hold]

I am kind of confused on how to differentiate this function. Here is the problem: $$P(t) = \frac{1}{(1 + e^{-t})} $$ Use the differentiation rules to compute the exact value of $P'(2)$. Certain ...
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1answer
40 views

Differentiation Commute with Lebesgue Integration

My question is simple: Given $f: \mathbb{R}^{n+m} \to \mathbb{R}$, $f\in C^{k}(\mathbb{R}^{n+m})$ , and $X \subset \mathbb{R}^{n}$. Write $f$ as $f(x_1, \ldots, x_n, t_1, \ldots t_m)$. When is ...
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3answers
28 views

Find the area between the given function , and two tangents off of the point (2,-2)

So here is a general graph of the first couple directions. $T_1$ and $T_2$ are supposed to be the points where the tangent line intersects the parabola. The tangent lines and points where the ...
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20 views

how can I prove that a derivative of an implicit function is bounded?

I have the following implicit function $V(\tau,\mu)$. The function is bounded and continuous and differentiable on $\mathbb{R}$. What other properties or assumptions should I make or what conditions ...
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21 views

From inequality on derivatives to inequality on functions

What is the set of differentiable functions $f$ that satisfy the following inequalities for all $x\geq 0$: $0\leq f'(x)\leq e^{-x}$ Initially, I thought I should just integrate the inequality and ...
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1answer
18 views

Piecewise $C_1$ and piecewise continuous

I would appreciate if the following questions could be clarified with your help. If a function is piecewise $C_1$, does this imply that it's also piecewise continuous? If a function is piecewise ...
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2answers
22 views

Finding Tangent Line Using Limit Definition

I'm supposed to get the equation of the tangent line to the graph of $f(x)= \frac{8}{x}$ at the point $(2,4)$. I started with $$\frac{\frac{8}{x+h} - \frac{8}{x}}{h},$$ then I cross multiplied: ...
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1answer
19 views

Conditions for convergence of derivatives from pointwise convergence

Let $\{f_n\}_n$ be a sequence of functions $\mathbb R \rightarrow \mathbb R$ which converges pointwise to $f$, ie: $$f_n(x) \rightarrow f(x) \hspace{10pt}\hbox{for all $x$}.$$ What additional ...
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1answer
22 views

Characterizing connected sets of $\mathbb R^n$ is terms of differentiable maps for which zero derivative everywhere implies constant

Let $U $ be an open subset of $ \mathbb R^n$ ; then how to prove that $U$ is connected iff for every differentiable function $f:U \to \mathbb R$ , $\nabla f(x)=0 \implies f $ is constant on $U$ ?
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0answers
12 views

General case for differentiation under the integral sign

What is the most convenient way to decide if we can differentiate under the integral sign? If the integrant is a smooth function, could we do so?
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1answer
32 views

Show that for each $a>0$ the function $e^{-ax}x^{a^2}$ has a maximum value, say $F(a)$, and that $F(x)$ has a minimum value $e^{-e/2}$

Show that for each $a>0$ the function $e^{-ax}x^{a^2}$ has a maximum value say $F(a)$,and that $F(x)$ has a minimum value $e^{-e/2}$. I differentiated the function $f(x)=e^{-ax}x^{a^2}$ to get ...
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3answers
32 views

The value of $a$ for which $f(x)=x^3+3(a-7)x^2+3(a^2-9)x-1$ have a positive point of maximum lies in the interval $(a_1,a_2)\cup(a_3,a_4)$

The value of $a$ for which $f(x)=x^3+3(a-7)x^2+3(a^2-9)x-1$ have a positive point of maximum lies in the interval $(a_1,a_2)\cup(a_3,a_4)$.find the value of $a_2+11a_3+70a_4$ I differentiated ...
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1answer
43 views

Proof of Green's identity

Can anyone explain to me how to prove Green's identity by integrating the divergence theorem? I don't understand how divergence, total derivative, and Laplace are related to each other. Why is this ...
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1answer
30 views

What is the radius of smallest circular disk large enough to cover every acute isosceles triangle of a given perimeter $L$?

What is the radius of smallest circular disk large enough to cover every acute isosceles triangle of a given perimeter $L$? Let $a,a,b$ are the sides of the isosceles triangle whose perimeter is ...
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1answer
38 views

Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area

Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area for the given constant length $l$ of the median drawn to its lateral side. I tried to solve this ...
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1answer
22 views

Find the values of the derivatives of the integral with a variable inside its limits.

$\require{cancel}$ Problem: I have the function $g: \mathbb{R} \to \mathbb{R}$ defined as $$ g(x)=\int^{(1+x^2)}_{-(1+x^2)} sin(t^3)\ dt,\ x \in \mathbb{R} $$ I would like to calculate values of ...
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1answer
27 views

If $f\in C^2(\mathbb R)$ then $M_1^2 \le 2M_0 M_2$, where $M_k = \text {sup}_x |(d/dx)^k f(x)|$ for $k=0,1,2.$

I wanna prove this problem. I tried it with Mean Value Theorem but cannot proceed to any plausible result. So could I have some hints?
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27 views

Find the maximum volume of the cylinder.

A cylinder is obtained by revolving a rectangle about the $x-$axis,the base of the rectangle lying on the $x-$axis and the entire rectangle lying in the region between the curve $y=\frac{x}{x^2+1}$ ...
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0answers
36 views

Questioning the differentiability of $f(x,y)$

$$f(x,y)=\begin{cases} y- \frac{e^{x^2+y^2}-x^2-y^2}{x^2+y^2},& x^2+y^2 \neq 0. \\ -1, & x=y=0 \end{cases}$$ I keep runnung into trouble with these types of questions. The way I do them is ...
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2answers
40 views

Let $p,q$ be real polynomials. Let $F: \Bbb R \to \Bbb R$ be differentiable, then $p=q$.

Let $p,q$ be real polynomials. Let $F: \Bbb R \to \Bbb R$ be differentiable, defined by: $$ F = \begin{cases} \hfill q \hfill & \text{X $\geq$ a} \\ \hfill p \hfill & ...
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3answers
29 views

Prove that the following functions is differentiable on $(-1,1) \times \mathbb R$

$$f(x,y)=\begin{cases} \frac{\tan x}{x}+y, & 0<|x|<1 \\ 1+y,& x=0 \\ \end{cases}$$ Prove that it is differentiable on $(-1,1) \times \mathbb R$. I use the Frechet definition of ...
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2answers
34 views

Differentiation using Chain Rule

Find $\frac{dy}{dx}$ if $y=7+5^{x^2+2x-1}$. So far I have done $\frac{dy}{dx}=(5^{x^2+2x-1})'$. Now, the RHS can be found by $(e^{\ln 5\cdot (x^2+2x-1)})'=e^{\ln 5\cdot (x^2+2x-1)}(x^2+2x-1)'\ln ...
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0answers
20 views

Computing Partial Derivatives (basic)

When I am asked to compute a partial derivative of $f_x$ for $f(x, y)=x \ln(xy)$, I treat this the same as $\frac{d}{dx} (x \ln(xy))$ which I then just simply apply the chain rule and get $\ln (xy) ...
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0answers
24 views

How to differentiate $\sum_{t=0}^{T}$ problem?

Define $F(C, H)=\sum_{t=0}^{T}\beta^t (logC_t+\mu log H_t)$ where $\beta$ and $\mu$ are constant. I expand this function like $F(C, H)=\beta^0 (logC_0+\mu logH_0)+ \beta^1(logC_1+\mu logH_1)+\dots$ ...
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1answer
54 views

why the integral of $\frac{dy}{y} =\ln(y)$?

I mean if I differentiate $\ln(y)$ the result will be $\frac{dy}{y}$ ? . What I know the diffential of $\ln(x)$ = $\frac{1}{x}$ right?. And following this idea what is going to happen if we ...
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1answer
62 views

How was this differentiated?

How red-circled function with 1/D is equal to green-circled? Note: D is equal to dy/dx. Update: Complete pic
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0answers
26 views

Derivative of mollification

This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
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2answers
58 views

Finding the Derivative of $\sqrt{x}$

How can I find the derivative of $\sqrt{x}$ using first principle. Specifically I'm having difficulty expanding $\sqrt{x + h}$ or rather $(x + h)^.5$. Is there any generalized formula for the ...
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1answer
35 views

Local extremes of $f(x) = (x-2)^{\frac{1}{5}}(x-7)^{\frac{1}{9}}$

The task is to find local extremes of $f: \mathbb R \to \mathbb R$, $f(x) = (x-2)^{\frac{1}{5}}(x-7)^{\frac{1}{9}}$ There is theorem that if $x_{0}$ is local extreme of $f(x)$ then $f'(x_0) = 0$ So ...
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1answer
11 views

Proof that a derivative's points of discontinuity are all essential

I'm reading Wikipedia's article on Darboux's theorem, and it says the following: "Every discontinuity of a Darboux function is essential, that is, at any point of discontinuity, at least one of the ...
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6answers
73 views

Derivation for the derivative of $a^{t}$ from The Equation

In Calculus, the Equation is known as: $$f'(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ This equation allow us to find the derivatives of functions. Let's try this with the exponential ...
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0answers
27 views

Use the alternative form of the derivative to find the derivative at $x=c$ [closed]

$f(x)= x^2-1,$ at $c=2$. $g(x)= \sqrt{|x|},$ at $c=0$. $h(x)= |x-5|,$ at $c=-5$.
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1answer
40 views

Value of the derivative of a function at a point depends only on the germ at that point

Suppose that f : I → R is a $C^∞$ function defined on an open subset I ⊆ R. How can I show that for $a \in I$ the value $f^n (s)$, n = 1, 2, 3, . . . of the derivative of $f$ of order n at s depends ...
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3answers
73 views

Implicit Differentiation. Please help me understand why!

I am trying to understand implicit differentiation; I understand what to do (that is no problem), but why I do it is another story. For example: $$3y^2=5x^3 $$ I understand that, if I take the ...
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1answer
66 views

-relationship between a function and a tangent line

$f: \mathbb{R} \rightarrow \mathbb{R}$ a continuous function at $x=a$. Show that $f$ has derivate at $x=a$ iff there's only a $L(x) = m(x-a)+b $ such that $$ \lim_{x \to a}\frac{f(x)-L(x)}{x-a} = 0 ...
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2answers
51 views

$n$-th derivative of $(ax+b)^{-m}$ [closed]

How to find the $n$-th derivative of $(ax+b)^{-m}$ ?
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1answer
58 views

Example of a function in $L^2(\mathbb{R})$ with derivative not in $L^2(\mathbb{R})$.

We know examples of a function which doesn't lie in $L^2(\mathbb{R})$ with derivatives in $L^2(\mathbb{R})$: $$f_1(x) = \mathrm{arctg}(x) \notin L^2(\mathbb{R}), \qquad f_1'(x) = \frac{1}{x^2+1}\in ...
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1answer
43 views

Prove that if $|a_1\sin x+a_2\sin2x+a_3\sin3x+\cdots+a_n\sin nx|\leq|\sin x|$ for $x\in R,$then $|a_1+2a_2+3a_3+…+na_n|\leq1$

Prove that if $|a_1\sin x+a_2\sin2x+a_3\sin3x+\cdots+a_n\sin nx|\leq|\sin x|$ for $x\in R,$then $|a_1+2a_2+3a_3+\cdots+na_n|\leq1$ When we try to differentiate it on both sides wrt $x$,then modulus ...
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2answers
82 views

Maxima/Minima of absolute function

Given $a_i=\{a_1,\dots,a_n\}$ and function $$f(x)=\sum_{i=1}^n{|x-a_i|}^3$$ I need to find minimum value of $f(x)$. As far my understanding goes the derivative is given by: $$f'(x) = ...
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4answers
56 views

If $x^4+7x^2y^2+9y^4=24xy^3$,show that $\frac{dy}{dx}=\frac{y}{x}$

If $x^4+7x^2y^2+9y^4=24xy^3$,show that $\frac{dy}{dx}=\frac{y}{x}$ I tried to solve it.But i got stuck after some steps. $x^4+7x^2y^2+9y^4=24xy^3$ ...
1
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1answer
29 views

$y=\tan^{-1}\frac{1}{x^2+x+1}+\tan^{-1}\frac{1}{x^2+3x+3}+\tan^{-1}\frac{1}{x^2+5x+7}+\tan^{-1}\frac{1}{x^2+7x+13}…$to n terms.

Prove that if $y=\tan^{-1}\frac{1}{x^2+x+1}+\tan^{-1}\frac{1}{x^2+3x+3}+\tan^{-1}\frac{1}{x^2+5x+7}+\tan^{-1}\frac{1}{x^2+7x+13}......$to n terms.Then ...
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3answers
29 views

Solve the following differential equation of one independent variable

I want to solve the following differential equation $$\frac{dy}{dx}=e^{x-y}(e^x-e^y)$$ I am trying to separate $x$ and $y$ in this way : $\frac{dy}{dx}=e^{x-y}(e^x-e^y)=e^x(e^{x-y}-1)$ Put $x-y=z$. ...
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3answers
40 views

If $f’(x) = \sin x + (\sin4x)(\cos x)$, then $f’(2x^2 + \pi/2) $is?

If $$f'(x) = \sin x + \sin4x \cdot \cos x,$$ then $$f'(2x^2 + \pi/2)$$ is? Given answer: $$4x\cos(2x^2) – 4x\sin(8x^2) \sin(2x^2)$$ I tried and I'm getting the answer as $\cos(2x^2) - ...