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

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Reference for Gradient expression of a function on matricies

I'm looking for a reference (I suppose the statement is correct) for the following formula: $$ \langle\nabla f(\rho)^\dagger,V\rangle=\left.\frac d{dt} f(\rho+tV)\right|_{t=0} $$ for any direction ...
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2answers
15 views

differentiation of both sides of the eqution

Why did this happen ? Why did differentiation of x^i become (i + 1)x^(i) instead of (i)x^(i-1)
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1answer
11 views

Expressing a set of discrete inequalities as a continuous differential equation

I'm trying to work out the solution to a problem of sequential inequalities. I believe the solution collapses to a set of differential equations, but I'm having trouble organizing things and I think I ...
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1answer
21 views

Show that this piece-wise function defines a differentiable solution

Show that $y(x) = \begin{cases}-x^4 & x < 0, \\ x^4 & x \geqq 0 \end{cases}$ defines a differentiable solution of $xy'=4y$ for all $x$, but is not of the form $y(x)=Cx^4$.
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0answers
10 views

Can I determine complex differentiability by differentiating wrt to z?

If I have a function in terms of $z\in C$ and need to determine the points where it is differentiable, can I simply find the derivative wrt z and see where it is defined? I know that one solution is ...
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1answer
27 views

Derivative and and function terminology

In mathematical parlance, we say "take the derivative of a function f" to indicate that we are computing a new function, which maps slopes, that derives from f. However, in physics, we say "take the ...
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0answers
14 views

Total derivative vs. partial derivative Legendre transformation

My question is about how to compute the total derivative for the function $f(x,y)$. In theory we have: $$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$ So, if somebody asks ...
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2answers
32 views

Binomial expansion derivative limit definition

Can someone help me with this? I am supposed to use a binomial expansion to calculate $\sqrt x$ directly from the limit definition of a derivative.
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2answers
41 views

$\frac{d^{100}}{dx^{100}}\left[\frac{f(x)}{g(x)}\right]=\frac{p(x)}{q(x)}$,then find the degrees of the polynomials $p(x)$ and $q(x)$

Let $g(x)=x^3-x$,and $f(x)$ be a polynomial of degree $\leq100$.If $f(x)$ and $g(x)$ have no common factor and $\frac{d^{100}}{dx^{100}}\left[\frac{f(x)}{g(x)}\right]=\frac{p(x)}{q(x)}$,then find the ...
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0answers
15 views

Using chain rule to find $\bigtriangledown ^2$ of spherically symmetric field

I have a scalar function $\psi = \psi(r)$, where $r$ is the radial distance. I see how, using the chain rule, $\bigtriangledown\psi = \psi'(r)$x/r. But then I need to find $\bigtriangledown^2\psi$, ...
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2answers
31 views

Sequence of functions and function series

For every $n \ge 0$ we define function $f_n:[-1;1]\rightarrow \mathbb{R}$ $f_n(x) = \sqrt[n+1]{n+1}(\frac{x+x^2}{2})^n$, $x\in[-1;1]$, $0^0=1$. a)determine whether sequence of functions $\{f_n\}$ ...
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0answers
25 views

Bounded derivative and Taylor's polynomial

Given function $f:\mathbb{R}\rightarrow \mathbb{R}$ of class $C^{\infty}$ such as $\forall_{n\ge2015}{\forall_{x\in\mathbb{R}}{|f^{(n)}(x)|\le7}}$ a)prove that sequence of functions $\{T_{n,f,0}\}$ ...
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1answer
20 views

Gradient of Norm 2 with chain rule

Assume $A$ is a matrix with size of $m*n$ and $b$ is a vector with size of m. If $f$ is a function which accepts a scalar and returns a vector with size of $n$. Now what is the gradient of following ...
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2answers
20 views

Integral differentiation with infinite bound (differentiation of expected value)

I am trying to proove the following: $$\frac{d}{dx}\int_x^{\infty}(z-x)f_Z(z)dz=1-F_Z(x)$$ Where $f_Z$ and $F_Z$ are resp. the probability density and cumulative distribution functions of a random ...
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1answer
39 views

Using epsilon and delta to compute a derivative

Let $a>1$, let $x\in\mathbb{Q}$, and define $f(x)=x^a$. I am interesting in computing $f'(0)$ if it exists. I claim that $f'(0)=0$. Attempt: Let $\epsilon > 0$. Suppose $0 < \lvert x-0 ...
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1answer
28 views

Appling Jensen's inequality

I have to prove that for every $a,b,c \in \mathbb{R}$ $$1+\sqrt[3]{e^{2a}}\sqrt[5]{e^b}\sqrt[15]{e^{2c}}\le \sqrt[3]{(1+e^a)^2}\sqrt[5]{(1+e^b)}\sqrt[15]{(1+e^c)^2}.$$ We can prove that ...
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0answers
23 views

What do the Stirling numbers of the first kind have to do with polylogarithms?

On a whim, I had decided to look into ways of evaluating series of the form $$\sum_{n\ge1}\frac{1}{n^k2^n}$$ which I learned has a more general form in terms of polylogarithms: ...
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1answer
31 views

How can I find the minimum of these two functions? [on hold]

Here are these two functions: $$P=x^{3}+y^{3}+3(xy-1)(x+y-2)$$, where $$x^{2}+y^{2}-8(x+y)+2xy\leq 0$$ and $$Q=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$, where $$x^{2}+y^{2}+z^{2}=1$$ I've no idea how ...
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1answer
28 views

derivative of an integral that's multiplied by a function [on hold]

I saw a post asking about how to integrate an integral function, I just need clarification about the derivative of an integral function that is multiplied by another function do I use the ...
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1answer
13 views

Particle's acceleration when it achieves maximum displacement in the +x direction

I'm not sure if this goes in the physics section or the mathematics section, but this question seems pretty math oriented, so I'll ask it here. $t\ge0$ and $v_x=24-3t^3$ I have to find the ...
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1answer
58 views

Conditions on $c$ such that the inequality dont hold.

I want to find conditions on $c$ such that the inequality don't hold. $$1-ac(a-2)(a-1)^2 < 0 \ \ \ \ \ \ \text{for } a>2, c>0$$ If $\phi(a) = ac(a-2)(a-1)^2 \Rightarrow \phi'(a) = ...
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4answers
89 views

How does $\int (\cos(x))^{-2}dx$ equal to $\tan(x)$?

How does $$\int \frac{1}{\cos^2(x)} dx= \tan(x)+ C$$ ?
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5answers
98 views

How do I integrate$ \int\frac{1}{e^{2x}+e^x} \,dx $ [on hold]

How do I integrate following function? $$ \int\frac{1}{e^{2x}+e^x} \,dx $$
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1answer
40 views

Derivative of $\frac{x}{\|x\|}$ w.r.t. x where $x\in \mathbb{R}$ ($x \neq \theta_n$)

I want to find the Hessian of a function. I have already computed the gradient of the function. So, I have to again differentiate it w.r.t. $x \in \mathbb{R}^n$ to get the hessian, but I am facing a ...
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1answer
22 views

Definition of differentiability at the point in multivariable calculus.

I'm self-studying the analysis from Zorich and the next definition of differentiability is given: $f:E\to \mathbb{R}^n$ is differentiable at the point $x$, which is a limit point of $E\subset ...
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1answer
34 views

How to determine the function from the following?

The graph of a certain function contains the point $ (0,2)$ and has the property that for each number 'p' the line tangent to $y = f(x)$ at $(p, f(p))$ intersect the x-axis at p + 2. Find $f(x)$ The ...
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1answer
26 views

uniformly convergent sequence of differentiable functions, series of derivative of terms not convergent

I am attempting to come up with a uniformly convergent sequence of differentiable functions $g_{n}:(0,1)\to \mathbb R$ such that the sequence $\{g_{n}'\}$ does not converge. I was thinking that ...
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0answers
23 views

Is this a “locally surjective” function?

I quote the "locally surjective" part because I haven't found any reference of that concept, but it kind of fits what I mean. Let $f:\mathbb{R}^N \to \mathbb{R}^M, f \in C^1, x_0 \in \mathbb{R}^N : ...
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3answers
61 views

What we exactly do when we take derivative of any function? [on hold]

When we take differentiation of any function then what actually we do with that function? Ex.d/dx of x^2 is 2x. So what we have actually done with x^2.
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1answer
28 views

Second derivative with implicit differentiation

Question: Determine whether the given relation is an implicit solution to the give differential equation. Assume that the relationship does define y implicitly as a function of x and use implicit ...
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0answers
16 views

Proof of Darboux's Theorem when the function has infinite derivatives at both endpoints.

I have a question about the statement in the NOTE above. It says that the Darboux's Theorem is also valid when one or both the one-sided derivatives are infinite. So say $f_{+}'(a)=-\infty, ...
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continuity of the piecewise functions [on hold]

$1$. $g(x)=0$,if $x$ is irrational and $g(x)=x$ if $x$ is rational Find all points of at which $f$ is continuous. $2$. Let $A$ and $B$ be compact sets. Define $A+B =$ ...
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1answer
12 views

Singular points while differentiating a function with respect to another function

I have $z(x) = \frac{df(x)}{dx}$ where $f(x)$ if a function of x. I'd like to have the derivative of $z(x)$ in respect to $f$: $\frac{dz}{df} = \frac{\partial f'(x)}{\partial x} \frac{dx}{df}$ ...
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1answer
80 views

A possible dumb question about derivative

I was solving some differentiation problems when I found the function $$g(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}.$$ So I thought: If I define the function $f:\mathbb{R_{x>0}}\to \mathbb{R}$ as ...
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3answers
94 views

Proving $\sin^2(x) + \cos^2(x) =1$ using calculus

Ok so the book in which I found this doesn't say mention the trigonometric functions by name but the question is: Let $s(x)$ and $c(x)$ be functions satisfying $s'(x)=c(x)$ and $c'(x)= -s(x)$ for ...
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2answers
33 views

L'Hopital's rule and limiting variables

I'm working some problems from a calculus text and came across this question: If $f(x)$ is a function that's differentiable everywhere, what is the value of the limit $$\lim\limits_{h \to ...
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3answers
30 views

Help with a derivative of integral please.

I'm supposed to calculate the derivative of $\frac{d}{dx}\int_{x^{2}}^{x^{8}}\sqrt{8t}dt$ the answer I got is $8x^7\cdot \sqrt{8x^8}$ but when I put this into the grading computer it is marked wrong. ...
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2answers
26 views

Let $f(x)=\arctan x -\frac{ln|x|}{2}$. Then $f(x)$ is increasing in?

Let $x<0$. $f'(x)=\frac{1}{1+x^2}+\frac{1}{2x}>0$ $$f'(x)=\frac{(1+x)^2}{2x(1+x^2)}$$ which is always negative. But answer is $x\in (-\infty,0)$.
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0answers
18 views

Proofs of the multiplication or chain rule for derivatives that invoke symmetry

Introductory calculus texts sometimes include direct proofs of the multiplication and chain rules for derivatives by: Introducing a pair of differences $D_f=\frac{f(x+h)-f(x)}{h}-f'(x)$ and ...
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1answer
25 views

Lowering the order of a linear differential equation

Let $$L(x) \equiv x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}(t)x'+a_n(t)x=0.$$ and let the following solutions be given: $x_1,x_2,...,x_m(m<n)$- linear independent solutions. Let's find: $x_{m+1}, ...
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1answer
45 views

Why do we write $df/dx$ instead of $df/dx(x)$?

I was just thinking about how, i.e., if $f\colon\mathbb R\to\mathbb R$ is defined by $f(x) = x^2$, then it's customary to write $$ \frac{df}{dx} = 2x. $$ But since the derivative is itself a function ...
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0answers
24 views

find the stationary points for $f(x)=x^{\frac 2 3}$.difference between the stationary point and critical point and one more called turning point.

Find the stationary points for $f(x)=x^{\frac 2 3}$. My work I realized the following $\spadesuit$ $f'(x)=\frac 2 3 x^{-\frac 1 3}$ which is not defined at $x=0$ $\spadesuit$ $f'(x)<0$ for ...
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2answers
534 views

Only once differentiable

Is there any example of a real function that is one-time-only differentiable, meaning there is $f'(x)$, but no $f''(x)$? I haven't been able to find any example... Of course it would be preferred if f ...
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2answers
56 views

To determine the existence and the value of $\lim_{x \to 0} \frac {2^x-1} x$

I would like to somehow firstly show that $$\lim_{x \to 0} \frac {2^x-1} x$$ exists and determine the value of the limit. My first ideas were by Monotone Convergence. I have been able to prove that ...
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2answers
23 views

The supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ when $x+y=2n$ for some fixed $n\in \mathbb N $

Let $S$ be the set of all tuples $(x,y)$ such that $x+y=2n$ for a fixed $n\in \mathbb N$. Then what is the supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ $?$ I substituted $y=2n-x$ ...
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0answers
33 views

Find a smooth path along which a given function on the plane is not differentiable at the origin

From Bamberg & Sternberg’s A Course In Mathematics For Students of Physics, Exercise 6.1d: Let $F(x,y) = \frac{x^3y}{x^2+y^2}$ for $(x,y) \neq (0,0)$ and $F(0,0)=0$. Invent a smooth curve ...
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2answers
71 views

Wolfram Alpha “x = derivative x”

Asking Wolfram Alpha $x = \text{derivative } x$, I was expecting $e^x$, being that the derivative of $e^x$ is $e^x$, Wolfram Alpha however yields $x = 1$. Is this stating that the derivative of a ...
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1answer
31 views

Infimum of $f(x): (0,\infty) \to \mathbb R$, $f(x) = \ln(e^x-1)+\frac2x-x$

Calculate infimum of $f(x): (0,\infty) \to \mathbb R$, $f(x) = \ln(e^x-1)+\frac2x-x$ I calculate derivative $$f'(x)= (\ln(e^x-1)+\frac2x-x)' = \frac1{e^x-1}e^x-\frac2{x^2}-1 = ...
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2answers
30 views

Find all points such that function has all partial derivatives in that point.

Find all points $(x,y) \in \mathbb{R}^2$ such that function has all partial derivatives in that point.$$ f(x,y) = \begin{cases} \frac{\sin(xy^2)}{y} &\mbox{if } y>0 \\ xy^2 & \mbox{if } y ...
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2answers
33 views

Derivative of implicit function with exponential functions of each other

We have the equation: $$ x^y = y^x +y $$ Which defines an implicit function $y(x)$ at the point $(2,1)$. I'm asked to find the derivative at $y'(2)$. I saw the answer in Wolfram: $$ y'(x) = \frac{y ...