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

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3
votes
1answer
33 views

Finding the (12k+2)th derivative

I've been working for a while now on this question and I'm stumped. Find: $$\frac{d^{12k+2}}{dt^{12k+2}}\left[e^{t\sqrt{3}/2}\cos(t/2)\right]$$ where k is an element of N (real integers). The ...
2
votes
1answer
21 views

Simple Question about Derivative property

Suppose $f:[-1,1] \to \mathbb{R}$ is twice differentiable and $f(-1) = f(1) = 0$ and $f(0) = 1$. Prove that there exists $x_0 \in (-1,1)$ with $f''(x_0) = -2$. I tried establishing this with ...
2
votes
2answers
25 views

Do roots lead to two antiderivatives that differ in their non-constant terms?

Consider the following example: $f'(x)= x^{-3/2}$ and $f(4)=2 $ $f'(x)= x^{-3/2}\Rightarrow \frac {x^{(-3/2) + 1}} {-1/2} \Rightarrow$ $\frac{-2}{\sqrt x} +C =f(x)$ This is where the problem ...
-1
votes
1answer
32 views

Why does the second derivative method allow us to classify stationary points?

I'm learning how to find the nature of a stationary point using the second derivative method. I understood the first derivative method where you pick two close points near the target point and find ...
1
vote
0answers
27 views

Interpreting notation for a differential equation

I'm looking at the following differential equation (see Appendix B of the following paper): $$ \partial_t\mu[V_E]=-\mu[V_E] + C_{EE}[x\nu_E+(1-x)v_{ext}]-C_{EI}J_{EI}\nu_I, $$ where $\mu[V_E]$ is ...
1
vote
0answers
21 views

Given two rates of change at two coordinates, can an exponential function be derived to fit?

I have an original exponential function: $1000 \cdot 2^{0.1x}$ At the point $\big( 18+6 \frac {\log5} {\log2} (\approx 31.93) , 9146 \big)$, I'm looking to know if it is possible to derive an ...
0
votes
0answers
41 views

If d/dx is an operation in functions, why do we need f(x)? [on hold]

This might be a little pedantic, but I need to sort out my terminology. Point 1:When mathematicians think of a function, they think of a mapping: $f:x \mapsto f(x)$. $f$ is a function that maps a ...
-1
votes
2answers
37 views

Converting the derivative into a particular form.

I have found the derivative of a function $$y = {(x+1)^2}{(x+2)^-}{^4}$$, the derivative is: $$\frac{dy}{dx} = -\frac{2x(x+1)}{(x+2)^5}$$ However how would you express this derivative into the form ...
-1
votes
0answers
30 views

$n$th derivative of $\sin(nx)$? [on hold]

Don't know how to solve this one please answer in a detailed manner. Thanks in advance... Answer is given in a recursive form. I have thought of using the expansion of $\sin(nx)$ and then ...
-1
votes
1answer
67 views

Find the $nth$ derivative of $y=\sin(x^2)$ [on hold]

Please help me and its not a homework.I have tried it a hell lot of many times and even asked my seniors but none had solved it ...So plzzz help me know step by step how to solve it..thanks in advance ...
1
vote
0answers
35 views

Finding critical points of $f(x) = \frac{\ln(x^2-1)}{x^2-x}$

In sketching the function below, I'm having a hard time finding the critical points of $f'(x)$, mostly because of factoring. The function is $$f(x) = \frac{\ln(x^2-1)}{x^2-x}$$ and it's derivative ...
2
votes
3answers
194 views

Derivative of integral in interval

Let $$F(x)=\int_{2}^{x^3}\frac{dt}{\ln t}$$ and $x$ is in $(2,3)$. Find $F'(x)$. Can somebody give me idea how to do this? Thank you
0
votes
0answers
24 views

Differentiation of Laplace Transform

It is known that The $s-$derivative rule states that $$ \mathcal{L} (t^{n} f) = (-1)^{n} F^{(n)} (s) $$ The proof for the laplace differentiation involves \begin{align*} F'(s) &= ...
0
votes
2answers
43 views

Finding the intervals where $f(x)=\frac{1}{|x-2|}-x$ is monotonous

Given $$f(x)=\frac{1}{|x-2|}-x.$$ I am interested in finding the intervals in $\mathbb{R}$ in which the function is monotonically increasing or decreasing. Usually I would take $f'(x)>0$ for the ...
2
votes
1answer
23 views

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 ...
0
votes
2answers
20 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)
1
vote
0answers
21 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 ...
1
vote
1answer
22 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 $y(x)=Cx^4$.
0
votes
0answers
17 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 ...
0
votes
1answer
29 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 ...
0
votes
0answers
25 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 ...
2
votes
2answers
33 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.
2
votes
2answers
42 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 ...
0
votes
0answers
18 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 $\bigtriangledown^2\psi$, ...
1
vote
2answers
32 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 $\{f_n\}$ ...
0
votes
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 $\{T_{n,f,0}\}$ ...
0
votes
1answer
21 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 ...
0
votes
2answers
22 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 ...
2
votes
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 ...
1
vote
1answer
29 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 ...
2
votes
0answers
24 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: ...
-1
votes
1answer
35 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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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) = ...
1
vote
4answers
89 views

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

How does $$\int \frac{1}{\cos^2(x)} dx= \tan(x)+ C$$ ?
4
votes
5answers
98 views

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

How do I integrate following function? $$ \int\frac{1}{e^{2x}+e^x} \,dx $$
2
votes
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 ...
2
votes
1answer
23 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 ...
2
votes
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 ...
1
vote
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 ...
3
votes
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 : ...
0
votes
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.
0
votes
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 ...
1
vote
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, ...
-4
votes
0answers
23 views

continuity of the piecewise functions [closed]

$1$. $g(x)=0$,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 $A+B =$ ...
1
vote
1answer
13 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 $f$: $\frac{dz}{df} = \frac{\partial f'(x)}{\partial x} \frac{dx}{df}$ ...
5
votes
1answer
81 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 ...
2
votes
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 ...
2
votes
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 ...