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

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3
votes
4answers
110 views

Derivability of a piecewise function

Let's say I have a continuous piecewise function of a single variable, so that $y = f(x)$ if $x < c$ and $y = g(x)$ if $x>=c$. Is it right to say that the derivative of the function at $x=c$ ...
2
votes
3answers
61 views

Finding the second derivative of an infinite series

I'm asked to find the 2nd derivative of $$f(x)=-2x+\frac{2x^3}{3!}-\frac{2x^5}{5!}+\cdots+(-1)^{n+1}\frac{2x^{n+1}}{(2n+1)!}+\cdots=\sum \limits_{n=0}^{\infty} \frac{(-1)^{n+1}2x^{2n+1} }{(2n+1)!}$$ ...
1
vote
1answer
25 views

Difference quotients are increasing for $f$ convex

Problem 11A.9(c) in Spivak's Calculus (4th edition) asks the following (I'm paraphrasing): Suppose $f$ is convex. Show that $f'(a)$ exists iff $f_+'(x)$ is continuous at $a$. ($f_+'(x)$ is the ...
9
votes
2answers
189 views

$f'$ exists, but $\lim \frac{f(x)-f(y)}{x-y}$ does not exist

Suppose $f$ is differentiable at $a$, i.e. $\lim_{x\to a}\frac{f(x)-f(a)} {x-a}$ exists. I wondered whether it was necessarily true that $$\lim_{\substack{x,y\to a\\x\neq y}}\frac{f(x)-f(y)}{x-y} ...
1
vote
1answer
16 views

Question concerning an example of higher derivatives and binomes

I got this exercise form OCW 18.03SC - problem 1G-5b: What is the solution of the following derivative?: $$\dfrac{d^{p+q}}{dx^{p+q}}x^p(1+x)^q$$ I used Leibniz' formula and the only non-zero term ...
0
votes
1answer
31 views

ODE with Laplace transform: the jump of $\dot y$

I solved this eq. using the Laplace Transform: $\ddot y+4\dot y+13 y=\delta(t-2\pi)-\delta(t-7\pi)$ The sol. is: $y(t)=\frac{1}{3} e^{2 t} (-e^{14 \pi} \theta(t-7\pi) sin(3 t)+e^{4 \pi} \theta(t-2 ...
1
vote
2answers
56 views

How to prove that a diffrensiation of a formula equals to another formula.

QUESTION 1) if $y =\dfrac{ \sin x-x\cos x}{x\sin x+\cos x}$ show that $\dfrac{dy}{dx}= \dfrac{x^2}{(x\sin x+\cos x)^2}$ QUESTION 2) if $y = \dfrac{\tan x+1}{\tan x-1}$ show that $\dfrac{dy}{dx}= ...
0
votes
0answers
37 views

Use of the anti-derivative

Given a velocity function $dx/dt$, I am asked to find when a certain particles changes direction. This would then be when $dx/dt=0$. Let's say that $dx/dt$ has roots at $t= -1$ and $ t = 3$. I am ...
0
votes
4answers
67 views

Derivative of $\frac{1}{(x+1)^{k-1}}$

How is it that the derivative of $\frac{1}{(x+1)^{k-1}}$ is $-\frac{k-1}{x^k}$ where $k$ is a parameter.
0
votes
2answers
25 views

Arithmetic simplification

I am asked to find $\frac{d^2y}{dx^2}$, and prove that $\frac{d^2x^2+y^2=a^2}{dx^2}$=$-\frac{a^2}{y^3}$, This is how I have proceeded: $2 y \frac{dy}{dx}+2 x=2 a \frac{da}{dx}$ => ...
0
votes
1answer
22 views

Show that $f$ is a contraction if and only if there exists $r \in (0, 1)$ such that $|f'(x)| \leq r$, for all $x \in [a, b]$.

Let $f : [a, b] \rightarrow [a, b]$ be differentiable. Show that $f$ is a contraction if and only if there exists $r \in (0, 1)$ such that $|f'(x)| \leq r$, for all $x \in [a, b]$. I managed ...
0
votes
0answers
31 views

Differentiating integral function

I need to take the partial derivative of a function for the purpose of finding a zero with Newton Raphson. The function to be zero-ed is $R=\frac{\int_{\phi \ge 0} Y(\phi)d'(\phi)d\phi }{\int_{\phi ...
0
votes
1answer
44 views

derivative B-spline with own knot set

Define the spline function of degree $q$ on the interval $[\xi_0,\xi_K]$ $$f(t)=\sum_{j=1}^{K+q}b_j B_j(t)$$ where $B_j$ are degree $q$ B-spline basis functions determined by the knots ...
1
vote
1answer
28 views

Find the largest $n\in \Bbb{N}$ answering the following terms

Let $$f(x) = -\frac{1}{12}x^4 + o(x^5)$$ Also, Let $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C & \mbox{if } x=0 \end{cases}$$ I need to find the largest $n\in\Bbb{N}$ ...
0
votes
0answers
8 views

Given a matrix $X$ what is the gradient of $log|X|$

Given a matrix $X$ Is it possible to obtain the gradient of $log|X|$? That is $\frac{\partial log|X|}{\partial X}$
1
vote
1answer
25 views

Related Rates Problem atan vs cos

This is another Khan Related Rates problem that I haven't been able to understand where I'm going wrong. GIVEN: A 13 meter ladder is leaning against a wall when its base begins to slide away. By ...
1
vote
0answers
58 views

what does the second derivative of a linear function mean?

So if I have a function f(x) = 7x-2 the first derivative is 7 which I'm inclined to think that the second derivative ...
2
votes
1answer
67 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
1
vote
0answers
43 views

Differentiation with respect to the index of the summation notion?

$$\sum_{t=1}^k \binom{N-1}{t-1} \int[1-F(s)]^{N-1}[F(s)]^{t-1}g(s)\,ds$$ $k\in\mathbb Z ^+$ If I want to find out the effects of changing $k$ (comparative statics), what can I do? Differentiation ...
5
votes
1answer
79 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
0
votes
2answers
41 views

Is the quotient rule needed in this case?

I need the partial derivative w.r.t. $r_{20}$, that in this function is only in the denominator, do I need to use the quotient rule? $\dfrac{\partial f}{\partial r_{20}} = \dfrac{r_{00}*u/w + ...
1
vote
0answers
50 views

Given a function $f$, find the largest $n$ such that $f(x)/x^n$ can be defined at $x=0$ to become differentiable there

Let $f(x) = \ln\left(\frac{x^2}{2}+1\right)+\cos x -1$. Find the largest $n\in\Bbb{N}$ such that there is $C\in\Bbb{R}$ such that: $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C ...
1
vote
2answers
26 views

Confirm right model

Have a Khan problem I've been working under the "Related Rates" category. GIVEN: A 2 meter tall boy "h" is rollerskating away from a 5 meter lantern at constant dx/dt = 2 meters per second. How ...
8
votes
4answers
128 views

How to find the derivative of a function defined by an integral? Namely, $f(y)=\int_0^{y^2} e^{-x^2y^2}dx$

Find at each point of its domain the derivative of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ $$f(y)=\int_0^{y^2} e^{-x^2y^2}dx$$ $$$$ Is the domain of the function $\mathbb{R}$ because of ...
3
votes
2answers
46 views

Prove that $f(a) \leq f(x) \leq f(b) $

If the following data are given, prove that $f(a) \leq f(x) \leq f(b) $ f is differentiable on [a,b] and f'(x) $ \geq 0 \forall x \in (a,b) $ Is the following argument correct? $f'(x) \geq 0 ...
1
vote
2answers
44 views

Finding derivative

$\lim\limits_{x\to\ 2}\frac{f(x)-f(2)}{x^2-4}=4$ where $f(x)$ is defined on $\Bbb R$ and $g(X)=\frac{f(x)e^x}{1-x}$. What is $g'(2)$?
0
votes
1answer
28 views

$f$ differentiable on $[a,b]$, but not Lipschitz

Question 11-37(d) of Spivak's Calculus, 4th ed., asks If $f$ is differentiable on $[a,b]$, is $f$ Lipschitz of order $1$ on $[a,b]$? The phrase "differentiable on $[a,b]$" is a little ...
3
votes
3answers
86 views

What is $\frac{d^n}{dx^n} \frac{e^{\lambda x}}{x}$?

I was wondering whether there is an explicit way to say what the derivative of $\dfrac{d^n}{dx^n} \dfrac{e^{\lambda x}}{x}$ for $n \in \mathbb{N}_0$is, where we assume that $\lambda \neq 0$.
2
votes
1answer
32 views

Strict local extremum without $f'$ “changing signs”

Let $f:\mathbb{R}\to \mathbb{R}$. Is it possible that $f$ has the following properties: $f$ is differentiable in a neighborhood of $a\in \mathbb{R}$ $a$ is a strict local minimum There is no ...
0
votes
0answers
10 views

Nomenclature for the function appearing in Carathéodory's criteria of differentiability

In my previous question Concerning Carathéodory's criteria of differentiability and a proof that differentiable implies continuous I stated the criteria as follows: There exists a function $\phi$ ...
1
vote
0answers
14 views

A question on the procedure of finding the matrix of a linear transformation of a polynomial and a combination of its derivatives

I'm trying to self-study Linear Algebra from Linear Algebra Done Wrong, but the book doesn't have solution manual so my question might be extremely easy, apologize in advance: The question is for the ...
1
vote
2answers
41 views

Using complex derivative to shows that a function is constant

If we know $\frac{\partial f}{\partial z} \equiv f'(z)=0$ where $f(z)=u(x,y)+iv(x,y)$ why do we need to check the Cauchy Riemann equations are all equal to zero, before concluding that $f$ is ...
0
votes
1answer
39 views

Specify the values of $p$ and $p'$ for a polynomial

Problem 10-26 from Spivak's Calculus, 4th edition: Let $a_1, \dotsc, a_n$ and $b_1, \dotsc, b_n$ be given numbers. If $x_1, \dotsc, x_n$ are distinct numbers, prove that there is a polynomial ...
1
vote
1answer
49 views

Limit of the derivative of a function

Under what conditions is true: If $$\lim_{x\rightarrow\pm\infty}\Phi(x,y)\rightarrow 0$$ then $$\lim_{x\rightarrow\pm\infty}\frac{\partial}{\partial x}\Phi(x,y)\rightarrow 0$$ Some time ago I ...
3
votes
1answer
21 views

Derivative of Binomial Coefficient wrt k

I've got $\binom{2N}{N-x}$ and I'd like to take the derivative with respect to x. I know that I can take the derivative of $\binom{n}{k}$ w.r.t. n using logarithmic differentiation, but that's not ...
0
votes
2answers
29 views

Differential of a shifted function

If I'm given the differential equation: $$\frac{d(12-24f(t))}{dt} = 5$$ How do I rearrange this so that it looks like a normal first order linear differential equation? e.g, so it looks something ...
0
votes
1answer
21 views

Concavity of a parametric curve: a formula for $d^2y/dx^2$

I am going through old math texts and this problem is suddenly giving me problems. We have two functions, $y(t)=t^3-3t$ and $x(t)=t^2$, and the the question asks for concavity of the curve. It ...
3
votes
4answers
85 views

Differential equation which has following solution $y=\frac{1}{1+\exp(ax)}$

Is there any linear differential equation which has following solution $$y=\frac{1}{1+\exp(ax)}$$ $a$ is constant. something like: $$ y'' + by' +cy + \alpha = 0$$ where $b$, $\alpha$ and $c$ are ...
1
vote
0answers
27 views

Zeros of the derivatives of a finite Blaschke product.

Let $B$ be an $n$ degree finite Blaschke product. By considering the level curves of $B$, one can show that $B'$ has $n-1$ critical points in the disk (counting multiplicity). Is anything known ...
4
votes
1answer
55 views

Where do I make mistake on this derivative containing e^x^2

My brother is preparing for the university and asked me the following multiple choice question. $$\frac{d}{dx}(x^3 * e^{x^2})$$ a) $e^{x^2}*x^2*(1+2x)$ b) $e^{x^2}*x^2*(3+2x)$ c) ...
1
vote
2answers
64 views

Summation of infinite series

If we know the series sum given below converges to a value $C$(constant) $$\sum_{n=0}^{\infty}a_n =C \tag 2$$ Can we generate following in terms of C. values of $a_n$ will tend to zero as n goes to ...
0
votes
1answer
36 views

Partial derivatives of $xy^2/(x^2+y^2)$ at the origin

I noticed that this is a big black hole in my understanding of partial derivatives at the point. I don't know how to count it: $$ f(x,y) = \frac {xy^2}{x^2+y^2} $$ $$ \frac {df}{dx}(0,0)=\lim_{t\to ...
2
votes
1answer
35 views

Solve 2 connected ODEs describing a domain

This problem confused me for a long time. I have 2 ODEs which describe part of our domain. They are connected at middle: $$ \frac{d^2}{dx^2} u = -a, x<x_0 $$ $$ \frac{d^2}{dx^2} u - \frac{u}{b^2}= ...
1
vote
4answers
67 views

Differential equation with the solution of $(1+ax/2)\exp(-ax)$

Is there any linear differential equation which has following solution $$y=(1+ax/2)\exp(-ax)$$ $a$ is constant.
0
votes
1answer
38 views

Two Strictly Convex Functions with Contact of Order 1

Let $f,g: \mathbb{R}\rightarrow \mathbb{R}$ be two strictly convex functions, where $f$ is differentiable, $g$ is smooth, and $f\geq g$. Suppose that for some $x_0\in \mathbb{R}$: ...
1
vote
1answer
17 views

Differentiable Strictly Convex Function on Interval

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable, strictly convex function. Let $I\subset \mathbb{R}$ be a closed, bounded interval such that $f'(x) \neq 0$ on $I$. Is $f$ strongly ...
0
votes
1answer
32 views

How do the existence and $L^p$ integrability of weak derivative affect the smoothness of Sobolev functions?

In the definition of Sobolev spaces, let us say the space $W^{1,p}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^n$. What contributes more to the smoothness of a function $f$: 1-The fact ...
0
votes
0answers
37 views

question on differentiable and continious function

How should the function $f(x)=x\operatorname{sgn} x$ be defined at $x=0$ so that it is continuous there? Is it then also differentiable? How should the function $g(x)=x^2 \operatorname{sgn} x$ be ...
2
votes
1answer
184 views

Finding the derivative using the definition?

Calculate the derivate of the given function directly from the definition of derivative, and express the result using differentials $$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ when $f(x)= ...
1
vote
0answers
31 views

Weird derivative computation

I found the following formulas in a control theory textbook : $$s(x,t)=\left(\frac{d}{dt}+\lambda\right)^{(n-1)}\varepsilon $$ where $\varepsilon(t)=T\left(\frac{e(t)}{p(t)}\right)$ and ...