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

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3
votes
2answers
88 views

Find the number of real roots of the derivative of $f(x)=(x-1)(x-2)(x-3)(x-4)(x-5)$ [duplicate]

Find out the number of real roots of equation $f'(x) = 0$, where $$f(x)=(x-1)(x-2)(x-3)(x-4)(x-5)$$ How can I differentiate this function without expanding it to the polynomial form. Am I ...
0
votes
0answers
26 views

$C^r(K,F)$ as a Banach space for $K$ compact, $F$ Banach space

Let $E$ and $F$ be Banach spaces and $K\subset E$ be compact. I want to understand what the "common definition" (if there is one) of the banach space $C^r(K,F)$ of $r$ times continuously ...
0
votes
1answer
27 views

Mean squares aproximation constant finding

Measurments $(t_k, C_k), t=1..n ; t_k, C_k > 0$ suggest that $C(t) = \frac{1}{At + Bsin(t) + 2}$. Using mean square approximation find probable values for constants $A$ and $B$. Should I start from ...
3
votes
2answers
115 views

Derivative of $x\cdot|x|$ on $x=0$?

$$f(x) = x |x|$$ Wolfram Alpha says is: $$f'(x) = \frac{2x^2}{|x|}$$ and thus $f'(0)$ is indeterminate, while an HP48 says that: $$f'(x) = |x| + x \operatorname{sgn} x,$$ which would yield $f'(0) ...
0
votes
1answer
17 views

Determine the derivative in an interval.

Determine the first derivative of $f(x)=\ln(\sin^2(x^2))$ in an interval: $(0, \infty)$ I determined the derivative just in the usual way, but then I saw I have to do this for this open interval. ...
1
vote
2answers
72 views

Derivative of multivariate normal distribution wrt mean and covariance

I want to differentiate this wrt $\mu$ and $\Sigma$ : $${1\over \sqrt{(2\pi)^k |\Sigma |}} e^{-0.5 (x-\mu)^T \Sigma^{-1} (x-\mu)} $$ I'm following the matrix cookbook here and also this answer . ...
1
vote
2answers
15 views

describing a decay process with exponentials and differential equations

I have a process of degradation of some material that proceeds like this across time $t$: $C_t = C_{t-1} + RC_{t-1}$ where $C_t$ is the amount of material at time $t$ and $R$ is a (negative) rate of ...
-1
votes
5answers
87 views

$f(x)=\frac{\sin 3x}{\sin x}$ is decreasing function [closed]

Let $f(x)=\dfrac{\sin 3x}{\sin x}$. How to prove that $f(x)$ is decreasing on $(0,\frac{\pi}{3})$? I tried using the derivative $f'$ but didn't get $f'<0$.
1
vote
1answer
67 views

Derivative of a partial derivative

Assuming that $\dot{x}=f(x,t,C)$, where $C$ is a parameter and $f(\cdot)$ is continuous, how do we show that $$\frac{d}{dt} \bigg[\frac{d x}{d C} \bigg]=\frac{\partial f}{\partial C}+ ...
0
votes
1answer
81 views

Theorems on substitution in indefinite integrals

I can't understand some facts on the substitutions in indefinite integrals. On my textbook is reported only the "standard" case (integral of a composed function and the derivative of the inner ...
0
votes
2answers
36 views

How do you know what variable to replace for a time derivative?

I am a avid reader of Physics articles, and enjoy the mathematics greatly. I have taught myself most of the needed mathematics however one part of it always messes me up. When you take a time ...
2
votes
1answer
51 views

finding the derivative of ${{\sqrt x}(x^2 - {\sqrt x})}$

I am trying to find the derivative of this expression ${{\sqrt x}(x^2 - {\sqrt x})}$ I would first of simplify the expression to: ${x^{1\over2}(x^2 - x^{1\over2})}$ And then apply ${x^{1\over2}}$ ...
0
votes
0answers
22 views

How do I investigate the function f(x)=xksin(x1/x) for x≠0, f(0)=0?

It's given function $f: \mathbb R \to \mathbb R$, $$f(n) = \begin{cases} x^k\sin x^{1/x}, & \text{if $x\neq 0$} \\ 0, & \text{if $x=0$}. \end{cases}$$ I need to show that (a) for k=1,2 this ...
11
votes
3answers
682 views

Why is there only one term on the RHS of this chain rule with partial derivatives?

I know that if $u=u(s,t)$ and $s=s(x,y)$ and $t=t(x,y)$ then the chain rule is $$\begin{align}\color{blue}{\fbox{$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial s}\times \frac{\partial ...
0
votes
1answer
48 views

Differentiating equation of water volume in the cone

I just studied the chain rule from Video on youtube (https://www.youtube.com/watch?v=U5_pClLuJj0) it says: $$ \frac{dy}{dx} = \frac{dy}{du} * \frac{du}{dx} $$ Then in the problem there is a cone ...
1
vote
1answer
48 views

Derivative of incomplete Gamma function : $\Gamma(c,t/a)=\int_{0}^{t/a}x^{c-1}e^{-x}dx$

How we can calculate derivative of incomplete Gamma. Can anyone give me these derivative. $\frac{\partial\Gamma(c,t/a)}{\partial a}$ and $\frac{\partial\Gamma(c,t/a)}{\partial c}$ where ...
-1
votes
0answers
103 views

Tangent line parallel to a line

I know how to do this type of question, but when I do it algebraically I get an answer, but not when I put it in the calculator. $f'(x) = \arctan(x^3-x)$ is the tangent line and the line is $y=2x$. So ...
1
vote
1answer
30 views

Derivative of: $f(x) = x^2e^{-x^2} $ for $|x| \le1$ and $ f(x) = \frac{1}{e} $ for $|x| > 1$.

I would like to get the derivative (one-sided if needed) of this function: $$f(x) =\begin{cases} x^2e^{-x^2} & \text{for $|x|\le 1$} \\[6px] \dfrac{1}{e} & \text{for $|x| > 1$} \end{cases} ...
0
votes
1answer
37 views

A question regarding the notions of left-right differentiability

Let $f:[a,b] \rightarrow \mathbb{R}$ be a continuous function. We can then say it is continuous at the points $a,b$ with unequivocality; we don't concern ourselves with right or left continuity. This ...
1
vote
1answer
52 views

Derivative of $f(x) = (\sin{x})^{|\cos{x}|}$

I got this function: $$f(x) = (\sin{x})^{|\cos{x}|}$$ and I want to determine for all points in the domain of the function whether (one-sided) derivative exists, and if it does, then what is it. Can ...
1
vote
1answer
71 views

Why are derivatives and antiderivatives defined only on intervals?

I've a question on derivatives and antiderivatives. From the definition I know$^{1}$ antiderivatives are defined only on intervals. If I got it, that's mainly because in a interval is possible to use ...
-1
votes
2answers
30 views

Finding the mixed second partial derivative of $\cos (xy^2)$

I am finding the second partials of $f(x,y)=\cos (xy^2)$. I have found $f_{xx}$ and $f_{yy}$ but I am stuck on how to find $f_{xy}$, could someone explain this please?
1
vote
0answers
48 views

differentiable map on a connected open set

Let $f$ be a differentiable mapping on a connected open set $E$ in $\mathbb{R}^n$ into $\mathbb{R}^m$ and if $f'(x)=0$ for every $x\in E$, prove that $f$ is constant in $E$. For simplicity, take ...
0
votes
1answer
52 views

Find the gradient of the curve $y² - 3xy = x²$

Given a curve $y² - 3xy = x²$, express $y$ in terms of $x$. Find the gradient of the curve when $x=4$. My attempt, $y^2-3xy-x^2=0$ $y=\frac{3x \pm \sqrt{13x^2}}{2}$ $\frac{dy}{dx}=\frac{1}{2}(3 ...
1
vote
1answer
23 views

Steps of finding an absolute extremum on an open interval

$$f(x)=\cot x-\sqrt 2 \csc x,\quad I=(0,\pi)$$ Show that the function $f$ has an absolute extremum on the given interval $I$ and find that value. I've found the local maximum point from the first ...
4
votes
2answers
245 views

Derivative cycles of length 8

Let's say I want to find derivative cycles, that is, a group of functions $\langle f_0, f_1, \ldots, f_{n-1} \rangle$ where ${f_0}^{(p)}(x)={f_{p\,\%\,n}}(x)$ where $\%$ is the modulo operator. For ...
1
vote
0answers
34 views

How to establish that $\frac{d^{n+1}}{dt^{n+1}} \prod_{i=0}^k (t-x_i)^{n_i+1}=(n+1)!$

How to establish the following identity? $$\frac{d^{n+1}}{dt^{n+1}} \prod_{i=0}^k (t-x_i)^{n_i+1}=(n+1)!$$ Where $n= k + n_0 + n_1 + ... + n_k$ Thanks.
1
vote
1answer
46 views

Can two functions have identical second derivatives and the same lateral limits?

I'm looking for two not identical functions $f(x)$ and $g(x)$ on $(a,b)$ (to simplify, assume that $a = 0$ and $b = 1$) where: 1) The second derivatives are the same on $(a,b)$, that is, $f''(x) = ...
3
votes
1answer
64 views

How is this integral equal to this natural logarithm?

I am trying to understand following problem: $$\int {\sin x \over \cos x}dx = -\int {d \cos x \over \cos x} = - \ln \lvert \cos x\rvert + k $$ I don't really get the final step, are they equal ...
0
votes
0answers
21 views

Cylindrical coordinate derivative of a vector field.

Considering the following identity transformation in cylindrical coordinate: $$\mathbf{v}(R,\theta,z)=R\;\mathbf{e}_{R}+\theta\;\mathbf{e}_{\theta}+z\;\mathbf{e}_{z} $$ Taking its derivative ...
0
votes
0answers
48 views

Where f(x) is increasing and where it is decreasing?

I have this function - $f(x)=(-1)^{3x}\cdot x + (-1)^{2x}\cdot x^{2}$ I have to find where its increasing and decreasing. Actually i dont know how to start, i trid to derivate but without success. ...
0
votes
2answers
50 views

Maximum and Minimum in a range?

I have a problem with this exercise. This is the text. The function $$f(x) = x-\ln (1 + 2x ^ 2)$$ in the interval $[1,3]$ has: two points of maximum and a minimum point relative or absolute a ...
0
votes
1answer
25 views

What is $\eta$ in this definition?

In this definition on differentials in my handbook there is a part that states the following: Since $$\lim\limits_{\Delta x \to 0 } {\Delta f\over \Delta x} = f'(x) $$ We can say that $${\Delta f ...
-1
votes
1answer
56 views

Derivative of the function $(x)!$. [duplicate]

I had been learning calculus. So what I was thinking about is what us differentiation if $(x)!$. I know. 'n 'th derivative of $x^n$ is $x!$ but it isn''t helping me to solve this problem.
2
votes
1answer
81 views

Derivatives of two functions

I need the limits ( for $x \rightarrow \pm \infty $) and the first three derivatives of the following two functions: 1) $f(x)=\dfrac{e^{-x}}{8-5x}$ 2) $f(x)=\dfrac{x-1}{\sqrt{x-2}}-2$ 1) ...
1
vote
1answer
29 views

construct a function with dense critical value

I am learning derivative of a function from $\mathbb R$ to $\mathbb R$. If $f$ has zero derivative at $x$, we call $x$ is a critical point of $f$, and $f(x)$ is critical value. If critical points ...
0
votes
2answers
41 views

A twice differentiable function such that $f''$ is continuous [closed]

Let $f$ be a twice differentiable function such that $f''$ is continuous and $f''(x)\neq0$ for all $x\in \Bbb R$. Let $f(0)=1$, $f(1)=0$ and $f'(0)=0$. Prove that $f'(1)\leq -1$.
2
votes
3answers
80 views

How do I show $f(x+2)-f(x)>2 \forall x$?

For the function $f(x)=x\cos{\frac{1}{x}}$, $x\geq1$, How do I show that $f(x+2)-f(x)>2 \forall x$?
1
vote
1answer
40 views

What to choose $g(x)$ as so that $f′′(x)−2f′(x)+f(x)≥e^x$?

Let $f:[0,1]→R$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f(0)=f(1)=0$ and satisfies $f′′(x)−2f′(x)+f(x)≥e^x$, $x∈[0,1]$.Prove that ...
2
votes
4answers
222 views

derivative of many roots

if $$ y=\frac{(1+2x)^{1/2}.(1+4x)^{1/4}.(1+6x)^{1/6} ... (1+100x)^{1/100}}{(1+3x)^{1/3}.(1+5x)^{1/5}.(1+7x)^{1/7} ... (1+101x)^{1/101}}$$ then find y' at x=0 Already tried to find a general term ...
1
vote
0answers
42 views

Deriving the wave equation in 3 dimensions and the history of it

I'm trying to find how the wave equation was derived in 3 dimensions. Surprisingly, there isn't much information available on this apart from wikipedia of all places ...
1
vote
1answer
51 views

If the second derivative of a function is zero, why is the second derivative test inconclusive?

2nd derivative test gives three possibilities: 1) greater than zero (strict local min) 2) less than zero (strict local max) 3) equal to zero - no information It is this third case that I do not ...
2
votes
3answers
69 views

I would like to get the derivative of this function: $ f(x)=(x-a)^2(x-b)^2 $ and $f(x) = \frac{1}{e}$

I want to get the derivative of this function: $$ f(x)=(x-a)^2(x-b)^2 $$ for $x ∈< a, b >$, $$f(x) = \frac{1}{e}$$ for all other $x$. Now I know the result is: $$ f'(x) = 2(x − a)(x − b)(2x − ...
9
votes
1answer
70 views

An Odd Mean Value Theorem Problem

If $f: [x_1,x_2] \to \mathbb{R}$ is differentiable, show for some $c \in (x_1,x_2)$ that $$ \frac{1}{x_1-x_2} \left| \begin{matrix} x_1 & x_2 \\ f(x_1) & f(x_2) \end{matrix} ...
2
votes
1answer
48 views

If $\sin y=a\sin(x+y)$ prove $\frac{\rm d y}{\rm d x}=\frac{\sin a}{1- 2x\cos a +x^2}$

If $\sin y=a\sin(x+y)$ prove $\frac{\rm d y}{\rm d x}=\frac{\sin a}{1- 2x\cos a +x^2}$ I am not finding any proper way even to express $y$ only in terms of $x$ too which could reduce bit ...
0
votes
1answer
34 views

Holder condition and differentability

Prove that if a function is $C^2$ on a closed interval, then it satisfies holder condition of order 2. Thanks
0
votes
1answer
83 views

What should I know about half vectorization and Kronecker product to do this matrix differentation?

I have a scalar function as follows: \begin{equation*} \ell(\beta, \Sigma, \mu, \Lambda) = \sum_{i=1}^{m} \left[\boldsymbol{y}_{i}^{T} \left(X_{i}\beta + Z_{1} \mu_{i} \right) - ...
2
votes
2answers
48 views

How to verify this identity?

From Weinstock, "Calculus of Variations", p.24: We have the readily verifiable identity \begin{align}\frac{d}{dx}\left(y'\frac{\partial f}{\partial y'}-f\right) = ...
1
vote
4answers
56 views

To check continuity and differentiability

Consider the function I am having problem with checking continuity because of y. Regarding differentiability i can apply Leibniz rule to get explicit formula.But then modulus part troubles me. Can ...
2
votes
1answer
64 views

Where is the function series $f(x)=\sum\limits_{n=0}^\infty\frac{e^{-nx}}{n^2+1}$ differentiable?

I was asked to analyze the convergence, continuity and differentiability intervals of the next function series: $f(x)=\sum\limits_{n=0}^\infty\frac{e^{-nx}}{n^2+1}$ I already know that this ...