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

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Expressing a function and differentiating.

Show that $$ 1+\cot^2x=\text{cosec}^2x $$ where $0<x<\frac{π}{2}$. By expressing $ y=\cot^{-1}x$ as $x=\cot y,$ obtain $dy/dx$ in term of $x.$
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1answer
66 views

signum function and derivation

I have these 2 functions: $f_1(x) = |x|^{4}$ and $f_2(x) = |x^{4}|$ Im supposed to find out whether or not these functions derivates are defined in $x = 0$. I start with $f_1'(x)$ .. If I say $u = ...
0
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2answers
76 views

Hard optimization problem - Maximum area of a tank with no lid

how would you solve this particular optimization problem (which seems harmless): A tank with a square base is more taller than it is wider. To build it, the sum of the perimeter of the base with ...
2
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0answers
17 views

Existence of a measurable $\theta$ such that $\frac{f(y)-f(x)}{y-x}=f'(\theta_{x,y})$?

Suppose $f:]a,b[\to\Bbb R$ is differentiable (possibly $C^1$, $C^2$ or a lot smoother, say $C^\infty$), and define $T=\lbrace(x,y)\mid a<x<y<b \rbrace$. Does there exist a somewhat regular ...
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0answers
20 views

Composition of non-differentiable functions to produce a differentiable one

What are examples of non-differentiable functions which when composed form a differentiable function? It would even better if they are $C^\infty$ I am hoping this will turn into a big list, if it ...
2
votes
1answer
20 views

Optimization Problem - Rectangle inscribed in a circumference and smallest angular coefficient of a tangent

I've been using a textbook to solve some Optimization problems and these two got my attention. The first one has no answer, just want to double check, and the second one is possibly wrong: What is ...
7
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3answers
97 views

Suppose that $f: \mathbb{R^+}\to \mathbb{R}$ satisfies $\lim_{x\to \infty} (f+f')(x)=0$. Show that $\lim_{x\to \infty} f(x)=0$. [duplicate]

Suppose that $f: \mathbb{R^+}\to \mathbb{R}$ satisfies $\lim_{x\to \infty} (f+f')(x)=0$. Show that $\lim_{x\to \infty} f(x)=0$. This is one solution I found to this problem. Solution: If $x=a$ is in ...
5
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2answers
80 views

How does this 'chain rule manipulation' work?

PS- this is a "physics" question, but mathematical in nature... if I should ask on physics SE instead, please let me know Looking back through my physics books, I found a derivation of Kinetic ...
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4answers
141 views

Derivative of the Square Root of X [closed]

Given $y = \sqrt x$ and nothing more, using the formula of a limit $$f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$ (that is, f prime of x equals the limit of h approaching zero with the equation ((f of ...
2
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4answers
53 views

Prove That $f(x)>f(y)+f'(y)(x-y)$ if $f''(x)>0$ For All $x$

Here's my question: Let $f$ be a function in a interval $I$, where $f''(x)>0$ for all $x\in I$. Prove that for every $x,y \in I$ $$f(x)>f(y)+f'(y)(x-y)$$ I'm sorry to say that but, ...
0
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0answers
41 views

Is $ f(z) = z^{3/2} $ holomorphic at $ z= 0$

Consider $f: \mathbb{C} \rightarrow \mathbb{C} $, where $f(z) = z^{3/2} $. Now evaluating derivative at $0$ using $$\lim_{h\to 0} (f(0+h)- f(0))/h$$, we get $f'(0)=0$. So $f$ is infinitely ...
3
votes
3answers
81 views

Inequality $|f(x)|' \le c|f'(x)| $?

Let $f:\mathbb{R} \ni x\mapsto f(x) \in \mathbb{C}$. I want to know that the following is true: $$\left| \frac{d|f(x)|}{dx}\right| \le \frac{|f_r f_r'| +|f_i f_i'|}{|f|} \le \frac{(|f_r| + ...
3
votes
2answers
65 views

Why is the cancellation of partial derivatives like fractions justified in this example?

Suppose we have two functions $Q=Q(q,p)$ and $p=p(q,Q)$ (the context is not important here, but if you're wondering $(p,q)$ arise as coordinates in a Hamiltonian system, and $(P,Q)$ are alternative ...
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2answers
71 views

Why does f'(0) not exist for this piecewise defined function? [closed]

Q: Does $f'(0)$ exist? $ f(x) = \begin{cases} 0 & x= 0 \\ x\sin(\frac{9}{x}) & x\ne 0 \end{cases} $ Why or why not? How about if the $x$ in front of sine changes to $x^3$ or ...
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1answer
29 views

How can I differentiate the complex-valued function $|f|$?

Let $f:\mathbb{R} \ni x\mapsto f(x) \in \mathbb{C}$. How can I differentiate the function $|f(x)|$ by $x$, namely, $$ \frac{d}{dx} |f(x)| = \frac{d}{dx} \sqrt{\operatorname{Re} (f(x))^2 + ...
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0answers
34 views

Why is the set of points that have injective derivative open?

Suppose $A$ and $B$ are finite dimensional vector spaces, $U\subseteq{A}$ and $f:U\rightarrow{B}$ is $C^{\infty}$. I need help proving that the set $\{a\in U:(Df)_a $is injective} is open. I think ...
0
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1answer
38 views

Calculate the Extrema of $ G(ω) = \frac{|15(1-ω^2)|}{\sqrt{25(3-ω^2)^2 +ω^2(11-ω^2)^2}} $

I want to calculate this without the use of software. I also want to find the points of Inflection but this might be too hard by hand, so let's forget that for now. So i know what is the procedure in ...
3
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2answers
37 views

When to use which derivative expanded function?

In our class we have learned the following two formulas for calculating derivative. $$1.\lim_{x\to a} \frac{f(x) - f(a)}{x-a}$$ $$2.\lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$$ When manually computing ...
2
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1answer
31 views

Derivative of a Product of a Variable Number of Terms

Take a function that is the product of degree $k$, such as $$f_k (x)=\prod_{n=2}^{k+1} g\left(\frac xn \right), k\ge 1.$$ What methods would on use to find $f^\prime_k(x)$ with respect to $x$ in a ...
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1answer
15 views

Compact set and function with no extremum

I have to find compact set $D\subset\mathbb{R}^2$ with non-empty interior and differentiable function $f:D\rightarrow\mathbb{R}$ which don't have local extremum in every interior point of $D$. Any ...
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1answer
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Differentiable function with negative derivative but non-decreasing

I have to give an example of differentiable function $f:[-1;1]\setminus\{0\}\rightarrow\mathbb{R}$ with negative derivative but this at the same time this function has to be non-decreasing. Is it ...
2
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1answer
43 views

Is continuity of first partials required for analyticity?

Let's cast the complex function $f(z) = u(z) + iv(z), z = x+iy$, as the multivariable function $F(x,y) = U(x,y) + iV(x,y) ; x,y \in R$. Thus, $$dF = F_x\,dx + F_y\,dy = U_x\,dx + iV_x\,dx + U_y dy + ...
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0answers
27 views

Deriving a function of two variables with some interaction

I hava function $F(X,Y)$. $X(t)$ and $Y(t)$ : both are functions of a third variable $t$. In addition $X(Y(t), t)$: $X$ is a function of $Y$, which is a function of $t$, and $t$. The third point is ...
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0answers
23 views

Problem finding $Df(x_{1},x_{2},\dotsc, x_{n})(h_{1},h_{2}, \dotsc, h_{n}) $

Let $E_{i}$ be banach spaces for $i=1,2,\dotsc, n$ . Let $f$ be a continuous multilinear function from $E_{1}\times E_{2}\times \dotsb \times E_{n}$ to another banach space $F$: ...
0
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1answer
42 views

Finding the $n$-th derivative and proving a given condition

The question that is getting me confused is: if $y = (\sin^{-1} x)^2$, prove that $(1-x^2) (y_{n+2}) - (2n+1)x y_{n+1} + n^2 y_{n} = 0$. Could you please explain the problem solving tricks used for ...
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1answer
301 views

How to find the nth derivative for $\cos^3(x)$?

Could you please explain it so that I can find nth derivatives for other terms such as $\sin^3(x)$, $x^2e^{5x}$. Or also $x^2\sin(5x)$? Thanks in advance. I understand Leibniz's theorem but I am not ...
3
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1answer
71 views

$f(x) \ge f(x + \sin x)$, nonconstant functions, infinite number of solutions to $f'(x) = 0$.

Let $\mathcal{F}$ be the set of all the differentiable functions $f: \mathbb{R} \to \mathbb{R}$, which have the property $f(x) \ge f(x + \sin x)$, for all $x \in \mathbb{R}$. Prove that ...
0
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1answer
15 views

Function non-differentiable with partial derivatives

I have to give an example of function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ non-differentiable in the point $a\in\mathbb{R}^2$ but with both partial derivatives equal to $1$. Any ideas?
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1answer
27 views

How do I solve this differential equation (involving Fourier series)

$$x(t) = 4y''(t)+4y'(t)+17y(t)$$ where, $$x(t) = \sum^\infty_{k=1}\left((-1)^{k}\left(-\frac{2\cdot}{k}\right)\sin kt\right) = \sum^\infty_{k=-\infty}\frac{j(-1)^k}{k}e^{jkt},\quad k\not=0$$ and, ...
2
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3answers
65 views

Calculating the $n^\text{th}$ derivative

How do we calculate the $n^{\text{th}}$ derivative for $$ \frac{x^3}{(x-a)(x-b)(x-c)}? $$ How can I obtain the partial fraction for the given term?
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1answer
33 views

continuity of derivative of continuous functions differentiable everywhere

For a continuous function if derivative exists everywhere is it necessary that the derivative itself is continuous. I am unable to think of any counterexample.
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0answers
66 views

Finding “Largest Possible Approximation” of rate of change from a table?

In this Calculus question, they give me a table: x = 0.9 , 1.0 , 1.2 f(x) = 1.226 , 1 , 0.754 I have to find the "largest possible approximation" of the rate of change of $f$ at $x=1$. I actually ...
1
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1answer
77 views

Show that $f$ is Gateaux differentiable

Define a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ as follows $$f(x,y)= \begin{cases} \frac{2y \exp(-x^{-2})}{y^2+ \exp(-2x^{-2})} & x \neq 0 \\ 0 & \text{otherwise} ...
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1answer
35 views

Multivariable chain rule confusion (Jacobian matrix).

I'm given the functions: $$g: \Bbb R^3\to \Bbb R^3:g=(u(x,y,z),v(x,y,z),w(x,y,z))\quad \text{and } f:\Bbb R^3\to \Bbb R.$$ And I'm asked to find the jacobian matrix of $h=f \circ g$, would that be: ...
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1answer
40 views

How are surface area or a revolution and arc length related in Calculus?

I understand that integrals is the area under the function f(x). and the integral f'(x) if the original f(x). I understand how to find the arc length on a given interval. I am looking to understand ...
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3answers
73 views

Which integration of tan x is correct?

Integration of tan x as given by http://www.wolframalpha.com is this: And my teacher suggested this one: Are these both right answers?
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1answer
33 views

To differentiate $f:GL_n(\mathbb R) \to GL_n(\mathbb R)$ given by $f(A)=A^{-1}$

I want to show that $f:GL_n(\mathbb R) \to GL_n(\mathbb R)$ given by $f(A)=A^{-1}$ is differentiable and find its derivative map ; now I'm convinced that the derivative map $Df_A :M_n(\mathbb R) \to ...
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1answer
57 views

Directional Derivative of a Function at a Point

How do I calculate the maximum value of the directional derivative of this function at the point $(0,0)$? $$f(x,y)=\sqrt[3]{x^2y}$$ I did some calculations, but my answer came out to be $(0,0)$ and ...
2
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3answers
77 views

Is it true that $e^{\sin(3.14)}e^{3.14} \le e^{\sin(3.15)}e^{3.15}$?

I have to determine whether is it true that $$e^{\sin(3.14)}e^{3.14} \le e^{\sin(3.15)}e^{3.15}$$ and whether it is a equality. I even don't know how to begin with it...
2
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1answer
28 views

To show that the function $f:M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A)=AA^t$ is differentiable and evaluate its derivative

How to show that the function $f:M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A)=AA^t$ is differentiable and how to find the total differential at a point $X$ i.e. how to find $D f_A(X)$ ?
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1answer
23 views

To show that the map $f:M_n(\mathbb R) \times M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A,B)=AB$ is differentiable and evaluate the derivative

How to show that the function $f:M_n(\mathbb R) \times M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A,B)=AB$ is differentiable and how to find the total differential at a point $(X,Y)$ i.e. how to ...
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0answers
49 views

Mean Value Theorem Problem - finding $\xi$ given $f$ and $[a,b]$

For the function $f(x) = x^{6}+x^{4}-1$ and the interval $[0,1]$, I need to find the number $\xi$ that occurs in the Mean Value Theorem: $\displaystyle \frac{f(b)-f(a)}{(b-a)} = f^{\prime}(\xi)$. ...
3
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1answer
50 views

Is there a rectangle with a maximum area which has two corners at the x axis, one corner at $y_1=e^x$ and one at $y_2=2e^{-x}$ .

The inverses of $y_1$ and $y_2$ are : $x_1=ln y$ and $x_2=-ln\frac{y}{2}$ we need them to to calculate the side $a$ of a rectangle. The area of a rectangle is defined as its one side multiplied ...
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3answers
70 views

Some clarification needed on the Relation between Total Derivative and Directional Derivative

I will consider here functions of several variables only. If both directional derivative $D_{v}f(x)$ at $x$ along $v$ and total derivative $D f(x)$ at $x$ exist then $$D_{v}f(x)=Df(x)(v).$$ ...
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2answers
61 views

Is this function differentiable at $x = 1$?

this present message was deleted
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1answer
25 views

Find a normal to a curve

I have this problem: "Find a number k such that the line $x + y = k$ is normal to the curve $y = x^{2}$ I did it like this: $y' = 2x \Rightarrow y'(a) = 2a$ $y(a) = a^{2}$ So I put this into the ...
0
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1answer
133 views

First derivative of RVM related matrix expression

Can somebody help me find the first of the following function $\mathcal{L}$ with respect to the elements $\phi_{mn}$ of the matrix $\mathbf{\Phi}$? \begin{equation} \mathcal{L} = ...
0
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2answers
40 views

Differentiation of norm

How do I differentiate the "norm" of $(x-μ)$, with respect to $μ$, where both $x$ and $μ$ are vectors ? How will I start and proceed ? Thank you in advance.
3
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1answer
50 views

The Derivative of polynomial function

Let $f$ be a generic polynomial function, defined by $$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + \dots + a_1x + a_0$$ ,$n \in \mathbb{N}$ If I define $f(x)$ using sigma notation, ...
3
votes
4answers
32 views

Tangent line help

If $f(x)= x^2+2$, find all the points on the graph of $f$ for which the tangent line passes through the origin $(0,0)$. So far I've used $2$ in place of $x$ and found the derivative is $y=4x-2$ but I ...