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
70 views

Derivative of $f(x) = \frac{x}{\sqrt{x}-x}$

How would I derivative this function? $$f(x) = \frac{x}{\sqrt{x}-x}$$ The numerator derivative is: $$1$$ The denominator derivative is: $$\frac{1}{2\sqrt{x}}-1$$ In the end, I get this weird ...
1
vote
2answers
72 views

Derivates of function and limits

I wonder if it's true $$\lim_{n \rightarrow \infty} \frac{d}{dx}f_n(x)=\frac{d}{dx} \lim_{n \rightarrow \infty} f_n(x)$$ When $\displaystyle f_n(x)$ is a sequence of function.
9
votes
8answers
429 views

What is the difference between partial and normal derivatives?

I have a clarifying question about this question: What is the difference between $d$ and $\partial$? I understand the idea that $\frac{d}{dx}$ is the derivative where all variables are assumed to be ...
1
vote
1answer
82 views

Why does a piecewise constant function not have a primitive function?

Let $$f(x)=\begin{cases} -1, \quad & a \leq x \leq 0 \\ 1, \quad & 0 \leq x \leq b \end{cases}$$ It says in my book because of the Darboux theorem: If $f:[a,b] \to \mathbb R$ is ...
5
votes
2answers
175 views

Proving a Polynomial Identity

Prove that $$\sum_{i=1}^{n} \dfrac{{r_{i}}^k P(x)}{P'(r_{i})(x-r_{i})} = x^k$$ where $P(x)$ is an $n$ degree polynomial having distinct roots $\{ r_{i} \}_{i=1}^{n}$ and $k$ is an ...
0
votes
1answer
57 views

Differentiability implies continuity in a norm vector space framework proof.

I am following Real Mathematical Analysis by Pugh. Here is the definition of derivative in the relevant setting and the proof that I have yet to fully understand I am not sure how this implies ...
0
votes
1answer
40 views

Find X in plateau of exponential decay

I have this equation: $$y=(a+b)\cdot e^{-KX} + c.$$ This is an exponential decay function. I need to get its derivative and find $X$ when derivative $= 0$. This function has a plateau at $y = c$. ...
0
votes
2answers
29 views

Calculus, specifically deriving the rule for exponents

I was watching the MIT lectures series on calculus 1(https://youtu.be/9v25gg2qJYE?t=29m19s), at 29minutes and 19 seconds he converts a^x to base e by e^((log(a))^x) Why does he convert base a to base ...
1
vote
1answer
38 views

Exchanging integrals, derivatives and series

I need to solve this integral: $$I = \sum_{n=0}^{+\infty}\int_0^{\frac{\pi}{4}} (-1)^n x^{2n} \arctan(x) dx.$$ Firstly, I know that $f_n(x) = (-1)^n x^{2n} \arctan(x)$ punctually converges to $f(x) ...
0
votes
0answers
17 views

From Newtonian systems to Lagrange mechanics using Euler - Lagrange equations

I'm asking the same question here: http://physics.stackexchange.com/questions/206758/from-newtonian-systems-to-lagrange-mechanics-using-euler-lagrange-equations Shoud I delete the question from ...
4
votes
1answer
113 views

What's the $n$-th derivative of $\ln(\sin(x))$?

I want to find the $n$-th derivative of $\ln(\sin x)$, i.e. $$ \frac{d^n\ln(\sin x)}{dx^n} $$ where $x\in (0,\pi/2)$ such that $\sin x>0$. To make the problem definitely, $x=\pi/4$ is assumed. In ...
0
votes
1answer
27 views

Is $g(x)\equiv f(x,1) = \frac{1}{1-x}+1$ increasing or decreasing? differentiable $x=1$?

We have $$g(x) \equiv f(x,1) = \frac{1}{1-x}+1$$. I would like to know whether $g(x)$ is increasing or decreasing. Re-writing $g(x) = \frac{2-x}{1-x}$, I obtain $$g'(x) = \frac{x-3}{(1-x)^{2}},$$ ...
1
vote
1answer
52 views

On an equivalent definition of the derivative.

I am following Real Mathematical Analysis by Pugh Pugh says that the statement $$f(x+h) = f(x) +f'(x)h + R(h) \implies \lim_{h \rightarrow 0} \frac{R(h)}{|h|} = 0 $$ is equivalent to the definition ...
0
votes
2answers
47 views

Use sum and difference formulas to find the derivative of the function. How to make my answer match the correct answer?

Find the derivative of the function. $y = 3x-1+\frac{1}{x}$ So I know we have to find the derivatives of the values with x. So for 3x, I did: $\frac{d}{dx}[3x] = 3\frac{d}{dx}[x]$ where if $x^n$ ...
3
votes
2answers
68 views

$f(x) = \|\textbf{x}\|\textbf{x}$, first derivative, second derivative.

Consider the function $f: \mathbb{R}^n \to \mathbb{R}^n$ given by $f(x) = \|\textbf{x}\|\textbf{x}$. Is $f$ differentiable at $\textbf{0}$? Do the second-order partial derivatives of $f$ exist at ...
-1
votes
3answers
20 views

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 $x.$
0
votes
1answer
73 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
votes
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
votes
0answers
19 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 ...
0
votes
0answers
24 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
21 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
votes
3answers
103 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
votes
2answers
91 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 ...
-1
votes
4answers
157 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
votes
4answers
55 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
votes
0answers
47 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
84 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 ...
-1
votes
2answers
83 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 ...
1
vote
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 + ...
0
votes
0answers
38 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
votes
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
votes
2answers
39 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
votes
1answer
42 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 ...
0
votes
1answer
18 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 ...
-1
votes
1answer
31 views

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
votes
1answer
49 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 + ...
1
vote
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 ...
0
votes
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
votes
1answer
43 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 ...
1
vote
1answer
355 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
votes
1answer
74 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
votes
1answer
17 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?
0
votes
1answer
28 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
votes
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?
1
vote
1answer
38 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.
2
votes
0answers
73 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
vote
1answer
85 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} ...
0
votes
1answer
46 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: ...
1
vote
1answer
46 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 ...