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

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14
votes
1answer
616 views

Inverse of a bijection f is equal to its derivative

Does there exist a differentiable bijection $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f'(x) = f^{-1}(x)$ ?
14
votes
2answers
448 views

Prove ${\large\int}_0^1\frac{\ln(1+8x)}{x^{2/3}\,(1-x)^{2/3}\,(1+8x)^{1/3}}dx=\frac{\ln3}{\pi\sqrt3}\Gamma^3\!\left(\tfrac13\right)$

Here is one more numerically discovered conjecture that I was not able to prove, and asking you for help: ...
14
votes
1answer
194 views

What's the minimal structure needed to define a notion of derivative?

I know that, for example, to define a limit all you need is the notion of "closeness" generated by a topology; and to define an integral you need a measure function and a sigma-algebra on which it is ...
14
votes
1answer
460 views

Definition of the nth derivative? [First post]

If the definition of the derivative is $$ f^\prime(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x} $$ Would it make sense that the nth derivative would be (I know that the 'n' in ...
14
votes
0answers
404 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
13
votes
8answers
11k views

Continuous versus differentiable

A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it. Until today, I thought these were merely two equivalent definitions of the same ...
13
votes
4answers
5k views

If a function has a finite limit at infinity, does that imply its derivative goes to zero?

I've been thinking about this problem: Let $f: (a, +\infty) \to \mathbb{R}$ be a differentiable function such that $\lim\limits_{x \to +\infty} f(x) = L < \infty$. Then must it be the case that ...
13
votes
3answers
671 views

FoxTrot Bill Amend Problems

So I found this on the Wolfram website today: So I was wondering about how one might be able to (if possible) solve those four problems by hand. Here are the problems, $\LaTeX$ed: $ \lim_{x \to ...
13
votes
2answers
231 views

How prove that there exists $\xi\in(a,b)$ with $f'(\xi)=\frac{f(\xi)-f(a)}{b-a}$

Let $f(x)$ be continuous on $[a,b]$, differentiable on $(a,b)$, and with some $c\in(a,b)$ such that $f'(c)=0$. Show: There exists $\xi\in(a,b)$ such that $$ f'(\xi)=\dfrac{f(\xi)-f(a)}{b-a} $$ ...
13
votes
6answers
143 views

Why does $n$-time differentiation of product have the same structure as raising sum to $n$th power?

A formula for differentiating a product is well known: $$(ab)'=a'b+ab'.$$ At first sight it doesn't resemble anything interesting. But what if we differentiate twice? We'll get ...
13
votes
2answers
592 views

Zeroes of the third derivative of an iterated sine.

I've been playing with the functions $$f_n:[0,\pi/2]\to[0,1]\\\begin{cases} f_1&=&\sin\\f_{n+1}&=&\sin\circ f_n\end{cases}.$$ A simple argument proves that $f_n(x)\to 0$ for $x\in ...
12
votes
5answers
1k views

Are all continuous one one functions differentiable?

I was reading about one one functions and found out that they cannot have maxima or minima except at endpoints of domain. So their derivative , if it exists, must not change it sign , i.e. , the ...
12
votes
3answers
506 views

The closed form of $\lim_{x\to\frac{4}{3}}\frac{\partial}{\partial x}\left[\,_2{\rm{F}}_1\left(\frac{1}{3},1;x;-1\right)\right]$

Do you think the following limit might have a closed form? Some hints or clues? $$\lim_{x\to\frac{4}{3}}\frac{\partial}{\partial x}\left[\,_2{\rm{F}}_1\left(\frac{1}{3},1;x;-1\right)\right]$$
12
votes
2answers
356 views

Finding the derivative of $x\uparrow\uparrow n$

I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general ...
12
votes
2answers
641 views

In what sense is the derivative the “best” linear approximation?

I am familiar with the definition of the Frechet derivative and it's uniqueness if it exists. I would however like to know, how the derivative is the "best" linear approximation. What does this mean ...
12
votes
3answers
301 views

Simplified form for $\frac{\operatorname d^n}{\operatorname dx^n}\left(\frac{x}{e^x-1}\right)$?

I have found the following formula: $$\frac{\operatorname d^n}{\operatorname ...
12
votes
2answers
591 views

Derivative of $x^{x^{\cdot^{\cdot}}}$?

The infinite tetration is defined as $$f(x)=x^{x^{\cdot^{\cdot}}}$$ This function is defined for $e^{-e} \leq x \leq e^{e-1}$. (Wikipedia image) Can one determine the derivative of this function? ...
12
votes
1answer
77 views

Prove that every such $f$ is $=0$ everywhere

Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable; let $0 \leq f'(x) \leq f(x)$ for all $x \in \mathbb{R}$; and let $f$ vanish at some point. Prove that $f = 0$ on $\mathbb{R}$. Since there is ...
12
votes
2answers
119 views

Second derivative of $f(f(\cdots f(x)\cdots )?$

For convenience, let's write $f_n(x)=f(f(\cdots f(x)\cdots )$ where $f$ is iterated $n$ times. Suppose: $$f(0)=0,\quad f'(0)=\alpha,\quad f''(0)=\beta$$ What is $f''_n(0)?$ I've found ...
12
votes
1answer
248 views

Zeroes of derivatives of high order

The problem is following. Let $f:(-1,1)\to [-1,1]$ has $n$ derivatives. Prove that there exists a number $\alpha_n$ (independent from $f$) such that condition $|f'(0)|\geq \alpha_n$ implies that ...
12
votes
3answers
406 views

Are there “differential equations” involving derivations in the sense of abstract algebra?

There is this abstract notion of a derivation, which really only cares about the property $$D(ab)=aD(b)+D(a)b,$$ where $a,b$ are elements of some algebra. This only tangents the ideas, which lead to ...
11
votes
6answers
953 views

Evaluating the derivative of $\large \;e^{e^x}$?

I know that the derivative of $\,e^x\,$ is $\,e^x$. But how do I evaluate $\dfrac{d}{dx}{\large\left(e^{e^x}\right)}\,$?
11
votes
3answers
331 views

Value of $f'(0)$ if $f(x)=\frac{x}{1+\frac{x}{1+\frac{x}{1+\ddots}}}$

Consider the function $$f(x)=\cfrac{x}{1+\cfrac{x}{1+\cfrac{x}{1+\ddots}}} $$ Determine the value of $f'(0)$. I tried to differentiate $f(x)$ but it is not subject to chain rule, and now I'm stuck. ...
11
votes
3answers
1k views

Are polynomials infinitely many times differentiable?

Are polynomials infinitely many times differentiable? If so, does it only mean that at some point we reach 0 and then we keep on getting 0? Thank you!
11
votes
5answers
2k views

Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$

I was messing around with the definition of the derivative, trying to work out the formulas for the common functions using limits. I hit a roadblock, however, while trying to find the derivative of ...
11
votes
4answers
270 views

Why does differentiating a polynomial reduce its degree by $1$?

This may seem a bit silly but I am wondering: can it intuitively be shown that the derivative of a polynomial is precisely 1 degree lower than itself? I understand the basics of calculus enough to ...
11
votes
2answers
253 views

Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, ...
11
votes
2answers
308 views

Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?

Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable? Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
11
votes
5answers
564 views

Closed form for $n$th derivative of exponential of $f$

What is the closed form for: $$\frac{\partial^n}{\partial x^n}\exp(f(x))=\exp(f(x))\cdot[????]$$
11
votes
1answer
145 views

Differentiation of a function $f:\mathbb{Q}\to \mathbb{Q}$(Rational Calculus)

Assume that $f:\mathbb{Q}\to \mathbb{Q}$ is given such that $\forall a\in \mathbb{Q}$ the following limit, exists \begin{equation} \lim_{x\to a} \frac{f(x)-f(a)}{x-a}\in \mathbb{R} ...
11
votes
1answer
297 views

Define second derivative ($f''$) without using first derivative ($f'$)

The question I'd like to ask is this: If $f''(0)$ exists, does $f'$ exist in a neighborhood of $0$? Of course, under the standard definition of $f''(0)$, we have already assumed that $f'$ exists ...
11
votes
3answers
508 views

A limit with an intuitive and wrong answer

In my last question I asked about a limit used in my exploration of tangent circles and whatnot. I decided to come up with a more direct approach to my problem, and now I only have to evaluate the ...
11
votes
3answers
513 views

$\nabla \cdot \color{green}{(\mathbf{F} {\times} \mathbf{G})} $ with Einstein Summation Notation [Stewart P1068 16.5.27]

$\nabla \cdot \color{green}{(\mathbf{F} {\times} \mathbf{G})} = \partial_h\color{green}{\epsilon_{hij}F_iG_j}$ $ = \epsilon_{hij}\partial_h[F_iG_j]$ $ = \color{purple}{\epsilon_{hij}G_j\partial_hF_i} ...
11
votes
3answers
160 views

Applications of functions of the form $f(x)^{g(x)}$

Early on in my calculus education, I learned how to take the derivative of $x^x$ by re-writing it in the form $e^{x\ln x}$. More generally, this technique is helpful in finding the derivative of ...
11
votes
1answer
368 views

Interchanging pointwise limit and derivative of a sequence of C1 functions

Assume the following: $f_n$ is a sequence of $C^1$ functions on $[0,1]$ $f_n(x) \rightarrow 0$ pointwise. $f'_n(x) \rightarrow g(x)$ pointwise. Is it true that $g(x) = 0$ almost everywhere? I think ...
11
votes
1answer
123 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
11
votes
1answer
140 views

If $f<1$, $f(0)^2 + f'(0)^2=4$, exists $x_0$ s.t. $f''(x_0) + f(x_0)=0$

Suppose $f:\mathbb{R}\to\mathbb{R}$ is $C^2$, $f < 1$ for all $x$, and $f(0)^2 + f'(0)^2=4$. Show that $\exists x_0$ s.t. $f''(x_0) + f(x_0)=0$. So far, I have let $\phi(x) = f(x)^2 + f'(x)^2$. ...
11
votes
2answers
470 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
10
votes
8answers
3k views

Why do you need to use the chain rule in differentiation of ln?

I understand application of chain rule in the differentiation of a random function $(x^2+3)^3$. But, why do you need to use chain-rule when differentiating something like $\ln(2x-1)$; why won't it ...
10
votes
7answers
1k views

What's wrong with these equations? [duplicate]

My friend Boris (Boryan) gave me a task, and completely refuses to give the answer what's wrong here. $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ $$(x^2)'=(x+\cdots+x)'$$ $$2x=1+\cdots+1$$ ...
10
votes
4answers
789 views

Showing $f'(x) = f(x)$ implies an exponential function [duplicate]

Possible Duplicate: Proof that $\exp(x)$ is the only function for which $f(x) = f'(x)$ How can I show the statement $f'(x) = f(x)$ implies the function is defined as $f: \mathbb{R} ...
10
votes
4answers
227 views

Computing $\frac{d^k}{dx^k}\left(f(x)^k\right)$ where $k$ is a positive integer

Does anyone know a formula for the derivative $$\frac{d^k}{dx^k}\left(f(x)^k\right)$$ where $k$ is some positive integer? I started trying to work it out but it got messy.
10
votes
3answers
1k views

Why does the condition of a function being differentiable always require an open domain?

Going through Spivak's Calculus on Manifolds and in his definition of a differentiable function from a subset $A$ of $\mathbb{R}^n$ to $\mathbb{R}^m$, $f$ is said to be differentiable if it can be ...
10
votes
6answers
9k views

Using the Limit definition to find the derivative of $e^x$

I was wondering how we could use the limit definition $$ \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ to find the derivative of $e^x$, I get to a point where I do not know how to simplify the ...
10
votes
5answers
775 views

Evaluate the general infinite square root [duplicate]

$$x = \sqrt{n\sqrt{n\sqrt{n}} \cdots}$$ I see that: $$x = \sqrt{nx}$$ $$x^2 -nx = 0$$ Them: $$x(x - n) = 0 \implies x \in \{0, n\}$$ How should I reject the $x = 0$ solution? (any level proof ...
10
votes
3answers
2k views

Derive or differentiate?

When the action is: Taking the derivative what verb should be used? to differentiate to derive I feel that deriving is not the correct word here. In my mind it's more a synonym of deducing. Am I ...
10
votes
5answers
760 views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
10
votes
3answers
126 views

Let $f$ be a continuous and differentiable function such that $f(a)=f(b)=0$ , show that $f'(c)=\pi f(c)$ for some $c \in (a,b)$

Let $f:[a,b] \rightarrow \mathbb{R} $ be a continuous function in $[a,b]$ and differentiable in $(a,b)$ such that $f(a)=f(b)=0$ . Show that $f'(c)=\pi f(c)$ for some $c \in (a,b)$ The initial ...
10
votes
2answers
131 views

Can you prove My conjecture about Invertiblity of the Derivative Matrix ?! (to use Inverse function Theorem)

In the Analysis2 midterm exam, we had the following problem: Let the equation $a_nx^n+\cdots+a_1x+a_0=0$ has $n$ simple real roots (distinct) $\{\alpha_1,\cdots,\alpha_n\}$. Prove that the above ...
10
votes
1answer
280 views

Maxima of the function $\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$

I am looking for extrema of the function $$g(a,b):=\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$$ where $a,b >0$ are real parameters. I already plotted this function and got the ...