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

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84
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1answer
4k views

How discontinuous can a derivative be?

There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many ...
63
votes
7answers
2k views

Does L'Hôpital's work the other way?

Hello fellows, As referred in Wikipedia (see the specified criteria there), L'Hôpital's rule says, $$ \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)} $$ As $$ \lim_{x\to ...
44
votes
1answer
1k views

Thurston's 37th way of thinking about the derivative

In Thurston's superb essay On proof and progress in mathematics, he makes this observation: Of course there is always another subtlety to be gleaned, but I would like to at least think that I ...
40
votes
1answer
1k views

The $100$th derivative of $(x^2 + 1)/(x^3 - x)$

I am reading a collection of problems by the Russian mathematician Vladimir Arnol'd, titled A Mathematical Trivium. I am taking a stab at this one: Calculate the $100$th derivative of the function ...
31
votes
7answers
9k views

What function can be differentiated twice, but not 3 times?

In complex analysis class professor said that in complex analysis if a function is differentiable once, it can be differentiated infinite number of times. In real analysis there are cases where a ...
30
votes
4answers
2k views

Is it necessary that every function is a derivative of some function?

I thought about this a lot and consulted a lot of people but everyone had contradicting answers. I am a high school student. please help.
30
votes
1answer
660 views

How to show that $f'(x)<2f(x)$

I would appreciate if somebody could help me with the following problem: Q: Let $f(x),f'(x),f''(x),f'''(x)>0$ , $f'''(x)$ is a continuous function and $f'''(x)<f(x)$ on $\mathbb{R}$ then ...
28
votes
5answers
2k views

Are there ways of finding the $n$-th derivative of a function without computing the $(n-1)$-th derivative?

Say we have a function $f(x)$ that is infinitely differentiable at some point. Is it possible to find $f^{(n)}(x)$ without having to find first $f^{(n-1)}(x)$? If so, does it take less effort than ...
25
votes
6answers
2k views

Uses of $\lim \limits_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}$

I have been wondering whether the following limit is being used somehow, as a variation of the derivative: $$\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} .$$ Edit: I know that this limit is defined in ...
20
votes
2answers
1k views

Proof derivative equals zero?

I know this must be wrong, but I am confused as to where the mathematical fallacy lies. Here is the 'proof': $$f '(x) = \lim_{ h\to0}\frac{f(x+h)-f(x)}{h}$$ L'Hôpital's Rule (The previous limit was ...
20
votes
2answers
401 views

When does $(uv)'=u'v'?$ [duplicate]

In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this dream product rule true? There are ...
20
votes
1answer
429 views

Derivative of the Meijer G-function with respect to one of its parameters

Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in ...
20
votes
1answer
574 views

$n^{th}$ derivative of a tetration function

I stumbled upon this very peculiar function last summer, namely: $f(x)=x^{x^{x^{...^{x}}}}$, where there is a number $n$ of $x$'s in the exponent, I tried to find the derivative for the function and I ...
19
votes
2answers
1k views

Is a differentiable function on $(-2, 4)$ always integrable on $[-2, 4]$?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
18
votes
5answers
398 views

Geometric interpretation of mixed partial derivatives?

I'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. ...
18
votes
2answers
601 views

What is the derivative of ${}^xx$

How would one find: $$\frac{\mathrm d}{\mathrm dx}{}^xx?$$ where ${}^ba$ is defined by $${}^ba\stackrel{\mathrm{def}}{=}\underbrace{ a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{\text{$b$ times}}$$ Work so ...
18
votes
2answers
444 views

Ramanujan style nested differential Equation

So I was exploring some math the other day... and I came across the following neat identity: Given $y$ is a function of $x$ ($y(x)$) and $$ y = 1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + ...
17
votes
4answers
1k views

Differentiating $y=x^{2}$

I am reading in a book about differentiating, but I am confused with one of the steps he takes. We start with: $$ \begin{align} y &= x^{2} \\ y + \mathrm{d}y &= (x + \mathrm{d}x)^2 \\ y + ...
16
votes
8answers
4k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
16
votes
2answers
790 views

A functional equation with no solution

Let $f:\mathbb{R}\to (0,\infty)$ be a differentiable function satisfying $$f(f(x))=f^\prime(x)$$for each $x$. Show no such function exists. I got this problem in an exam. I haven't done anything ...
16
votes
1answer
488 views

“Converse” of Taylor's theorem

Let $f:(a,b)\to\mathbb R$. We know that for every $c\in(a,b)$ we can write $f(t)=\sum_{i=0}^k a_i(c)(t-c)^i+o\left((t-c)^k\right)$ and $\forall i$ $a_i(c)$ is continuous (with respect to $c$). Can we ...
15
votes
4answers
1k views

$\int^{1}_{0} f^{-1} = 1 - \int^1_0 f$

One more from hard to believe facts, which I'm curious why are true. Let $f : [0,1] \rightarrow [0,1] $ is a continuous, monotonically increasing and surjective function Then $$\int^{1}_{0} f^{-1} ...
15
votes
4answers
467 views

Derivative of ${x^{x^2}}$

Studying past exam problems for my exam in ~$4$ weeks, and I came across this derivative as one of the questions. I actually have no idea how to solve it. $$\frac{d}{dx} (x^{x^2})$$ Using the chain ...
15
votes
3answers
282 views

Finding $f'(0)$ when $f(x)=\int\limits_0^x\sin\left(\frac{1}{t}\right)dt$

I need to show that $f'(0)=0$ for $$ f(x)=\int\limits_0^x\sin\left(\frac{1}{t}\right)dt $$ But fundamental theorem of calculus is unapplicable here. What should I do?
15
votes
1answer
345 views

Is there any meaning to an “infinite derivative”?

I've been thinking about this: say you have an infinitely differentiable function. Then you can form a sequence $f(x), f'(x), f''(x), \cdots, f^{(n)}(x), \cdots$ and attempt to take its limit. For ...
15
votes
3answers
1k views

Fractional Derivative Implications/Meaning?

I've recently been studying the concept of taking fractional derivatives and antiderivatives, and this question has come to mind: If a first derivative, in Cartesian coordinates, is representative of ...
14
votes
3answers
433 views

Calculate:$y'$ for $y = x^{x^{x^{x^{x^{.^{.^{.^{\infty}}}}}}}}$ and $y = \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+…\infty}}}}$

(1) If $y = x^{x^{x^{x^{x^{.^{.^{.^{\infty}}}}}}}}$ (2) If $y = \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+....\infty}}}}$ then find $y'$ in both cases (3)If $ y= ...
14
votes
1answer
245 views

How to calclulate a derivate of a hypergeometric function w.r.t. one of its parameters?

Is it possible to take a derivative of a hypergeometric function w.r.t. one of its parameters and express it in a closed form? I am particularly interested in this case: ...
14
votes
3answers
441 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} ...
14
votes
1answer
311 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 ...
13
votes
4answers
8k views

Discontinuous derivative.

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is ...
13
votes
4answers
4k 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
2answers
220 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
2answers
382 views
13
votes
6answers
117 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
0answers
353 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) = ...
12
votes
8answers
9k 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 ...
12
votes
2answers
328 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
314 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
3answers
562 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 ...
12
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 ...
12
votes
3answers
293 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
1answer
586 views

Any ideas on how I can prove this expression?

I don't have a lot of places to turn because i am still in high school. So please bear with me as i had to create some notation. In order to understand my notation you must observe this identity for ...
12
votes
2answers
116 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
227 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
2answers
539 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 ...
11
votes
4answers
232 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
292 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
1answer
508 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 $f'(x) = f^{-1}(x)$ ?
11
votes
5answers
425 views

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

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