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

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79
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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 ...
60
votes
6answers
1k 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 ...
37
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 ...
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
622 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 ...
29
votes
7answers
8k 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 ...
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
1k 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
359 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
385 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
1answer
367 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 ...
18
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'Hopital's Rule (Previous limit was ...
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
1answer
465 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 ...
16
votes
1answer
391 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 + ...
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
8answers
3k 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 ...
15
votes
4answers
442 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
280 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
285 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
2answers
465 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 ...
14
votes
3answers
389 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
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 ...
13
votes
3answers
398 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= ...
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
213 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
1answer
176 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: ...
13
votes
1answer
217 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 ...
12
votes
2answers
299 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
3answers
290 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
111 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
223 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
497 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
8answers
6k 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 ...
11
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. $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ $$(x^2)'=(x+\cdots+x)'$$ $$2x=1+\cdots+1$$ ...
11
votes
4answers
214 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
283 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
442 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
351 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
105 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
418 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? ...
11
votes
3answers
146 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
242 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
3answers
345 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
1answer
131 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
0answers
324 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) = ...
10
votes
8answers
2k 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
4answers
5k 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 ...
10
votes
4answers
586 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} ...