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

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144
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6answers
10k views

Why does this matrix give the derivative of a function?

I happened to stumble upon the following matrix: $$ A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix} $$ And after trying a bunch of different examples, I noticed the ...
120
votes
1answer
6k 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 ...
113
votes
5answers
3k views

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...
87
votes
6answers
10k views

What did Alan Turing mean when he said he didn't fully understand dy/dx?

Alan Turing's notebook has recently been sold at an auction house in London. In it he says this: Written out: The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find extremely difficult ...
68
votes
7answers
3k 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 ...
57
votes
4answers
15k views

Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$

Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$. This question is just after the definition of differentiation and the theorem that if $f$ is finitely ...
54
votes
1answer
3k 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 ...
49
votes
8answers
25k views

Why is the derivative of a circle's area its perimeter (and similarly for spheres)?

When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle. Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is ...
44
votes
1answer
2k 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 ...
43
votes
7answers
5k views

Is there a function whose antiderivative can be found but whose derivative cannot?

The question is in the title. To rephrase, does a function, $f(x)$, exist such that $\int f(x) dx $ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if ...
43
votes
15answers
9k views

What are “instantaneous” rates of change, really?

Here's how I see it (please read the following if you can, because I address a lot of arguments people have already made): Let's take instantaneous speed, for example. If it's truly instantaneous, ...
40
votes
11answers
3k views

Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator?

$1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$. I got this question in an Application of Derivatives test. I think log might be used here to compare the values, but even then the values ...
38
votes
3answers
3k views

Why, although these functions have the same derivative, do they not differ by a constant?

I calculated the derivative of $\arctan\left(\frac{1+x}{1-x}\right)$ to be $\frac{1}{1+x^2}$. This is the same as $(\arctan)'$. Why is there no $c$ that satisfies $\arctan\left(\frac{1+x}{1-x}\right) ...
36
votes
7answers
11k 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 ...
36
votes
16answers
4k views

Why does the derivative of sine only work for radians?

I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm ...
33
votes
13answers
3k views

Why do differentiation rules work? What's the intuition behind them? (Not asking for proofs)

Differentiation rules have been bugging me ever since I took Basic Calculus. I thought I'd develop some intuitive understanding of them eventually, but so far all my other math courses (including ...
31
votes
6answers
3k views

Function whose third derivative is itself.

I'm looking for a function $f$, whose third derivative is $f$ itself, while the first derivative isn't. Is there any such function? Which one(s)? If not, how can we prove that there is none? Notes: ...
31
votes
1answer
696 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 ...
30
votes
5answers
2k views

Notation of the second derivative - Where does the d go?

In school I was taught that we use $\frac{du}{dx}$ as a notation for the first derivative of a function $u(x)$. I was also told that we could use the $d$ just like any variable. After some time we ...
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
2k views

How is the derivative geometrically inverse of integral?

I know that derivative is the slope of the tangent line, and that integral is the area under the curve. My question is that how these two distinct concepts are geometrically related? What is the ...
28
votes
2answers
17k 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 ...
28
votes
7answers
3k views

Why does the fundamental theorem of calculus work?

I've known for some time that one of the fundamental theorems of calculus states: $$ \int_{a}^{b}\ f'(x){\mathrm{d} x} = f(b)-f(a) $$ Despite using this formula, I've yet to see a proof or even a ...
28
votes
5answers
3k 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 ...
26
votes
3answers
4k views

Do “other” trigonometric functions than Tan Sin Cos and their derivatives exist?

I remember my physics teacher mentioning that other trigonometric functions exist apart from the Sin Cos and Tan, he mentioned a few and they did not sound familiar, nothing like Sec Csc and Cot. I ...
26
votes
10answers
11k 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 ...
26
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 ...
24
votes
6answers
4k views

Is there any function which grows 'slower' than its derivative?

Does a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f'(x) > f(x) > 0$ exist? Intuitively, I think it can't exist. I've tried finding the answer using the definition of ...
23
votes
6answers
25k views

What exactly is the difference between a derivative and a total derivative?

I am not too grounded in differentiation but today, I was posed with a supposedly easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to ...
23
votes
1answer
425 views

Function's analytic continuation is its own derivative

This is the question we were asked at the university by our professor for complex analysis. Not as an exam, but as a challenge. I don't think he knew the answer himself. Find a nontrivial example of ...
22
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 ...
22
votes
2answers
8k views

Differentiating an Inner Product

If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...
22
votes
5answers
2k 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. ...
22
votes
1answer
554 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 ...
22
votes
2answers
787 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 ...
22
votes
1answer
1k 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 ...
21
votes
4answers
4k views

“What if” math joke: the derivative of $\ln(x)^e$

Randall Munroe, the creator of xkcd in his latest book What if writes (p. 175) that the mathematical analog of the phrase "knock me over with a feather" is seeing the expression $ \ln( x )^{e}$. And ...
21
votes
3answers
564 views

What is the derivative of: $f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$?

I happened to ponder about the differentiation of the following function: $$f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$$ Now, while I do know how to manipulate power towers to a certain ...
20
votes
2answers
665 views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
20
votes
3answers
2k 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 ...
20
votes
7answers
1k views

Can someone provide the formal definition of the tangent line to a curve?

I was recently explaining differentiation from first principles to a colleague and how differentiation can be used to obtain the tangent line to a curve at any point. While doing this, my colleague ...
20
votes
2answers
485 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
2answers
575 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 + ...
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
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 + ...
18
votes
4answers
1k views

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

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} ...
18
votes
8answers
287 views

Intuitively, why should the coefficient of the derivative of $x^n$ be $n$?

I am able to differentiate $x^n$ with respect to $x$ from first principles using the definition of differentiation. Also it seems natural that the gradient of a finite polynomial will be one order ...
18
votes
2answers
1k 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 ...
18
votes
2answers
7k views

Is $ \frac{\mathrm{d}{x}}{\mathrm{d}{y}} = \frac{1}{\left( \frac{\mathrm{d}{y}}{\mathrm{d}{x}} \right)} $?

In calculus, is $ \dfrac{\mathrm{d}{x}}{\mathrm{d}{y}} = \dfrac{1}{\left( \dfrac{\mathrm{d}{y}}{\mathrm{d}{x}} \right)} $? I’m so confused about this matter. What would be a proof of it? Edit: By the ...
18
votes
1answer
742 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 ...