# Tagged Questions

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

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### 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 ...
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### 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 "...
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### 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 ...
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### 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 ...
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### 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: ...
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### How is the derivative truly, literally the “best linear approximation” near a point?

I've read many times that the derivative of a function $f(x)$ for a certain $x$ is the best linear approximation of the function for values near $x$. I always thought it was meant in a hand-waving ...
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### 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 ...
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### 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 extent,...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...