# Tagged Questions

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

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### 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 ...
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### How many non-differentiable functions exist?

The size of the set of functions that map $\mathbb{R}\to \mathbb{R}$ equals $(\#\mathbb{R})^{\#\mathbb{R}}$. How many non-differentiable functions are there in this set?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Is calculating the summation of derivatives “mathematically sound”?

I have just discovered that if you take the following series: $$1 + x + x^2 + x^3 + x^4 + \cdot \cdot \cdot = \sum_{n = 0}^\infty x^n$$ and replace each term in the series with the derivative of them, ...
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### Proof that the range of a map is determined by its behaviour on the boundary.

Let f be a mapping from an open neighbourhood of the 3-dimensional unit ball to the 2-dimensional plane. Suppose that f is smooth (infinitely continuously differentiable on its domain) and regular (it'...
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### “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 ...
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### Derivative/integral relationship appears to disprove the fundamental theorem of calculus!!!

Consider the floor function: $$f(x) = \lfloor x \rfloor$$ The indefinite integral of f is: $$\int_0^x f(x) dx = x\lfloor x \rfloor - \frac {\lfloor x \rfloor^2 + \lfloor x \rfloor} 2$$ This should ...
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### Find the value of a function whose derivative is zero

The initial function is $$h(x)=\arcsin x + \arccos x$$ The derivative of this function is $0$ since $$h'(x)=\frac{1}{\sqrt{1-x^2}}-\frac{1}{\sqrt{1-x^2}}\equiv0$$ This means that $h(x)$ is a ...
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### $f$ be a smooth function on real line , $f(0)=0$ , $f(x)>0, \forall x \ne 0$ and any $f^{(n)}(0)=0$ ; is $\sqrt f$ smooth?

Let $f: \mathbb R \to \mathbb R$ be an infinitely differentiable function such that $f(0)=0$ , $f(x)>0 , \forall x \ne 0$ and $f^{(n)}(0)=0$ ( the $n$-th derivative ) $, \forall n \in \mathbb N$ ...
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### 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 ...
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### 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 ...
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### Why use the derivative and not the symmetric derivative?

The symmetric derivative is always equal to the regular derivative when it exists, and still isn't defined for jump discontinuities. From what I can tell the only differences are that a symmetric ...
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### 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 $$f'(\xi)=\dfrac{f(\xi)-f(a)}{b-a}$$ My ...
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### 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? $$\lim_{x\to\frac{4}{3}}\frac{\partial}{\partial x}\left[\,_2{\rm{F}}_1\left(\frac{1}{3},1;x;-1\right)\right]$$
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### Derivative of the $f(x,y)=\min(x,y)$

I just encountered this function $f(x,y)=\min(x,y)$. I wonder what the partial derivatives of it look like.
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### 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!
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 ...
### 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 ...