# Questions tagged [derivatives]

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

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### Smallest $c$ such that $f'<cf$ holds for all $f$ such that $f,f',f'',f'''>0$ and $f''' \le f.$

Let $f: \mathbb{R} \to \mathbb{R}$ be a $C^3$ function such that $f,f',f'',f'''>0$ and $f''' \le f.$ What is the smallest $c$ such that we can guarantee $f'<cf$? Since $f(x)=e^x$ works, we must ...
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### How to prove this lemma related to Rolle's theorem

For any function $f$ denote by $Z(f)$ and $Z_o(f)$ the cardinalities of $f^{-1}(0)\cap[0,1]$ and $f^{-1}(0)\cap(0,1)$, respectively. Let $H=\{f\in C^\infty(\mathbb{R}): \text{supp}(f) = [0,1]\}$ From ...
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### Existence of function satisfying $f(f'(x))=x$ almost everywhere

My project is to Study the existence of a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ differentiable almost everywhere satisfying $f\circ f'(x)=x$ almost everywhere $x \in \mathbb{R}$...
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### Number of zeros of the second derivative of composition of sigmoid-like functions

Consider the bounded, positive and monotonic functions $f(x) = \frac{x^k}{a^k+x^k}$ and $g(x) = \frac{b^h}{b^h+x^h}$ with $a,b>0,$ and $k,h > 1$ and defined for $x\in \mathbb{R}^+$. Using ...
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### Is $\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}$ completely monotone?

Recall the definitions of the sine and cosine integrals: $$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos t}t dt.$$ Both functions are oscillating, ...
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### "Angle-preserving" equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be angle-...
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### Is it feasible to use Operator Calculus to solve for $f(t)=\underbrace{\exp(\exp(\dots\exp}_{t}(0)\dots))$ over $\mathbb{R}$

Consider the function $f(t)=\exp^{(t)}(0)$ where $\exp^{(0)}(0)=0$ and $\exp^{(t+1)}(0)=\exp(\exp^{(t)}(0))$. That is, $$f(t)=\underbrace{\exp(\exp(\dots\exp}_{t}(0)\dots)).$$ Such a function is not ...
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### How to prove using elementary methods that this function is everywhere continuous but nowhere differentiable?

Let $f$ be the function defined on all of $\mathbb{R}$ by the formula $$f(x) \colon= \sum_{n=0}^\infty \frac{1}{2^n} \cos \left( 3^n x \right).$$ How to show (rigorously but through elementary ...
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### Quantify the similarity between a polynomial roots and the roots of its derivatives

On $\mathbb{C}[X]$, many theorems and conjectures deal with relations between a polynomial roots and the roots of its derivatives. When looking at a graph, the derivative roots distribution somewhat ...
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### Is there a generalized solution to the "Freshman's dream" of derivatives equation?

Michael Penn shows that $(f(x)g(x))^{\prime} = f^{\prime}(x) g^{\prime}(x)$ has solutions of the form $$f(x) = C \exp{\int \frac{g^{\prime}(x)}{g^{\prime}(x)-g(x)}dx}$$ for a given function $g(x)$. ...
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### Asymptotic distribution for non differentiable functions of estimators

is there kind of a standard tool to derive the distribution of $f(\theta)$ if f is non differentiable (so no Delta Method available) and $\theta$ is asymptotically normal distributed? Thanks a lot!
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### limit of a region of integration in $\mathbb{R}^2$ approaches a line

I am trying to follow the derivation of derivatives in a paper published in some japanese journal but there seems to be a mistake in the proof. I will present the problem in 2D and in 2 variables so ...
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