# Questions tagged [derivatives]

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

4,347 questions
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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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### 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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### 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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### Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ can ...
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### Is there any continuous function that is only differentiable on $\mathbb{Q}$?

I am looking for a continuous function $f: \mathbb R \rightarrow \mathbb R$ so that $f$ is differentiable in $x$, if and only if $x \in \mathbb Q$. I already know there is no function that is ...
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### Are there any other fields other than $\mathbb{R},\mathbb{C}$, rich enough to have analysis built on them?

I've been thinking about this, I don't know how to look up anything similar, so here I am asking a question. Specifically, is there any space $X$ with the following properties: Algebraic structure: ...
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### An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
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### Different Definitions on the Differentiability of Functions on a closed set.

I have encountered three different definitions on the differentiablity of functions on a closed set. In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The ...
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### “Simple” proof of Lebesgue's Differentiation Theorem for dimension 1?

Lebesgue Differentiation Theorem for $\mathbb{R}$: Let $f:[a,b]\to \mathbb{R}$ be intergable and $F(x)=\int_a^xf$. Then $F$ is differentiable almost everywhere in $[a,b]$ and $F'=f$ a.e. Is there a (...
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### Are most physics books wrong about the covariant derivative and connection?

I have always read in many physics books that a valid way of intuitively introducing the covariant derivative and the connection was the following: (example in GR but same thing for gauge theories) ...
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### (How) can one visualise the derivative of the function $A \mapsto A^{-1}$, where $A$ is a matrix?

In class, we have shown: Let $V$ be a finite-dimensional Banach-space, then the general linear group $$\mbox{GL}(V) := \{ A \in L(V, V): A \text{ is invertible } \}$$ is open in $L(V,V)$ and the ...
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### About the differential notation in measure theory

Is there any good reason for which integrating according to a measure includes a $\mathrm d$ as in $\int f\mathrm d\mu$ ? Or is it just a manner to keep formal consistency with the traditional ...
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### chain rule for derivations

Off we go. So let $b:X\rightarrow Y$ be a function from $X$ to $Y$ endowed with as much structure as it needs to make sense of the question :) and $a:Y\rightarrow \mathbb R$ a function into the reals. ...
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### $\sum_{k=0}^n a_kx^k$ splits $\Rightarrow \sum_{k=0}^n \frac {a_k} {k!}x^k$ splits over reals

Suppose that $a_0, a_1, \ldots a_n \in \mathbb R$ and the polynomial $P(x) = \sum_{k=0}^n a_kx^k$ has all real roots. I'm supposed to show that $$Q(x) = \sum_{k=0}^n \frac {a_k} {k!}x^k$$ also has ...
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### Properties of a continuous function that satisfies $f(tx)=t^2 f(x)$

Let $f:\mathbb{R}^{n} \to \mathbb{R}$ be a continuous function such that $f(x) > 0$ for each $x \neq 0$ e, moreover, $$f(tx)=t^2 f(x)$$ for any $x \in \mathbb{R}^{n}$ and $t \in \mathbb{R}$. ...
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### A rigorous yet intuitive summary of inflection and critical points for beginning calculus?

I haven't done these in awhile. While my analysis covered continuity but not differentiability, I have so far not revisited these in learning geometry or algebra. I am trying to help a calculus ...