query: Elementary proof that $2^x$ is derivable
positive_passages.docid: 22686
positive_passages.text: Show $\lim_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log
negative_passages.docid: 150139
negative_passages.text: Derivable product of derivable and non-derivable function, not satisfying product rule
negative_passages.docid: 39082
negative_passages.text: Proving $x^\alpha-\alpha x \le 1- \alpha $
negative_passages.docid: 148585
negative_passages.text: Show that, for any $\epsilon>0$, there exist two rationals such that $q < x < q'$ and $|q-q'|<\epsilon$
negative_passages.docid: 3820
negative_passages.text: Prove that $||x|-|y||\le |x-y|$
negative_passages.docid: 112631
negative_passages.text: Prove that if $d$ divides $n$, then $2^d -1$ divides $2^n -1$
negative_passages.docid: 67234
negative_passages.text: The function $G: x \mapsto 2^{x^2}$ maps $\mathbb{R}$ onto $\{ x \in \mathbb{R} : x \geq 1 \}$
negative_passages.docid: 141909
negative_passages.text: Prove ${2}^{n}\cdot(2m+1)-1 $ is invertible
negative_passages.docid: 52113
negative_passages.text: Proof by Induction $\sum_{i=0}^{k-1} 2^i = 2^k-1$
negative_passages.docid: 142513
negative_passages.text: Prove that $2xy\mid x^2+y^2-x$ implies $x$ is a perfect square.
negative_passages.docid: 72044
negative_passages.text: prove $x \mapsto x^2$ is continuous
negative_passages.docid: 21923
negative_passages.text: Prove that $\exp(x)>0$ using only formal definition of exp
negative_passages.docid: 58622
negative_passages.text: Intersection points of $2^x$ and $x^2$
negative_passages.docid: 158681
negative_passages.text: How to show that $e^x$ is differentiable?
negative_passages.docid: 62177
negative_passages.text: Prove that: $2^n < n!$ Using Induction
negative_passages.docid: 20964
negative_passages.text: Proof that $n^2 < 2^n$
negative_passages.docid: 115712
negative_passages.text: Application of Derivatives rigorous proof