query: $g(f(x))=f'(x)$ and $f$ is differentiable. Prove that $f$ is monotonic.
positive_passages.docid: 153580
positive_passages.text: Solutions of autonomous ODEs are monotonic
negative_passages.docid: 15727
negative_passages.text: show that if $f$ is uniformly differentiable then prove that $f'$ is continuous
negative_passages.docid: 18734
negative_passages.text: Showing that a function is differentiable
negative_passages.docid: 136714
negative_passages.text: Proving that $\lim_{x \to 0} \frac{f(x)}{|x|}=1$ implies $f$ is not differentiable at $0$.
negative_passages.docid: 152877
negative_passages.text: Is a monotone differentiable function continuously differentiable?
negative_passages.docid: 33523
negative_passages.text: If $f(0)=0$ and $f'$ is increasing, then $\frac{f(x)}{x}$ is increasing.
negative_passages.docid: 57193
negative_passages.text: Prove that $f$ has derivatives of all orders at $x=0$, and that $f^{(n)}(0)=0$ for $n=1,2,3\ldots$
negative_passages.docid: 145424
negative_passages.text: If $f''(x)+f(x)>0$ and $f(x)>0$ $\forall x\in(a,b)$; $f(a)=f(b)=0$; prove that $b-a>\pi$.
negative_passages.docid: 158681
negative_passages.text: How to show that $e^x$ is differentiable?
negative_passages.docid: 105084
negative_passages.text: Let f be a function twice differentiable and with derivatives continuous on an interval $[a,b]$ containing $0$. Prove the following statement:
negative_passages.docid: 150968
negative_passages.text: Differentiable function has measurable derivative?
negative_passages.docid: 71857
negative_passages.text: Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
negative_passages.docid: 107727
negative_passages.text: $m\{x\in [0,1]:f'(x)=0\}>0$
negative_passages.docid: 48592
negative_passages.text: Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
negative_passages.docid: 111470
negative_passages.text: Proof of differentiability of $f(x)$
negative_passages.docid: 53767
negative_passages.text: How to prove that derivatives have the Intermediate Value Property
negative_passages.docid: 95684
negative_passages.text: Show that $g'(x)+g(x)-2e^x=0$