query: $f'(x) = g(f(x)) $ where $g: \mathbb{R} \rightarrow \mathbb{R}$ is smooth. Show $f$ is smooth.
positive_passages.docid: 142344
positive_passages.text: The solution of ODE $k'(x) = r(k(x))$ is infinitely differentiable if $r$ is
negative_passages.docid: 108060
negative_passages.text: Suppose $f:[0,\infty]\rightarrow$ to R and has continuous $f'(x)$ and $f''(x)$ f(x) → 0 as x → ∞. Show that f′(x)→0 as x→∞
negative_passages.docid: 21339
negative_passages.text: IF $f(x) = \int_{0}^{\phi (x)} g(t) dt$, How could we find $f'(x)$?
negative_passages.docid: 111458
negative_passages.text: Prove that $f$ is differentiable
negative_passages.docid: 143491
negative_passages.text: Show that $f$ is differentiable on $(−1, 1)$.
negative_passages.docid: 27366
negative_passages.text: Prove that $g$ is tangent to the graph of $f$
negative_passages.docid: 65889
negative_passages.text: Showing $\Phi(x)=\int_0^x (x-t)f(t)\,dt$ is twice differentiable and finding $\Phi''(x)$
negative_passages.docid: 112923
negative_passages.text: Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}$
negative_passages.docid: 59956
negative_passages.text: Show that the function $g(x) = x^2 \sin(\frac{1}{x}) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$
negative_passages.docid: 94944
negative_passages.text: Differentiable functions satisfying $f'(f(x))=f(f'(x))$
negative_passages.docid: 3387
negative_passages.text: Derivative of $f(x)=|x|$
negative_passages.docid: 158681
negative_passages.text: How to show that $e^x$ is differentiable?
negative_passages.docid: 7945
negative_passages.text: Approximate $|x|$ with a smooth function
negative_passages.docid: 135634
negative_passages.text: Prove that there exists a smooth map $g\colon R\to R$ such that $f(\cos(t),\sin(t))=(\cos(g(t)),\sin(g(t)))$ and satisfying $g(2π)=g(0)+2\pi q$.
negative_passages.docid: 31685
negative_passages.text: Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$
negative_passages.docid: 133405
negative_passages.text: $\lim_{x \rightarrow 0} f'(x) = L$ exists, does it follow that $f'(0)$ exists?
negative_passages.docid: 69378
negative_passages.text: The derivative of $x!$ and its continuity