query: Differentiability of $f(x)={x^2\sin{(\dfrac{1}{x})}}$ at $x=0$
positive_passages.docid: 59956
positive_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: 154881
negative_passages.text: Question on differentiability of a continuous function
negative_passages.docid: 107727
negative_passages.text: $m\{x\in [0,1]:f'(x)=0\}>0$
negative_passages.docid: 94193
negative_passages.text: determine whether $f(x, y) = \frac{xy^3}{x^2 + y^4}$ is differentiable at $(0, 0)$.
negative_passages.docid: 134871
negative_passages.text: Using definition of derivative to differentiate $f(x)= \sqrt{x}+1$
negative_passages.docid: 30533
negative_passages.text: Showing that this function is infinitely differentiable
negative_passages.docid: 145173
negative_passages.text: $f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$, then $x \cdot \nabla f(x) = 2 f(x)$
negative_passages.docid: 49705
negative_passages.text: Is $f(x,y)=(x+2y+y^2+|xy|, 2x+y+x^2+|xy|)$ differentiable?
negative_passages.docid: 63191
negative_passages.text: What is the $k-$th derivative of $ \frac{1}{1-x^2}$?
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: 99795
negative_passages.text: Derivative of piece-wise function given by $x\sin\frac1x$ at $x=0$
negative_passages.docid: 75932
negative_passages.text: Differentiability of $g:=f(\sqrt{x^2+y^2})$ for a $C^1$ function with $f'(0)=0$ NBHM $2008$
negative_passages.docid: 11167
negative_passages.text: A counterexample of a differentiable function on an interval whose derivative is discontinuous
negative_passages.docid: 119209
negative_passages.text: Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$
negative_passages.docid: 141423
negative_passages.text: The continuity of $f(x,y) = \frac {\sin (xy)} {\sqrt {x^2 + y^2}}$ at $(0,0)$
negative_passages.docid: 34136
negative_passages.text: $f$ is differentiable in $[0,1]$ ,$\sup_{x\in[0,1]}|f'(x)| \le M\lt+\infty $
negative_passages.docid: 38192
negative_passages.text: Derivative of inverse function at a point.