query: If $\lim_{x\to x_0}f'(x)$ exists, then $f$ differentiable at $x_0$ and $f'(x_0)=\lim_{x\to x_0} f'(x)$
positive_passages.docid: 1371
positive_passages.text: Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
negative_passages.docid: 138930
negative_passages.text: What does $\lim\limits_{x \to \infty} f(x) = 1$ say about $\lim\limits_{x \to \infty} f'(x)$?
negative_passages.docid: 20273
negative_passages.text: Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
negative_passages.docid: 144013
negative_passages.text: If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.
negative_passages.docid: 111458
negative_passages.text: Prove that $f$ is differentiable
negative_passages.docid: 31051
negative_passages.text: Is $f(x,y)=(xy)^{2/3}$ differentiable at $(0,0)$?
negative_passages.docid: 25467
negative_passages.text: Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$
negative_passages.docid: 38923
negative_passages.text: What can we say about differentiability of $f$ at $c$?
negative_passages.docid: 18734
negative_passages.text: Showing that a function is differentiable
negative_passages.docid: 94944
negative_passages.text: Differentiable functions satisfying $f'(f(x))=f(f'(x))$
negative_passages.docid: 113343
negative_passages.text: $f$ is constant if derivative equals zero
negative_passages.docid: 71127
negative_passages.text: How do I find $f(0)$, $f'(0)$, and $f'(x)$ given $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$?
negative_passages.docid: 151109
negative_passages.text: Verification that $\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$ and $\lim_{h\to 0}\frac{f(a+h) -f(a)}{h}$ are equivalent definitions of the derivative.
negative_passages.docid: 74583
negative_passages.text: Prove that $f'(0)=L$.
negative_passages.docid: 127114
negative_passages.text: Differentiability implying continuity
negative_passages.docid: 151958
negative_passages.text: How to show $f'(0)$ exist and is equal to $1$?
negative_passages.docid: 34136
negative_passages.text: $f$ is differentiable in $[0,1]$ ,$\sup_{x\in[0,1]}|f'(x)| \le M\lt+\infty $