$f$ is differentiable. If $\lim_{x \to c}f'(x)$ exists, then this limit must be $f'(c)$.
1371
Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
33304
Let $f$ be a continuous function satisfying $\lim \limits_{n \to \infty}f(x+n) = \infty$ for all $x$. Does $f$ satisfy $f(x) \to \infty$?
117338
Counterexample to second derivative test when f''(x) is not continuously differentiable
85623
Proof of continuity implied by differentiability
152877
Is a monotone differentiable function continuously differentiable?
114747
If $\lim_{x \to \infty}f'(x)=+\infty$ then $\lim_{x \to \infty}(f(x)-f(x-1))=+\infty$ and $\lim_{x \to \infty}f(x)=+\infty$.
20675
$L=\lim_{x\to\infty}(f(x)+f'(x))$ exists . Which of the following statements is\are correct?
135693
Suppose f is a twice differentiable function with $f(0)=0$ and $f(1)=1$, $f'(0)=f'(1)=0$. Then show that $|f''(x)| > 4$ for some $x$ in $(0,1)$
149764
If $f$ s twice differentiable and satisfies the following constraints, prove $f'(0)>-\sqrt 2$
103706
If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$
138544
For which values of $x$, will $f (x) . f '(x) < 0$?
129716
Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
135610
How to give an example of a $f$ differentiable in a deleted neighborhood of $x_0$ such that $\lim_{x\to x_0}f^\prime(x)$ doesn't exist?
143354
Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
16375
Is the function differentiable at $0$?
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
15032
Let $f: \Bbb R \to \Bbb R$ be a differentiable function such that $\sup_{x \in \Bbb R}|f'(x)| \lt \infty$. Then