Assuming that $f''(a)$ exists, show that: $f''(a)=\lim_{h\to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}$
30723
Second derivative "formula derivation"
32743
How to prove that $\lim_{h\to 0} \, \frac{e^h-1}{h}$
151109
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.
150699
Show that if $f'$ is strictly increasing, then $\frac{f(x)}{x}$ is increasing over $(0,\infty)$
28399
Showing $f$ is constant
139453
Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$
83707
Suppose that $f$ is differentiable on $\mathbb{R}$ and $\lim_{x\to \infty}f'(x)=M$. Show that $\lim_{x\to \infty}f(x+1)-f(x)$ exists and find it.
44370
If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
15262
Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.
16418
Showing that $\frac{f(x)-f(x-h)}{h} \le f'(x) \le \frac{f(x+h)-f(x)}{h}$ given that f' is increasing
127125
Can the limit $\lim_{h \to 0}\frac{f(x + h) - 2f(x) + f(x - h)}{h^2}$ exist if $f'(x)$ does not exist at $x$?
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
109099
Prove that $\lim_{h \to 0}\frac{1}{h}\int_0^h{\cos{\frac{1}{t}}dt} = 0$
109907
How to prove that $\lim\limits_{h \to 0} \frac{a^h - 1}{h} = \ln a$
17154
Does a non-trivial solution exist for $f'(x)=f(f(x))$?
8854
Prove $f $ is identically zero if $f(0)=0$ and $|f'(x)|\le|f(x)|$