$f$ differentiable, $c\in(a,b)$ $\Rightarrow \exists x_1,x_2$, $a\leq x_1<x_2\leq b$ : $f'(c)=\frac{f(x_2)-f(x_1)}{x_2-x_1}$.
116027
Is inverse use of mean value theorem right?
106013
Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable
154881
Question on differentiability of a continuous function
1371
Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
56936
$\frac{\partial f_i}{x_j}=\frac{\partial f_j}{x_i}\implies(f_1,\ldots,f_n)$ is a gradient
33523
If $f(0)=0$ and $f'$ is increasing, then $\frac{f(x)}{x}$ is increasing.
96973
If $F(0,0)=0$ and $F(x,y)= \frac{xy}{x^2+y^2}$ for $(x,y)\neq (0,0)$ then $F$ is differentiable at $(0,0)$?
38923
What can we say about differentiability of $f$ at $c$?
20273
Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
48592
Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
151819
A twice differentiable function $f$ satisfies $f′′(x)+f(x)=−xg(x)f′(x)$, $\forall x\ge 0$.
111186
Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$
52858
Find $f$ if $ f(x)+f\left(\frac{1}{1-x}\right)=x $
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
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