Differentiation

Answer all the questions below and press submit to see how many you got right.

id: 17580

A function is called differentiable in point $a=12$ if which of the following is true?

  • The limit $\lim \limits_{x\to 0}\frac{f(12+x)-f(12)}{x}$ exists.
    $\lim \limits_{x\to 0}\frac{f(x+12)-f(x)}{x}=f(12).$
    $\lim \limits_{x\to 12}f(x)=f(12).$
    The limit $\lim \limits_{x\to a}f(x)$ exists.
id: 17641

Which of the following statements are true for two differentiable functions $f$ and $g$?

  • $2\cdot(f+g)$ is differentiable.
    $f\circ g$ is differentiable.
    $f\cdot g$ is differentiable.
    $f+15$ differentiable.
id: 17607

$$\lim \limits_{x\to0} \frac{(1+x)^2-1^2}{x}=$$ $$=\lim \limits_{x\to0} \frac{1^2+2\cdot 1\cdot x +x^2-1^2}{x}=$$ $$=\lim \limits_{x\to0} \frac{2\cdot 1\cdot x +x^2}{x}=$$ $$=\lim \limits_{x\to0} 2\cdot 1 +x=?$$