Differentiation

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id: 17635

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

  • $f+20$ differentiable.
    $20\cdot(f+g)$ is differentiable.
    $f\cdot g$ is differentiable.
    $f\circ g$ is differentiable.
id: 17610

$$\lim \limits_{x\to0} \frac{(4+x)^2-4^2}{x}=$$ $$=\lim \limits_{x\to0} \frac{4^2+2\cdot 4\cdot x +x^2-4^2}{x}=$$ $$=\lim \limits_{x\to0} \frac{2\cdot 4\cdot x +x^2}{x}=$$ $$=\lim \limits_{x\to0} 2\cdot 4 +x=?$$

id: 17576

A function is called differentiable in point $a=1$ if which of the following is true?

  • The limit $\lim \limits_{x\to a}f(x)$ exists.
    $\lim \limits_{x\to 0}\frac{f(x+1)-f(x)}{x}=f(1).$
    $\lim \limits_{x\to 1}f(x)=f(1).$
    The limit $\lim \limits_{x\to 0}\frac{f(1+x)-f(1)}{x}$ exists.