Given the function graphed, which is the graph of the derivative?

True or False: If  $f\left(x\right)\ =\ x^2\ +\ x$  , then  $f'\left(x\right)$  exists for every real number x. 

True or False: If $\lim_{x\rightarrow0^-}\ f'\left(x\right)\ =\ \lim_{x\rightarrow0^+}\ f'\left(x\right)$ , then  $f$  is differentiable at 0. 

Let $f\left(x\right)\ =\ 1-3x^2$ . Which of the following is equal to  $f'\left(1\right)$  ?

True or False: If   $f$  has a derivative at  $x\ =\ a$  , then  $f$  is continuous at  $x\ =\ a$  ?

If $f\left(x\right)\ =\ 10x\ +64$  then  $f'\left(x\right)\ =\ ?$  

Find $\frac{dy}{\text{}dx}$ , If $\ y=\ \frac{x^3}{3}\ +\ \frac{x^2}{2}+x$ .

Let $f\left(x\right)\ =\ \left|x+1\right|$  . Which of the following statements about  $f$  is or are true?
I.  $f$  is continuous at x = -1
II.  $f$  is differentiable at x = -1
III.  $f$  has a corner at x = -1

Find $\frac{dy}{\text{}dx}$ , If  $\ y=\left(\ x^2+1\right)\left(x^3+3\right)$  .