Basic Derivatives

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

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

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

• True

• False

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

• False

• True

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

• 6

• -5

• -2

• -6

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

• False

• True

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

• 64

• x

• 10

• 10x

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

• $x^2+x$

• $x^2+x+1$

• $3x^2+2x+1$

• $x^3+x^2\ +1$

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

• I only

• III only

• I and III only

• I and II only

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

•  $2x\ +\ 3x^2$  

•  $6x^3$  

•  $2x^2+2x\ +3x^{5\ }+\ 9x^2$  

•  $5x^4+3x^2+6x$