# Question 59f5a

Feb 6, 2018

$\setminus$

$\setminus m b \otimes \left\{A n s w e r\right. \setminus \quad \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{x}^{2} + \left(2 {x}^{3} + 2 x\right) \arctan \left(x\right)}{{x}^{2} + 1} .$

#### Explanation:

$\setminus$

$\setminus m b \otimes \left\{T h i s b a s i c a l l y w i l l \ne e d 3 \in g red i e n t s\right. \setminus \setminus \setminus m b \otimes \left\{\prod u c t r \underline{e} , \arctan r \underline{e} , s i m p l \mathmr{if} i c a t i o n .\right\}$

$\setminus m b \otimes \left\{W e a r e g i v e n\right. \setminus q \quad y = {x}^{2} \arctan \left(x\right) .$

$\setminus m b \otimes \left\{T h u s\right.$

 \mbox{1) product rule:} \qquad {dy}/{dx} = x^2 d/{dx} [ arctan(x) ] + arctan(x) d/{dx} [ x^2 ] 
$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus = {x}^{2} \frac{d}{\mathrm{dx}} \left[\arctan \left(x\right)\right] + \left[\arctan \left(x\right)\right] \left[2 x\right]$
$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus = {x}^{2} \frac{d}{\mathrm{dx}} \left[\arctan \left(x\right)\right] + 2 x \arctan \left(x\right) .$

 \mbox{2) arctan rule:} \qquad \quad {dy}/{dx} = x^2 ( 1 / { x^2 + 1 } ) + 2x arctan(x). 

 \mbox{3) simplification:} \quad {dy}/{dx} = x^2 / { x^2 + 1 } + { 2x ( x^2 + 1 ) arctan(x) } / { x^2 + 1 } 

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus = \frac{{x}^{2} + 2 x \left({x}^{2} + 1\right) \arctan \left(x\right)}{{x}^{2} + 1}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus = \frac{{x}^{2} + \left(2 {x}^{3} + 2 x\right) \arctan \left(x\right)}{{x}^{2} + 1} .$

 \mbox{4) Final Answer:} \quad {dy}/{dx} = { x^2 + ( 2 x^3 + 2 x ) arctan(x) } / { x^2 + 1 }. #