## Problem Statement

Find the derivative.

$$
y=x^{2} \arccos x-\frac{x^{2}+2}{3} \sqrt{1-x^{2}}
$$
## Solution

$y^{\prime}=\left(x^{2} \arccos x-\frac{x^{2}+2}{3} \sqrt{1-x^{2}}\right)^{\prime}=$
$=2 x \cdot \arccos x+x^{2} \cdot \frac{-1}{\sqrt{1-x^{2}}}-\frac{2 x}{3} \sqrt{1-x^{2}}-\frac{x^{2}+2}{3} \frac{1}{2 \sqrt{1-x^{2}}} \cdot(-2 x)=$

$=2 x \cdot \arccos x-\frac{x^{2}}{\sqrt{1-x^{2}}}-\frac{2 x\left(1-x^{2}\right)}{3 \sqrt{1-x^{2}}}+\frac{\left(x^{2}+2\right) x}{3 \sqrt{1-x^{2}}}=$

$=2 x \cdot \arccos x-\frac{3 x^{2}+2 x-2 x^{3}-x^{3}-2 x}{3 \sqrt{1-x^{2}}}=$

$=2 x \cdot \arccos x-\frac{3 x^{2}-3 x^{3}}{3 \sqrt{1-x^{2}}}=2 x \cdot \arccos x-\frac{x^{2}(1-x)}{\sqrt{1-x^{2}}}=$

$=2 x \cdot \arccos x-x^{2} \cdot \sqrt{\frac{1-x}{1+x}}$

## Problem Kuznetsov Differentiation 14-26