## Problem Statement

Find the derivative.

$$
y=\sqrt{1-3 x-2 x^{2}}+\frac{3}{2 \sqrt{2}} \arcsin \frac{4 x+3}{\sqrt{17}}
$$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(\sqrt{1-3 x-2 x^{2}}+\frac{3}{2 \sqrt{2}} \arcsin \frac{4 x+3}{\sqrt{17}}\right)^{\prime}= \\
& =\frac{1}{2 \sqrt{1-3 x-2 x^{2}}} \cdot(-3-4 x)+\frac{3}{2 \sqrt{2}} \cdot \frac{1}{\sqrt{1-\left(\frac{4 x+3}{\sqrt{17}}\right)^{2}}} \cdot \frac{4}{\sqrt{17}}= \\
& =\frac{-3-4 x}{2 \sqrt{1-3 x-2 x^{2}}}+\frac{3 \sqrt{2}}{\sqrt{17-(4 x+3)^{2}}}= \\
& =\frac{-3-4 x}{2 \sqrt{1-3 x-2 x^{2}}}+\frac{3 \sqrt{2}}{\sqrt{8-16 x^{2}-24 x}}= \\
& =\frac{-3-4 x}{2 \sqrt{1-3 x-2 x^{2}}}+\frac{3 \sqrt{2}}{2 \sqrt{2} \cdot \sqrt{1-2 x^{2}-3 x}}= \\
& =\frac{-3-4 x+3}{2 \sqrt{1-3 x-2 x^{2}}}=-\frac{4 x}{2 \sqrt{1-3 x-2 x^{2}}}=-\frac{2 x}{\sqrt{1-3 x-2 x^{2}}}
\end{aligned}
$$

## Problem Kuznetsov Differentiation 14-10