## Task Condition

Find the derivative.

$$
y=3 \arcsin \frac{3}{4 x+1}+2 \sqrt{4 x^{2}+2 x-2}, 4 x+1>0
$$
## Solution

$$
y^{\prime}=\left(3 \arcsin \frac{3}{4 x+1}+2 \sqrt{4 x^{2}+2 x-2}\right)^{\prime}=
$$

$$
\begin{aligned}
& =\frac{3}{\sqrt{1-\left(\frac{3}{4 x+1}\right)^{2}}}+\frac{2}{2 \sqrt{4 x^{2}+2 x-2}} \cdot(8 x+2)= \\
& =\frac{3(4 x+1)}{\sqrt{(4 x+1)^{2}-3^{2}}}+\frac{8 x+2}{\sqrt{4 x^{2}+2 x-2}}=\frac{3(4 x+1)}{\sqrt{16 x^{2}+8 x+1-9}}+\frac{8 x+2}{\sqrt{4 x^{2}+2 x-2}}= \\
& =\frac{3(4 x+1)}{2 \sqrt{4 x^{2}+2 x-2}}+\frac{8 x+2}{\sqrt{4 x^{2}+2 x-2}}=\frac{12 x+3}{2 \sqrt{4 x^{2}+2 x-2}}+\frac{16 x+4}{2 \sqrt{4 x^{2}+2 x-2}}= \\
& =\frac{28 x+7}{2 \sqrt{4 x^{2}+2 x-2}}=\frac{7 \cdot(4 x+1)}{2 \sqrt{4 x^{2}+2 x-2}}
\end{aligned}
$$

## Problem Kuznetsov Differentiation 14-5