## Task Condition

Find the derivative.

$$
y=2 \arcsin \frac{2}{3 x+4}+\sqrt{9 x^{2}+24 x+12}, 3 x+4>0
$$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(2 \arcsin \frac{2}{3 x+4}+\sqrt{9 x^{2}+24 x+12}\right)^{\prime}= \\
& =2 \cdot \frac{1}{\sqrt{1-\left(\frac{2}{3 x+4}\right)^{2}}}+\frac{1}{2 \sqrt{9 x^{2}+24 x+12}} \cdot(18 x+24)= \\
& =2 \cdot \frac{3 x+4}{\sqrt{(3 x+4)^{2}-4}}+\frac{6(3 x+4)}{2 \sqrt{9 x^{2}+24 x+12}}= \\
& =\frac{6 x+8}{\sqrt{9 x^{2}+24 x+12}}+\frac{3(3 x+4)}{\sqrt{9 x^{2}+24 x+12}}= \\
& =\frac{6 x+8+18 x+24}{\sqrt{9 x^{2}+24 x+12}}=\frac{8(3 x+4)}{\sqrt{9 x^{2}+24 x+12}}
\end{aligned}
$$

## Problem Kuznetsov Differentiation 14-7