## Task Condition

Find the derivative.

$$
y=\left(3 x^{2}-4 x+2\right) \sqrt{9 x^{2}-12 x+3}+(3 x-2)^{4} \cdot \arcsin \frac{1}{3 x-2}, 3 x-2>0
$$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(\left(3 x^{2}-4 x+2\right) \sqrt{9 x^{2}-12 x+3}+(3 x-2)^{4} \cdot \arcsin \frac{1}{3 x-2}\right)^{\prime}= \\
& =(6 x-4) \sqrt{9 x^{2}-12 x+3}+\left(3 x^{2}-4 x+2\right) \cdot \frac{1}{2 \sqrt{9 x^{2}-12 x+3}} \cdot(18 x-12)+ \\
& +4(3 x-2)^{3} \cdot 3 \cdot \arcsin \frac{1}{3 x-2}+(3 x-2)^{4} \cdot \frac{1}{\sqrt{1-\left(\frac{1}{3 x-2}\right)^{2}}} \cdot\left(\frac{-1}{(3 x-2)^{2}}\right) \cdot 3= \\
& =\frac{(6 x-4) \cdot\left(9 x^{2}-12 x+3\right)}{\sqrt{9 x^{2}-12 x+3}}+\frac{\left(3 x^{2}-4 x+2\right) \cdot(9 x-6)}{\sqrt{9 x^{2}-12 x+3}}+ \\
& +12(3 x-2)^{3} \cdot \arcsin \frac{1}{3 x-2}-\frac{3(3 x-2)^{3}}{\sqrt{(3 x-2)^{2}-1}}= \\
& =\frac{(3 x-2) \cdot\left(18 x^{2}-24 x+6+9 x^{2}-12 x+6-3(3 x-2)^{2}\right)}{\sqrt{(3 x-2)^{2}-1}}+12(3 x-2)^{3} \cdot \arcsin \frac{1}{3 x-2}= \\
& =\frac{(3 x-2) \cdot\left(27 x^{2}-36 x+12-27 x^{2}+36 x-12\right)}{\sqrt{(3 x-2)^{2}-1}}+12(3 x-2)^{3} \cdot \arcsin \frac{1}{3 x-2}= \\
& =12(3 x-2)^{3} \cdot \arcsin \frac{1}{3 x-2}
\end{aligned}
$$

Kuznetsov Differentiation 13-16