## Task Condition

Find the derivative.

$$
y=\frac{x-3}{2} \sqrt{6 x-x^{2}-8}+\arcsin \sqrt{\frac{x}{2}-1}
$$
## Solution

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

$$
\begin{aligned}
& =\frac{6 x-x^{2}-8}{2 \sqrt{6 x-x^{2}-8}}+\frac{3 x-x^{2}-9+3 x}{2 \sqrt{6 x-x^{2}-8}}+\frac{1}{2 \sqrt{4 x-8-x^{2}+2 x}}= \\
& =\frac{6 x-x^{2}-8+3 x-x^{2}-9+3 x}{2 \sqrt{6 x-x^{2}-8}}+\frac{1}{2 \sqrt{6 x-x^{2}-8}}= \\
& =\frac{12 x-2 x^{2}-16}{2 \sqrt{6 x-x^{2}-8}}=\frac{6 x-x^{2}-8}{\sqrt{6 x-x^{2}-8}}=\sqrt{6 x-x^{2}-8}
\end{aligned}
$$

## Problem Kuznetsov Differentiation 10-17