## Task Condition

Find the derivative $y_{x}^{\prime}$.

$$
\left\{\begin{array}{l}
x=\sqrt{2 t-t^{2}} \\
y=\arcsin (t-1)
\end{array}\right.
$$
## Solution

$x_{t}^{\prime}=\left(\sqrt{2 t-t^{2}}\right)^{\prime}=\frac{1}{2 \sqrt{2 t-t^{2}}} \cdot(2-2 t)=\frac{1-t}{\sqrt{2 t-t^{2}}}$

$y_{t}^{\prime}=(\arcsin (t-1))^{\prime}=\frac{1}{\sqrt{1-(t-1)^{2}}}=\frac{1}{\sqrt{1-t^{2}-2 t-1}}=\frac{1}{\sqrt{-t^{2}-2 t}}$

We obtain:

$$
\begin{aligned}
& y_{x}^{\prime}=\frac{y_{t}^{\prime}}{x_{t}^{\prime}}=\left(\frac{1}{\sqrt{-t^{2}-2 t}}\right) /\left(\frac{1-t}{\sqrt{2 t-t^{2}}}\right)=\frac{\sqrt{2 t-t^{2}}}{(1-t) \sqrt{-t^{2}-2 t}}= \\
& =\frac{1}{(1-t)} \cdot \sqrt{\frac{2 t-t^{2}}{-t^{2}-2 t}}=\frac{1}{(1-t)} \cdot \sqrt{\frac{2-t}{-t-2}}=\frac{1}{(1-t)} \cdot \sqrt{\frac{t-2}{t+2}}
\end{aligned}
$$

## Problem Kuznetsov Differentiation 16-6