## Problem Statement

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

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

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

$=-\frac{1}{\sqrt{1-1+t^{2}}} \cdot \frac{t}{\sqrt{1-t^{2}}}=-\frac{1}{t} \cdot \frac{t}{\sqrt{1-t^{2}}}=-\frac{1}{\sqrt{1-t^{2}}}$

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

We obtain:

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