## Problem Statement

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

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

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

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

We obtain:

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

## Problem Kuznetsov Differentiation 16-23