## Problem Statement

Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.

$$
\left\{\begin{array}{l}
x=t+\sin t \\
y=2-\cos t
\end{array}\right.
$$
## Solution

$y_{x}^{\prime}=\frac{y^{\prime}(t)}{x^{\prime}(t)}$

$y_{x^{2}}^{\prime \prime}=\frac{\left(y_{x}^{\prime}\right)_{t}^{\prime}}{x^{\prime}(t)}$

$y^{\prime}(t)=\sin t$.

$x^{\prime}(t)=1+\cos t$.

$y_{x}^{\prime}=\frac{\sin t}{1+\cos t}$

$y_{x^{2}}^{\prime \prime}=\left(\frac{\sin t}{1+\cos t}\right)^{\prime} \cdot \frac{1}{1+\cos t}=\frac{\cos t(1+\cos t)-\sin t(-\sin t)}{(1+\cos t)^{3}}=$

$=\frac{\cos t+\cos ^{2} t+\sin ^{2} t}{(1+\cos t)^{3}}=\frac{1}{(1+\cos t)^{2}}$.

Kuznetsov Differentiation Problem 20-5