## Problem Statement

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

$$
\left\{\begin{array}{l}
x=2(t-\sin t) \\
y=4(2+\cos t)
\end{array}\right.
$$
## Solution

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

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

$y^{\prime}(t)=4(2+\cos (t))^{\prime}=-4 \sin (t)$

$x^{\prime}(t)=2(t-\sin (t))^{\prime}=2(1-\cos (t))$

$y_{x}^{\prime}=\frac{-4 \sin (t)}{2(1-\cos (t))}=\frac{-2 \sin (t)}{1-\cos (t)}$

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

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

## Problem Kuznetsov Differentiation 20-27