Задача 71445 Найти у''_(xx) от функции, заданной...
Условие
{ x = sin2t
{ y = cos^2t
Решение
[m]\left\{\begin{matrix}x`_{t}=(cos2t)\cdot (2t)`\\ y`_{t}=2cos2t\cdot(cos2t)`\end{matrix}\right.[/m]
[m]\left\{\begin{matrix}
x`_{t}=2cos2t\\ y`_{t}=-4cos2t\cdot sin2t\end{matrix}\right.[/m]
[m]y`_{x}=\frac{y`_{t}}{x`_{t}}=\frac{2cos2t}{(-4cos2t\cdot sin2t)}[/m]
[m]y`_{x}=-\frac{1}{2\cdot sin2t}[/m]
[m]y``_{xx}=\frac{(y`_{x})`_{t}}{x`_{t}}=\frac{(-\frac{1}{2\cdot sin2t})`_{t}}{4cos2t\cdot sin2t}=\frac{(-\frac{1}{2})\cdot(-\frac{1}{ sin^22t})\cdot (sin2t)`_{t}}{4cos2t\cdot sin2t}=\frac{\frac{1}{2})\cdot\frac{1}{ sin^22t})\cdot (2cos2t)}{4cos2t\cdot sin2t}=\frac{1}{4sin^32t}[/m]