Задача 60165 Вычислить производную...
Условие
математика ВУЗ
229
Решение
★
[m]y`_{t}=\frac{t`\cdot (\sqrt{1-t^2})-t\cdot (\sqrt{1-t^2})`}{(\sqrt{1-t^2})^2}=\frac{1\cdot (\sqrt{1-t^2})-t\cdot \frac{1}{2\sqrt{1-t^2}}\cdot (1-t^2)`}{1-t^2}=\frac{\sqrt{1-t^2}-t\cdot \frac{(-2t)}{2\sqrt{1-t^2}}}{1-t^2}=\frac{\sqrt{1-t^2}+ \frac{(t^2}{\sqrt{1-t^2}}}{1-t^2}=[/m]
[m]\frac{ \frac{\sqrt{1-t^2}\cdot \sqrt{1-t^2}+(t^2}{\sqrt{1-t^2}}}{1-t^2}=\frac{1-t^2+t^2}{\sqrt{1-t^2}\cdot (1-t^2)}=\frac{1}{\sqrt{1-t^2}\cdot (1-t^2)}[/m]
[m]x`_{t}=2arcsint\cdot (arcsint)`=2arcsint \cdot \frac{1}{\sqrt{1-t^2}}[/m]
⇒
[m]y`=\frac{\frac{1}{\sqrt{1-t^2}\cdot (1-t^2)}}{2arcsint \cdot \frac{1}{\sqrt{1-t^2}}}=\frac{1}{2(1-t^2)arcsint}[/m]