1) Найти производную (dy)/(dx) от функции, заданной параметрически.
х=√(1-t²)
{
y=tg√(1+t)
2)Найти производную (d²y)/(dx²) от функции, заданной параметрически
x=cost+sint
{
y=sin2t
y`_(x)=y`_(t)/x`_(t)
y`_(t)=(tgsqrt(1+t))`=(1/cos^2sqrt(1+t))*(sqrt(1+t)`=
=(1/cos^2sqrt(1+t))*(1/2sqrt(1+t))*(1+t)`=
=1/(2sqrt(1+t)cos^2sqrt(1+t))
x`_(t)=(1/2sqrt(1-t^2))*(1-t^2)`=(1/2sqrt(1-t^2))*(-2t)
y`_(x)= -sqrt(1+t)/(2tcos^2sqrt(1+t))
2.
y`_(x)=y_`(t)/x`_(t)
y`_(t)=(sin2t)`=(cos2t)*(2t)`=2cos2t
x`_(t)=(cost+sint)`=(cost)`+(sint)`=-sint+cost=cost-sint
y`_(x)=2cos2t/(cost-sint)
y``_(xx)=(y`(x))`_(t)/x`_(t)
(y`(x))`_(t)=(2cos2t/(cost-sint))`_(t)=
=((2cos2t)`*(cost-sint)-2cos2t*(cost-sint)`)/(cost-sint)^2=
=(-4sin2t*cost+4sin2t*sint-2cos2t(-sint)-2cos2t*(-cost))/(cost-sint)^2=
y``_(xx)=(-4sin2t*cost+4sin2t*sint-2cos2t(-sint)-2cos2t*(-cost))/(cost-sint)^3