1. Вычислить значения частных производных f’x (M0), f’y (M0), f’z (M0) для данной функции...

1)
$f(x,y,z)=arcsin( \frac{x^2}{y} - z)$

$f'_x= \frac{1}{ \sqrt{1-(\frac{x^2}{y} - z)} } * \frac{2x}{y}$
$f'_x(M_0)=\frac{1}{ \sqrt{1-(\frac{2^2}{5} - 0)} } * \frac{2*2}{5} = \frac{4}{5}* \frac{1}{ \sqrt{1- \frac{4}{5} } } = \frac{4}{5}* \frac{1}{ \sqrt{\frac{1}{5} } } =0.8*2.2361=1.79$

$f'_y= \frac{1}{ \sqrt{1-(\frac{x^2}{y} - z)} } * (-\frac{x^2}{y^2} )$
$f'_y(M_0)=\frac{1}{ \sqrt{1-(\frac{2^2}{5} - 0)} } * (-\frac{2^2}{5^2} )= \frac{1}{ \sqrt{ \frac{1}{5} } } * (-\frac{4}{25} )=-0.16*2.2361=-0.36$

$f'_z=-\frac{1}{ \sqrt{1-(\frac{x^2}{y} - z)} }$
$f'_z(M_0)=-\frac{1}{ \sqrt{1-(\frac{2^2}{5} - 0)} }= \frac{1}{ \sqrt{ \frac{1}{5} } } =2.24$

2)
$u=x^2e^{-y}=sin^2t*e^{-sin^2t}$
$u'=2sint*cost*e^{-sin^2t}-2sint*cost*e^{-sin^2t}*sin^2t=$
$=2sint*cost*e^{-sin^2t}(1-sin^2t)=2sint*cost*e^{-sin^2t}*cos^2t=$
$=2sint*cos^3t*e^{-sin^2t}$

$u'(t_0)=2sin( \frac{ \pi }{2} )*cos^3(\frac{ \pi }{2} )*e^{-sin^2(\frac{ \pi }{2} )}=2*1*0*e^{-1}=0$