Question

Алгебра. Решите пожалуйста

Answer 1

6.    $z = arctg{ frac{x}{y} } ; x = e^{2t} + t^5 , y = e^{2t} - 1 ;$

$frac{dz}{dt} = frac{d}{dt} arctg{ frac{x}{y} } = frac{1}{ 1 + (x/y)^2 } cdot frac{d}{dt} frac{x}{y} = frac{y^2}{ x^2 + y^2 } cdot frac{ x'_t y - y'_t x }{ y^2 } = \\ = frac{ ( e^{2t} + t^5 )'_t y - ( e^{2t} - 1 )'_t x }{ x^2 + y^2 } = frac{ 2e^{2t} y + 5t^4 y - 2e^{2t} x }{ x^2 + y^2 } = \\ = frac{ 2e^{2t} ( y - x ) + 5t^4 y }{ [ e^{2t} + t^5 ]^2 + [ e^{2t} - 1 ]^2 } = frac{ 2e^{2t} ( -1 - t^5 ) + 5t^4 ( e^{2t} - 1 ) }{ e^{4t} + 2 e^{2t} t^5 + t^{10} + e^{4t} - 2 e^{2t} + 1 } = \\ = frac{ -2e^{2t} - 2e^{2t} t^5 + 5t^4 e^{2t} - 5t^4 }{ 2e^{4t} + 2 e^{2t} ( t^5 - 1 ) + t^{10} + 1 } = frac{ 5t^4 ( e^{2t} - 1 ) - 2e^{2t} ( 1 + t^5 ) }{ 2e^{4t} + 2 e^{2t} ( t^5 - 1 ) + t^{10} + 1 } ;$

$frac{dz}{dt} = frac{ 5t^4 ( e^{2t} - 1 ) - 2e^{2t} ( 1 + t^5 ) }{ 2e^{4t} + 2 e^{2t} ( t^5 - 1 ) + t^{10} + 1 } ;$




7.    $z = x^2 y^2 ; x = u + 2v ; y = frac{v}{u} ;$

$z = (xy)^2 = ( ( u + 2v ) frac{v}{u} )^2 = ( u cdot frac{v}{u} + 2v cdot frac{v}{u} )^2 = ( v + frac{ 2v^2 }{u} )^2 ;$


$frac{ partial z }{ partial u } = frac{ partial }{ partial u } ( v + frac{ 2v^2 }{u} )^2 = 2 ( v + frac{ 2v^2 }{u} ) cdot frac{ partial }{ partial u } ( v + frac{ 2v^2 }{u} ) = \\ = 2 ( v + frac{ 2v^2 }{u} ) cdot ( -frac{ 2v^2 }{u^2} ) = -4 ( frac{v}{u} )^2 ( v + frac{ 2v^2 }{u} ) ;$


$frac{ partial z }{ partial v } = frac{ partial }{ partial v } ( v + frac{ 2v^2 }{u} )^2 = 2 ( v + frac{ 2v^2 }{u} ) cdot frac{ partial }{ partial v } ( v + frac{ 2v^2 }{u} ) = \\ = 2 ( v + frac{ 2v^2 }{u} ) cdot ( 1 + frac{ 4v }{u} ) = 2 ( v + frac{ 2v^2 }{u} ) ( 1 + frac{ 4v }{u} ) ;$