Question

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Answer 1

$1)\ \ z=sin2x+y^2x^3+\dfrac{y}{x}+5\\\\z'_{x}=2cos2x+3x^2y^2-\dfrac{y}{x^2}\ \ ,\ \ \ \ z'_{y}=2x^3y+\dfrac{1}{x}\\\\\\z''_{xx}=-4sin2x+6xy^2-\dfrac{-2xy}{x^4}=-4sin2x+6xy^2+\dfrac{2y}{x^3}\\\\z''_{xy}=6x^2y\ \ \ ,\ \ \ \ z''_{yy}=2x^3\ \ \ ,\ \ \ z''_{yx}=6x^2y$

$2)\ \ u=cos^2x+xz+xy^2-10\\\\u'_{x}=2cosx\cdot (-sinx)+z+y^2=-sin2x+z+y^2\\\\u'_{y}=2xy\ \ ,\ \ \ u'_{z}=x$

$3)\ \ z=e^{y}\cdot \Big(3^{x}+\dfrac{y^2}{e^{y}}\Big)\\\\z'_{x}=e^{y}\cdot 3^{x}\, ln3\ \ \ ,\\\\z'_{y}=e^{y}\cdot \Big(3^{x}+\dfrac{y^2}{e^{y}}\Big)+e^{y}\cdot \dfrac{2y\, e^{y}-y^2\, e^{y}}{e^{2y}}=e^{y}\cdot \Big(3^{x}+\dfrac{y^2}{e^{y}}\Big)+\dfrac{y\, e^{y}(2-y)}{e^{y}}=\\\\=e^{y}\cdot \Big(3^{x}+\dfrac{y^2}{e^{y}}\Big)+y(2-y)=e^{y}\cdot 3^{x}+y^2+2y-y^2=e^{y}\cdot 3^{x}+2y\\\\z'_{xx}=e^{y}\cdot 3^{x}\, ln^23\\\\z''_{xy}=e^{y}\cdot 3^{x}\, ln3\\\\z''_{yy}=e^{y}\cdot 3^{x}+2$$d^2z=z''_{xx}\cdot dx^2+2\, z''_{xy}\cdot dx\, dy+z''_{yy}\cdot dy^2\\\\d^2z=\Big(e^{y}\cdot 3^{x}\, ln^23\Big)dx^2+\Big(e^{y}\cdot 3^{x}\, ln3\Big)\, dx\, dy+\Big(e^{y}\cdot 3^{x}+2\Big)\, dy^2$