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Question Number 7939 by tawakalitu last updated on 24/Sep/16

f(x, y) = xy^3  + 5xy^2  + 2x + 1  find:  f_x  , f_y  , f_(xx)  , f_(yy)  , f_(xy)  , f_(yx)

$${f}\left({x},\:{y}\right)\:=\:{xy}^{\mathrm{3}} \:+\:\mathrm{5}{xy}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{1} \\ $$$${find}:\:\:{f}_{{x}} \:,\:{f}_{{y}} \:,\:{f}_{{xx}} \:,\:{f}_{{yy}} \:,\:{f}_{{xy}} \:,\:{f}_{{yx}} \\ $$

Commented by Rasheed Soomro last updated on 25/Sep/16

What′s meant by f_(xy)  ,f_x  etc?

$${What}'{s}\:{meant}\:{by}\:{f}_{{xy}} \:,{f}_{{x}} \:{etc}? \\ $$

Commented by tawakalitu last updated on 25/Sep/16

Is partial derivative

$${Is}\:{partial}\:{derivative} \\ $$

Commented by tawakalitu last updated on 25/Sep/16

or  f(x), f(xy), f(xx) ......

$${or}\:\:{f}\left({x}\right),\:{f}\left({xy}\right),\:{f}\left({xx}\right)\:...... \\ $$

Answered by prakash jain last updated on 02/Oct/16

f(x, y) = xy^3  + 5xy^2  + 2x + 1  f_x =y^3 +5y^2 +2  f_y =3xy^2 +10xy  f_(xx) =0  f_(yy) =6xy+10x  (∂/∂y)((∂f/∂x))=3y^2 +10x  (∂/∂x)((∂f/∂y))=3y^2 +10y

$${f}\left({x},\:{y}\right)\:=\:{xy}^{\mathrm{3}} \:+\:\mathrm{5}{xy}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{1} \\ $$$${f}_{{x}} ={y}^{\mathrm{3}} +\mathrm{5}{y}^{\mathrm{2}} +\mathrm{2} \\ $$$${f}_{{y}} =\mathrm{3}{xy}^{\mathrm{2}} +\mathrm{10}{xy} \\ $$$${f}_{{xx}} =\mathrm{0} \\ $$$${f}_{{yy}} =\mathrm{6}{xy}+\mathrm{10}{x} \\ $$$$\frac{\partial}{\partial{y}}\left(\frac{\partial{f}}{\partial{x}}\right)=\mathrm{3}{y}^{\mathrm{2}} +\mathrm{10}{x} \\ $$$$\frac{\partial}{\partial{x}}\left(\frac{\partial{f}}{\partial{y}}\right)=\mathrm{3}{y}^{\mathrm{2}} +\mathrm{10}{y} \\ $$

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