3 - Second-order PDs

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Calculus of Multivariable Functions

\(f(x) = x^{2} + 4xy + y^{2}\)
\(f(x,y) = x^{2}(3x + 2y)\)
\(f(x,y) = 10x^{2}y - 5xy^{2}\)
\(f(x,y) = x^{0.4}y^{0.4}\)
\(f(x,y) = x^{0.4}y^{0.6}\)

\(f_{x} = 2x + 4y\)

\(\;f_{y} = 4x + 2y\)

\(f_{xx} = 2\)

\(f_{yy} = 2\)

\(f_{xy} = f_{yx} = 4\)
\(f_{x} = 9x^{2} + 4xy\)

\(\;f_{y} = 2x^{2}\)

\(f_{xx} = 18x + 4y\)

\(f_{yy} = 0\)

\(f_{xy} = f_{yx} = 4x\)
\(f_{x} = 20xy - 5y^{2}\)

\(\;f_{y} = 10x^{2} - 10xy\)

\(f_{xx} = 20y\)

\(f_{yy} = -10x\)

\(f_{xy} = f_{yx} = 20x - 10y\)
\(f_{x} = 0.4x^{-0.6}y^{0.4}\)

\(\;f_{y} = 0.4x^{0.4}y^{-0.6}\)

\(f_{xx} = -0.24x^{-1.6}y^{0.4}\)

\(f_{yy} = -0.24x^{0.4}y^{-1.6}\)

\(f_{xy} = f_{yx} = 0.16x^{-0.6}y^{-0.6}\)
\(f_{x} = 0.4x^{-0.6}y^{0.6}\)

\(\;f_{y} = 0.6x^{0.4}y^{-0.4}\)

\(f_{xx} = -0.24x^{-1.6}y^{0.6}\)

\(f_{yy} = -0.24x^{0.6}y^{-1.6}\)

\(f_{xy} = f_{yx} = 0.24x^{-0.6}y^{-0.4}\)