Let f(x, y) = sin(xy²) + \(e^{{x^3}+5y}\) for all ∈ R². Calculate \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ { \partial y\partial x } } \)and \(\frac { { \partial }^{ 2 }f }{ { \partial x\partial y } } \)

Let w(x, y) = xy+\(\frac { { e }^{ y } }{ { y }^{ 2 }+1 } \) for all (x, y) ∈ R². Calculate \(\frac { { \partial }^{ 2 }w }{ { \partial y\partial x } } \) and \(\frac { { \partial }^{ 2 }w }{ { \partial x\partial y } } \)

For each of the following functions find the f_x, f_y, and show that f_xy = f_yx
f(x, y) = \(\frac { 3x }{ y+sinx \ } \) 

If U(x, y, z) = log (x³ + y³ + z³), find \(\frac { \partial U }{ \partial x } +\frac { \partial U }{ \partial y } +\frac { \partial U }{ \partial z } \)

If V(x,y) = e^x(x cos y - y siny), then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } =\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } \) = 0