Let f (x, y) = 0 if xy ≠ 0 and f (x, y) = 1 if xy = 0.
Calculate: \(\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).\)

Evaluate \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}cos=\left( \frac { { e }^{ x }siny }{ y } \right) \), if the limit exists.

For each of the following functions find the f_x, f_y and show that f_xy = f_yx
f(x, y) = cos (x² - 3xy)

For each of the following functions find the g_xy, g_xx, g_yy and g_yx.
g(x, y) = xe^y + 3x²y

For each of the following functions find the g_xy, g_xx, g_yy and g_yx.
g(x, y) = log (5x + 3y)

Let z(x, y) = x²y + 3xy⁴, x, y ∈ R. Find the linear approximation for z at (2, -1).

Let W(x, y, z) = x² - xy + 3 sin z, x, y, z ∈ R. Find the linear approximation for U at (2, -1, 0).

Verify the above theorem for F(x, y) = x² - 2y² + 2xy and x(t) = cos t, y(t) = sin t, t ∈ [0, 2\(\pi\)]

If w (x, y, z) = x² + y² + y², x = e^t, y = e^t sin t, z = e^t cos t, find \(\frac{dw}{dt}\)

If u(x, y) = \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ \sqrt { x+y } } \), prove that \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } =\frac { 3 }{ 2 } u\)