Consider the function $g ( x , y ) = x ^ { \frac { 1 } { 5 } } y ^ { \frac { 1 } { 5 } }$ (a)Find g_x(x, y)and g_y(x, y)for (x, y) $\neq$ (0, 0).
(b)Use the limit definition of partial derivative to show that g_x(0, 0)= 0 and g_y(0, 0)= 0.
(c)Are the functions g_x and g_y continuous at (0, 0)? Explain.
(d)Is g differentiable at (0, 0)? Explain.

Question

Consider the function

g ( x , y ) = x ^ { \frac { 1 } { 5 } } y ^ { \frac { 1 } { 5 } }

(a)Find g_x(x, y)and g_y(x, y)for (x, y)

\neq

(0, 0).
(b)Use the limit definition of partial derivative to show that g_x(0, 0)= 0 and g_y(0, 0)= 0.
(c)Are the functions g_x and g_y continuous at (0, 0)? Explain.
(d)Is g differentiable at (0, 0)? Explain.

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The Answer of Consider the function $g ( x , y ) = x ^ { \frac { 1 } { 5 } } y ^ { \frac { 1 } { 5 } }$ (a)Find g_x(x, y)and g_y(x, y)for (x, y)$\neq$(0, 0). (b)Use the limit definition of partial derivative to show that g_x(0, 0)= 0 and g_y(0, 0)= 0. (c)Are the functions g_x and g_y continuous at (0, 0)? Explain. (d)Is g differentiable at (0, 0)? Explain.

Consider the Function g(x,y)=x15y15g ( x , y ) = x ^ { \frac { 1 } { 5 } } y ^ { \frac { 1 } { 5 } }g(x,y)=x51​y51​

Consider the Function $g ( x , y ) = x ^ { \frac { 1 } { 5 } } y ^ { \frac { 1 } { 5 } }$