Question

How can check derivative exist or not in two variable?

Answer 1

First one ff is the ratio of two differentiable functions, the denominator one not vanishing in the neighborhood of the origin. Hence ff is differentiable at the origin.

Second one Using a theorem stating that if ff is continuous in an open set UU and has continuous partial derivatives in UU then ff is continuously differentiable at all points in UU.

Third one Using the definition of the derivative, prove that

lim(h,k)→(0,0)f(h,k)−f(0,0)+h−kh2+k2−−−−−−√=0

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