Doubt regarding partial derivatives

How can we check if a function has a Partial Derivative at all points of its domain of definition?

For example, we can check if a function is Differentiable at a point by checking if the Right Hand and Left Hand Derivatives are equal. But what do we do when we have to check for the existence of Partial Derivatives? Could somebody please shed light on this? Any help would be truly appreciated.

Note by Ishan Dasgupta Samarendra
4 years, 2 months ago

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It works the same way as the normal derivative works. Just apply the definition of a derivative that is the first principle such that all the other variables are assumed to be a constant and the limit is taken over only the variable with which the function is to be differentiated. So if there is a function ff in variables xx and yy, then the partial derivative of ff with respect to xx at a point (x0,y0) (x_0 , y_0) is given by f(x,y)x=limh0f(x0+h,y0)f(x0,y0)h \frac{\partial f(x,y)}{\partial x} = \lim_{h \to 0} \frac{f(x_0 +h , y_0) - f(x_0 , y_0)}{h} . Yet again, the partial derivative is said to exist iff the limit exists. Similarly, you can write the partial derivative wrt yy.

Sudeep Salgia - 4 years, 2 months ago

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Alright Sir, thanks very much!

Ishan Dasgupta Samarendra - 4 years, 2 months ago

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Happy to help ! Please don't call me Sir. You can call me Sudeep.

Sudeep Salgia - 4 years, 2 months ago

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@Sudeep Salgia OK, Sudeep:) It sounds a bit strange though since you're my senior.

Ishan Dasgupta Samarendra - 4 years, 2 months ago

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@Sudeep Salgia I have another doubt:( How would we check if the partial derivative exists over a given interval?

Ishan Dasgupta Samarendra - 4 years, 2 months ago

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It is just like the way you would have checked the differentiability of a single variable function over an interval. If the above limit exists for all points in the interval, then the partial derivative exists at all points.

Sudeep Salgia - 4 years, 2 months ago

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@Sudeep Salgia Alright, many thanks as always!

Ishan Dasgupta Samarendra - 4 years, 2 months ago

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@Sandeep Bhardwaj Sir @Sudeep Salgia Sir, please help me.

Ishan Dasgupta Samarendra - 4 years, 2 months ago

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