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
3 years, 1 month 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 \(f\) in variables \(x \) and \(y\), then the partial derivative of \(f\) with respect to \(x\) at a point \( (x_0 , y_0) \) is given by \[ \frac{\partial f(x,y)}{\partial x} = \lim_{h \to 0} \frac{f(x_0 +h , y_0) - f(x_0 , y_0)}{h} \] . Yet again, the partial derivative is said to exist iff the limit exists. Similarly, you can write the partial derivative wrt \(y\).

Sudeep Salgia - 3 years, 1 month 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 - 3 years, 1 month 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 - 3 years, 1 month ago

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@Sudeep Salgia Alright, many thanks as always!

Ishan Dasgupta Samarendra - 3 years, 1 month ago

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Alright Sir, thanks very much!

Ishan Dasgupta Samarendra - 3 years, 1 month ago

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

Sudeep Salgia - 3 years, 1 month ago

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

Ishan Dasgupta Samarendra - 3 years, 1 month ago

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

Ishan Dasgupta Samarendra - 3 years, 1 month ago

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