# If #f= x^2 y^2+2xy^2#, then show that #(partial ^2f)/(partial x partial y) = (partial ^2f)/(partial y partial x)#. obtain the value of #f_(xy)(1,1)# ?

# f_(xy)(1,1) = 8 #

We have:

We compute the first partial derivatives (by differentiating wrt to specified variable and treating all other variables as constants):

Next we compute the second partial derivatives:

And:

And indeed we can verify that:

Using this result we then calculate:

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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