# What is #f_x# when #f (x,y) = sin^2 (x^2y^2)# ?

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To find ( f_x ) when ( f(x, y) = \sin^2(x^2y^2) ), we need to differentiate ( f ) with respect to ( x ) while treating ( y ) as a constant.

Using the chain rule and the power rule, we first differentiate the outer function ( \sin^2(u) ) with respect to its inner function ( u = x^2y^2 ), and then multiply by the derivative of ( u ) with respect to ( x ).

So, we get: [ f_x = 2\sin(x^2y^2)\cos(x^2y^2)(2xy^2) = 4xy^2\sin(x^2y^2)\cos(x^2y^2) ]

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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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