I will assume that you want d/(dx)(x^2y^2)
By implicit differentiation:
d/(dx)(x^2y^2)=2xy^2+2x^2y(dy)/(dx)
We are assume that y is some function or functions of x. Think of it as
x^2y^2=x^2*("some function")^2
To differentiate this, you'd need the product rule and the chain rule.
d/(dx)[x^2*("some function")^2]=2x*("some function")^2+x^2*2("some function")*"the derivative of the function"
That looks kinda complicated, but we have names for the "some function" and for its derivative. We call them y and (dy)/(dx).
So
d/(dx)(x^2y^2)=2xy^2+x^2 2y(dy)/(dx)=2xy^2+2x^2y(dy)/(dx)
.
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If my assumption was mistaken:
If you wanted to find the partial derivatives of the function f(x,y)=x^2y^2, then you will not need the product rule.
(del)/(del x)(x^2y^2)=2xy^2
because y^2 is constant relative to x
and
(del)/(del y)(x^2y^2)=2x^xy
because x^2 is constant relative to y
