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So with the function $$f(x,y)=\frac{x^{2}-y}{y^{2}-x}$$ I don't like the quotient rule because it seems like it makes way more sense to just find$$\frac{\partial}{\partial x}(x^{2}-y)(y^{2}-x)^{-1}$$$$=2x(y^{2}-x)+(-1)(y^{2}-x)^{-2}(-1)$$ $$=2x(y^{2}-x)+(y^{2}-x)^{-2}$$

But wolfram alpha is getting $$\frac{-2x^{2} y+x+y^{2}}{(x-y^{2})^{2}}$$

I've tried jumping through a bunch of algebra hoops to see if my answer can be simplified to its but I can't pull it off. Is this somehow wrong or is Wolfram doing some weird simplification that I can't do?

So with the function $$f(x,y)=\frac{x^{2}-y}{y^{2}-x}$$ I don't like the quotient rule because it seems like it makes way more sense to just find$$\frac{\partial}{\partial x}(x^{2}-y)(y^{2}-x)^{-1}$$$$=2x(y^{2}-x)+(-1)(y^{2}-x)^{-2}(-1)$$ $$=2x(y^{2}-x)+(y^{2}-x)^{-2}$$

But wolfram alpha is getting $$\frac{-2x^{2} y+x+y^{2}}{(x-y^{2})^{2}}$$$$\frac{x^{2} -2xy^{2}+y}{(x-y^{2})^{2}}$$

I've tried jumping through a bunch of algebra hoops to see if my answer can be simplified to its but I can't pull it off. Is this somehow wrong or is Wolfram doing some weird simplification that I can't do?

So with the function $$f(x,y)=\frac{x^{2}-y}{y^{2}-x}$$ I don't like the quotient rule because it seems like it makes way more sense to just find$$\frac{\partial}{\partial x}(x^{2}-y)(y^{2}-x)^{-1}$$$$=2x(y^{2}-x)+(-1)(y^{2}-x)^{-2}(-1)$$ $$=2x(y^{2}-x)+(y^{2}-x)^{-2}$$

But wolfram alpha is getting $$\frac{x^{2} -2xy^{2}+y}{(x-y^{2})^{2}}$$

I've tried jumping through a bunch of algebra hoops to see if my answer can be simplified to its but I can't pull it off. Is this my answer somehow wrong or is Wolfram doing some weird simplification that I can't do?