Question: Breaking up a large expression

Say I have an expression like this:

E = 1/(x^2 + y^2) + (x^2+y^2)

And let's say I want to make a substitution for t = (x^2 + y^2);

E = 1/t + t

Now I want to be able to differentiate this.  eg,

diff(E, x)

I should get:

- dt/dx / t^2 + dt/dx

How can I make Maple provide output in this form?  ie, I do not want to see an output of - 2*x/(x^2+y^2)^2 + 2*x

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