The definition of the derivative of a function f(x) is given by lim_(h->0)(f(x+h)-f(x))/h.
For this function, its derivative is lim_(h->0)(((x+h)^2+1/(x+h)+7)-(x^2+1/x+7))/h=lim_(h->0)((x+h)^2+1/(x+h)+7-x^2-1/x-7)/h=lim_(h->0)((x+h)^2+1/(x+h)-x^2-1/x)/h.
Expand (x+h)^2 and combine 1/(x+h)-1/x into a single function. This gives lim_(h->0)(x^2+2xh+h^2-x^2+(x-x-h)/(x(x+h)))/h=lim_(h->0)(2xh+h^2-h/(x(x+h)))/h.
Some of the h's cancel out: lim_(h->0)2x+h-1/(x(x+h)).
Now, we can substitute h=0. This gives our final answer: 2x-1/x^2.
