Proofs of differential calculus
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1) Sum Rule- If f and g are differentiable at x_0, then f+g is also differentiable at x_0, and (f+g)'(x_0)=f '(x_0) + g '(x_0)
2) Product Rule- If f and g are differentiable at x_0, then fg is also differentiable at x_0, and (fg)' (x_0)= f '(x_0)g(x_0)+f(x_0)g '(x_0)
3) If g is differentiable at x_0, and g is non-zero on X (i.e. g(x) does not equal 0 for all x belonging to X) then 1/g is also differentiable at x_0, and (1/g)'(x_0)= -(g'(x_0)/g(x_0)^2)
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