For f of the function x-3 over x+3 simplfy the difference quotient

Question

The formula is f(x+h)-f(x) over h

Answer 1

Difference quotient

(x+h-3)/(x+h+3)-(x-3)/(x+3) is ((x+h-3)(x+3)-(x-3)(x+h+3))/(x+3)(x+h+3). This becomes:

(x^2+xh-3x+3x+3h-9 - (x^2+xh+3x-3x-3h-9))/((x+3)^2+h(x+3)), so we're left with:

6h/((x+3)^2+h(x+3)). When h is incremental, the denominator approaches (x+3)^2 and the difference quotient becomes the expression divided by h=6/(x+3)^2.