Determine f`(x) from first principles : f(x)= -3/x^2

Question

Involve the limit h→0

Answer 1

f(x+h)=-3/(x+h)^2=-3/(x^2+2hx+h^2), where h is a very small change in x. f(x+h)-f(x)=-3[1/(x^2+2hx+h^2)-1/x^2]=-3[1/(x^2(1+2h/x+h^2/x^2))-1/x^2]. In the limit as h→0, h^2/x^2→0.

f(x+h)-f(x)=-3[1/(x^2(1+2h/x))-1/x^2]=-3[(1-2h/x)/x^2-1/x^2], because as h→0, 1/(1+2h/x)→(1-2h)/x.

The change in f=-3[1/x^2-2h/x^3-1/x^2]=6h/x^3. So the rate of change is 6h/x^3÷h=6/x^3, the derivative as h→0.