DIFFERENTIATE FROM FIRST PRINCIPLES

Question

(I)3X
(ii)X^2+2X+3
(iii)1/x^2

Answer 1

I will do the middle one, you do others, looks like an assignment.

f(x) = x^2 - 2x + 3

dy/dx = LIM [ f(x+h) - f(x) ]/h as h---> 0

f(x+h) = (x+h)^2 + 2(x+h) + 3
= x^2 + 2xh + h^2 + 2x + 2h + 3

Lim [ f(x+h) - f(x) ]/h
= lim (x^2 + 2xh + h^2 + 2x + 2h + 3 - (x^2 + 2x + 3)/h
= lim (2xh + h^2 + 2h)/h
= lim 2x + h + 2
= 2x + 2

(each of the "lim" has h--> 2 under it)