Prove that 1/(2x+1) = -2/((2x+1)^2) using first principles.

Question

Must be proven through first principles, question from IB HL Math 2.

Answer 1

I think you mean: If f(x) = 1/(2x+1), then prove that f'(x) = -2/((2x+1)^2).

f'(x) = lim (h->0) [(f(x+h) - f(x)) / h]

= lim (h->0) [(1 / (2x+2h+1) - 1/(2x+1)) / h]

= lim(h->0) [(-2h / ((2x+2h+1)(2x+1)) / h]

= lim(h->0) [ -2 /( (2x+2h+1)(2x+1))] = -2(2x+1)^2