First principles here is the definition of a derivative:
#f'(x) = lim_(epsilon rightarrow0) (f(x+epsilon) - f(x))/epsilon #
We can apply it here:
#f'(x) = lim_(epsilon rightarrow 0) 1/epsilon (1/(2(x+epsilon) + 1) - 1/(2x + 1)) #
#= lim_(epsilon rightarrow 0) 1/epsilon ((2x+1)/((2x+1)(2(x+epsilon) + 1)) - (2(x+epsilon) + 1)/((2(x+epsilon) + 1)(2x + 1))) #
#= lim_(epsilon rightarrow 0) 1/epsilon ((2x+1 - 2(x + epsilon) - 1)/((2x+1)(2(x+epsilon) + 1)) ) #
#= lim_(epsilon rightarrow 0) 1/epsilon ((- 2epsilon)/((2x+1)(2(x+epsilon) + 1)) ) #
#= lim_(epsilon rightarrow 0) ((- 2)/((2x+1)(2(x+epsilon) + 1)) ) #
#= ((- 2)/((2x+1)(2x + 1)) ) = -2/(2x+1)^2 #
We can also do this by an #epsilon-delta# proof, but I just will assume that's not what is intended.
