Derivative of a reciprocal linear function from first principles and a formula for its nth derivative by induction

Starting from the definition \(\mathrm{f}'(x)=\lim_{h\to0}\left(\frac{\mathrm{f}(x+h)-\mathrm{f}(x)}{h}\right)\), show that when \(\mathrm{f}(x)=\frac{1}{2x+1}\), its derivative is \(\frac{-2}{(2x+1)^2}\).[4]

Use mathematical induction to prove that, for every positive integer \(n\), the \(n^{\mathrm{th}}\) derivative of \((2x+1)^{-1}\) is equal to \((-1)^n\frac{2^n n!}{(2x+1)^{n+1}}\).[9]