Using the definition of a derivative as \(\mathrm{f}'(x)=\lim_{h\to0}\left(\frac{\mathrm{f}(x+h)-\mathrm{f}(x)}{h}\right)\), show that the derivative of \(\frac{1}{2x+1}\) is \(\frac{-2}{(2x+1)^2}\).[4]
Prove by induction that the \(n^{\mathrm{th}}\) derivative of \((2x+1)^{-1}\) is \((-1)^n\frac{2^n n!}{(2x+1)^{n+1}}\).[9]