If \(y=[(x+1)(2 x+1)(3 x+1) \ldots .(\mathrm{n} x+1)]^{\mathrm{n}}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=0\) is

\(y=[(x+1)(2 x+1)(3 x+1) \ldots(\mathrm{n} x+1)]^{\mathrm{n}} \)
\( \Rightarrow \log y=\operatorname{nlog}[(x+1)(2 x+1)(3 x+1) \)
\( \Rightarrow \log y=\mathrm{n}[\log (x+1)+\log (2 x+1) \) \(+\log (3 x+1)+\ldots+\log (\mathrm{n} x+1)] \)
Differentiating both sides w.r.t. \(x\), we get
\(\frac{1}{y} \cdot \frac{\mathrm{d} y}{\mathrm{~d} x}=\mathrm{n}\left(\frac{1}{x+1}+\frac{2}{2 x+1}+\frac{3}{3 x+1}+\ldots+\frac{\mathrm{n}}{\mathrm{n} x+1}\right) \)
\( \Rightarrow \frac{1}{1} \cdot\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)_{x=0}=\mathrm{n}(1+2+3+\ldots+\mathrm{n})\)
\(\ldots[\) At \(x=0, y=1]\)
\(\Rightarrow\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)_{x=0}=\mathrm{n}\left[\frac{\mathrm{n}(\mathrm{n}+1)}{2}\right]=\frac{\mathrm{n}^2(\mathrm{n}+1)}{2}\)