Choose the correct alternative in the following:
If $f(x)=\left(\frac{x^{1}}{x^{m}}\right)^{1+m}\left(\frac{x^{m}}{x^{n}}\right)^{m+n}\left(\frac{x^{n}}{x^{1}}\right)^{n+1}$, then $f^{\prime}(x)$ is equal to
A. 1
B. 0
C. $x^{l+m+n}$
D. none of these
$f(x)=\left(\frac{x^{1}}{x^{m}}\right)^{1+m}\left(\frac{x^{m}}{x^{n}}\right)^{m+n}\left(\frac{x^{n}}{x^{1}}\right)^{n+1}$
$f(x)=\frac{\left(x^{1}\right)^{1+m} \cdot\left(x^{m}\right)^{m+n} \cdot\left(x^{n}\right)^{n+1}}{\left(x^{m}\right)^{1+m} \cdot\left(x^{n}\right)^{m+n} \cdot\left(x^{1}\right)^{n+1}}$
$=\frac{(x)^{1^{2}+m} \cdot(x)^{m^{2}+n} \cdot(x)^{n^{2}+1}}{(x)^{1+m^{2}} \cdot(x)^{m+n^{2}} \cdot(x)^{n+l^{2}}}$
$\Rightarrow \mathrm{f}(\mathrm{x})=\frac{(\mathrm{x})^{1^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}+\mathrm{m}+\mathrm{n}+1}}{(\mathrm{x})^{1^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}+\mathrm{m}+\mathrm{n}+1}}=1$
Differentiating w.r.t $x$
$\Rightarrow \frac{d y}{d x}=0$