Question

If $f\left(x\right)={\left(\frac{x^l}{x^m}\right)}^{l+m}{\left(\frac{x^m}{x^n}\right)}^{m+n}{\left(\frac{x^n}{x^l}\right)}^{n+1}$, the f' (x) is equal to
(a) 1
(b) 0
(c) $x^{l+m+n}$
(d) none of these

Answer 1

(b) 0
We have, $f\left(x\right)={\left(\frac{x^l}{x^m}\right)}^{l+m}{\left(\frac{x^m}{x^n}\right)}^{m+n}{\left(\frac{x^n}{x^l}\right)}^{n+1}$

$$\begin{gathered} \Rightarrow f\left(x\right)=x^{\left(l-m\right)\left(l+m\right)}\times x^{\left(m-n\right)\left(m+n\right)}\times x^{\left(n-l\right)\left(n+l\right)} \\ \Rightarrow f\left(x\right)=x^{l^2-m^2}\times x^{m^2-n^2}\times x^{n^2-l^2} \\ \Rightarrow f\left(x\right)=x^{\left(l^2-m^2+m^2-n^2+n^2-l^2\right)} \\ \Rightarrow f\left(x\right)=x^0 \\ \Rightarrow f\left(x\right)=1 \\ \Rightarrow f\text{'}\left(x\right)=0 \end{gathered}$$