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 _____________ .

Options

• 1

• 0

• \[x^{l + m + n}\]

• none of these

Answer 1

`0`

 

We have,

\[f\left( x \right) = \left( \frac{x^l}{x^m} \right)^{l + m} \left( \frac{x^m}{x^n} \right)^{m + n} \left( \frac{x^n}{x^l} \right)^{n + 1}\]

\[\Rightarrow f\left( x \right) = x^{\left( l - m \right)\left( l + m \right)} \times x^{\left( m - n \right)\left( m + n \right) }\times x^{\left( n - l \right)\left( n + l \right) }\]
\[ \Rightarrow f\left( x \right) = x^{l^2 - m^2} \times x^{m^2 - n^2} \times x^{n^2 - l^2} \]
\[ \Rightarrow f\left( x \right) = x^\left( l^2 - m^2 + m^2 - n^2 + n^2 - l^2 \right) \]
\[ \Rightarrow f\left( x \right) = x^0 \]
\[ \Rightarrow f\left( x \right) = 1\]
\[ \Rightarrow f'\left( x \right) = 0\]